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Hindawi Publishing CorporationAdvances in Optical TechnologiesVolume 2013 Article ID 469389 18 pageshttpdxdoiorg1011552013469389
Research ArticleModeling and Simulation of Piecewise Regular Multimode FiberLinks Operating in a Few-Mode Regime
Anton Bourdine
Department of Communication Lines Povolzhskiy State University of Telecommunications and Informatics (PSUTI)77 Moscow Avenue Samara 443090 Russia
Correspondence should be addressed to Anton Bourdine bourdineyandexru
Received 29 June 2013 Accepted 22 September 2013
Academic Editor Zoran Ikonic
Copyright copy 2013 Anton Bourdine This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This work presents an alternative model of multimode fiber links with conventional silica weakly-guiding graded-index irregularmultimode fibers under a few-mode optical signal propagation generated by laser source The proposed model is based onthe piecewise regular representation It takes into account launch conditions differential mode delay both lower- and higher-order mode chromatic dispersion differential mode attenuation and mode mixing and power diffusion occurring due to realfiber irregularity and micro- and macrobends We present some results of introduced model approbation with following pulsepropagation simulations A close matching with measured pulse responses at the output of test fibers is noticed
1 Introduction
Silica multimode graded-index fibers are used in a widevariety of applications ranging from on-board to in-premisesnetworks links with length not more than 1ndash2 km SinceIEEE 8023z standard was ratified and commercial avail-able SFP transceivers with Vertical Cavity Surface EmittingLasers (VCSELs) appeared on telecommunication marketmultimode fibers became very popular both for in-buildingstructural cabling systems (SCS) and typical distributed net-works with backbonevertical fiber cabling systems privatenetworks in premises and campus environments Nowadaysthe most top applications of multimode fibers are associatedwith data center SCS high bit-rate storage area networksand radio-over-fiber (RoF) techniques over already installedmultimode fiber infrastructure inside building [1ndash3]
Modern commercial multi-Gigabit transmission tran-sceivers are realized on Vertical Cavity Surface EmittingLasers (VCSELs) or single-mode Fabry-Perot laser diodes(LD) [1ndash4] Because emission from the conventional VCSELusually consists of about 5 or 6 transversal modes 119871119875119897119898 withmaximal azimuthal order 119897 not more than 3 and single-modeLD injects just fundamental and lower-ordermodes 11987111987501 and11987111987511 only several guided modes are excited in multimodefiber link [5ndash7] Therefore optical signal propagates over
multimode fiber link in a so-called few-mode regime andpassage to the simulation of a few-mode pulse transmissionover multimode fiber requires taking into account bothldquoindividualrdquo dispersion parameters of mode component withparticular order (amplitude attenuation delay chromaticdispersion etc) and mode coupling
A variety of methods have been developed for modelingand simulation of laser-based multi-Gigabit data transmis-sion over multimode fibers Monograph [4] can be con-sidered as fundamental complete basis work The followinggroups of publicationsmdash[8ndash15]mdashshould also be noted theirauthors are affiliated to IEEE 8023z 8023ae 8023aq and8023ba Task Forces Other works are also concerned withthe design and simulation ofmulti-Gigabitmultimode opticalcommunication systems Thus model [16ndash18] was realizedin commercial RSoft tool ModeSYS [19 20] are based onvector methods and [21] applies ray-tracing analysis Allmentioned models [4ndash20] are primarily assigned for the esti-mation of differential mode delay (DMD) which is the mainissue of pulse distortion for laser-based multi-Gigabit datatransmission over multimode fiber links and is concernedwith theoretical research of DMD dependence on sourceparameters launch conditions and real fiber refractive indexprofile defects and deviations from the optimum graded-index profile form They also provide accurate simulation
2 Advances in Optical Technologies
technique of signal generation conversion and processing intransceivers of multi-Gigabit communication systems
A generalized chromatic dispersion parameter is appliedin some models mentioned above It is based on well-known empiric formula which is cited in the most of fiberspecifications and data sheets [22] However this approachdoes not differ of lower- and higher-order modes and doesnot solve the problem of ldquoindividualrdquo chromatic dispersionestimation for guidedmode with particular order Alsomodecoupling is neglected in most of the listed above It is justifiedby the short length of multimode fiber link In some workscorrection factors or modal noise penalty are applied butonly mode power redistribution at fiber interconnections infiber optic distribution systems consolidation points patch-panels and so forth is recalculated without taking intoaccount both real fiber irregularities and fiber bends
Here we introduce an alternative model of multimodefiber links with conventional silica weakly-guiding graded-index irregular multimode fibers under a few-mode opti-cal signal propagation generated by laser source Proposedmodel is based on the piecewise regular representation Ittakes into account launch conditions differential mode delayboth lower- and higher-order mode chromatic dispersiondifferential mode attenuation and mode mixing and powerdiffusion occurring due to real fiber irregularity and microandmacro-bendsWe also present some results of introducedmodel approbation with following pulse propagation simu-lations and matching with measured pulse responses at theoutput of test fibers
2 Brief Overview of Methods forSimulation of Pulse Propagation overIrregular Optical Fibers
A several methods have been developed for computing elec-tromagnetic wave fields during propagation over irregularwaveguidesMonograph [23] is one of the fundamental worksof this direction Effects of wave interaction in irregularwaveguides are exact described by the coupled mode theoryComplete overview description classification and list ofapplications of coupled mode theory methods can be foundin monograph [24]
Generally coupled mode theory is based on represen-tation of mode fields along longitudinal cross-section 119911
of irregular optical waveguide in the kind of mode fieldsuperposition of complete system of guided and leaky modesin regular optical waveguide This expansion varies along thefiber longitudinal axis and it is described by systemof coupledmode equations The system of equations is formulated bysubstitution of mentioned irregular optical waveguide modefield representation to Maxwell equations [25] or Helmholtzequation [26 27] Result of substitution is infinite system ofdifferential equations of the first or second order (it dependson initial equationsmdashMaxwell or Helmholtz) for each modeamplitude depending on longitudinal coordinate 119911
The system of equations exactly describes mode fields inlongitudinal cross-section of irregular optical waveguide butit has not general analytical solution Problem can be much
simplified for the limited number of modesmdashjust one ortwo [28ndash30] For example coupled mode theory is effectivelyapplied for analysis of the fundamental mode propagationover irregular optical fiber under taking into account thebirefringence effect [25ndash27 31] However few-mode regime ofsignal transmission over multimode fibers requires increaseof mode components that much complicates solutions of thesystem of equations This fact limits coupled mode theoryapplication in the case of random fluctuations of fiber coregeometry parameters along the longitudinal axis 119911
In [32 33] mode components of few-mode signal arecombined in mode groups with roughly equal propagationconstants In works [34ndash36] a passage to infinite number ofmode components (mode continuum) is proposed followedby replacing the sum of discrete mode component fields byintegration By neglecting core geometry variation randomfiber macrobends is considered as the main factor of modecoupling in [37ndash39] Also in [40ndash44] a special curvilinearcoordinates are applied to describe mode coupling via fiberbends In [45ndash47] mode mixing and power diffusion issimulated by combination of coupled mode theory and ray-tracing analysis
The system of incoherently coupled nonlinear Schro-dinger equations is applied for mathematical description ofpulse propagation overmultimode fiber in [48ndash54] Althoughthe generalized system of equations was written for infinitenumber of modes by taking into account fiber nonlinearitythemain application of thismethod is used just for simulationof coupling between two orthogonally polarized modes insingle mode fibers occurring due to birefringence
Piecewise regular representation [25 26 55] of irregularoptical waveguide is an alternative method to coupled modetheory It is basic for direct methods like finite-differencetime-domain (FDTD) methodmdashone of the most accurateuniversal and ldquopopularrdquo numerical methods which is widelyused for analysis of electromagnetic wave propagation overwaveguides with arbitrarily complicated configuration (eg[56ndash62] etc) Also well-known direct rigorous methods aremethod of moments [58 62 63] finite-difference frequency-domain (FDFD) method [55 58 62] finite element method[55 58 62] wavelets [64ndash66] and their modifications andcombinations with elements of coupled mode theory [41ndash44 67ndash73]
Rigorous methods provide accurately computing of thetransversal mode field distribution at the particular distanceof emission source and time interval under the lowesterror The main issue of their application is complicatedformulation of boundary and initial conditions according toindividual problem and waveguide properties and configura-tion They are very effective for the modeling of waveguidestructures with quite short dimensionsmdashnot more than(103 10
4) sdot 120582mdashlike integral optics components However
simulation of pulse propagation over conventional fiberlink (with length of hundreds or even tens meters) by thementioned direct rigorousmethods requires extra computingresources and is almost impossible
In terms of piecewise regular representation of both LANMAN (metropolitan area network) and long-haul fiber links
Advances in Optical Technologies 3
50
25
0
minus25
minus50
0 002 004 006 008 01 012 014 016 018 02Length of regular span Δz
(i minus 1)th regular span
iminus1
i i+1
(i + 1)th regular span
Cor
e dia
met
er
r(120583
m)
L (m)
Figure 1 Piecewise regular representation of 200m irregular mul-timode optical fiber with varying core diameter
matrix methods [62 74ndash80] recurrent methods [81ndash85] andquasi-analytic time-domain models based on quasianalyticmethods [86ndash89] and spatial transformmethods [55 86 90ndash95] are widely used The most of them were developed forsimulation of pulse propagation over single mode fibers
From this point of view method described in [96ndash98]can be considered as an exception It combined coupledmode theory with piecewise regular representation and wasdeveloped formodeling of signal transmission processes overlong irregular multimode fiber links Here a random fiberbends are preset and supposed as the main reason of modecoupling occurring Also it takes into account birefringenceHowever the method neglects core geometry variation alonglongitudinal fiber axis and considers only multimode fiberswith ideal infinite 120572-profiles
Therefore the most considered methods above for mod-eling and analysis of pulse propagation over optical fibersshould be adapted for irregular multimode fiber link oper-ating in a few-mode regime Model has to take into accountthe whole total selected excited guided mode compositionof transferred few-mode optical signal and their ldquoindividualrdquoparameters during propagation over multimode fibers withreal graded-index profiles differing form ideal 120572-profiles bylocal defects and refractive index fluctuation Moreover inthis case division ofmode composition on lower- and higher-order mode groups with averaged propagation constant isunacceptable Also model has to take into account launchconditions and mode coupling due to both variation ofcore geometry parameter along fiber length and micro- andmacrobends of fiber occurring in real cable links
3 Model of Piecewise RegularMultimode Fiber Link
In this work we also apply piecewise regular representationcombined with general approach of split-step method [99]to simulate a processes of mode mixing and power diffusiondue to mode coupling (Figure 1) Here single silica weaklyguiding circular multimode fibers with arbitrary axial sym-metric refractive index profile with single continue outercladdings are considered According to piecewise regularrepresentation the fiber is divided into regular spans withlength Δ119911 Inside the span fiber geometry parameters areconsidered as constant and modes propagate independently
without interaction and mixing It is supposed that eachexcited guided mode with propagation constant varyingfrom one regular span to another span satisfies to cut-offcondition for whole regular spans composing fiber Also it isassumed that at the link transmitter end each excited guidedmode starts transferring single optical pulse in particularform identical to input signal (eg Gaussian) During pulsepropagation over regular span its amplitude decreases dueto mode attenuation The signal is mainly distorted due todifference between group velocities and amplitudes ofmodesthat is DMD effect Also transferred by each mode pulsespreading due to chromatic dispersion for particular modeis taken into account
Boundaries of regular spans can be represented gener-ally as ideal axially alignment splice of two almost similaroptical fibers with mismatching parameters However itis correct only for ldquostraightrdquo fibers Therefore it is pro-pose to simulate fiber bends by representation of bound-aries in the form of splice of two mismatched fibers withsome low angular misalignment Mode power redistribu-tion between the amplitudes of signal components as aresult of mode interaction is estimated by mode couplingcoefficient matrix computing at the joints of regular spansHere only guided modes are considered as the main issueunder pulse dynamics research during propagation overmultimode fiber link in a few-mode regime However in factpower loss due lo component transformation from guidedto leaky mode and reflections are also indirectly taken intoaccount
At the receiver end the resulting pulse envelope is con-sidered as superposition of all mode components of signalHere it is propose to apply a well-known expression for thefrequency response of the signal transferred by 119872 modecomponents 119871119875119897119898 over regular multimode fiber with length119911 [4 8ndash15 18 99ndash101]
119867119877119909119905 (120596 119911) = 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(0)
119901exp (minus120572119901119911)
times exp [minus119895 (120596 minus 1205960) 120573(119901)
1119911]
times exp [minus11989512(120596 minus 1205960)
2120573(119901)
2119911]
(1)
where 119865 is direct Fourier transform ℎ119879119909(119905) is initial pulse atthe transmitter end 119860(0)
119901and 120572119901 are starting amplitude and
mode attenuation of 119901th guided mode 119871119875119897119898 (119901 = 1 119872)120573(119901)
1and 120573
(119901)
2are first- and second-order dispersion param-
eters These dispersion parameters are elements of well-knownTaylor series expansion approximation of propagationconstant frequency dependence 120573(120596) [2 55 99]
120573 (120596) = (120596 minus 1205960) 1205731 +1
2(120596 minus 1205960)
21205732
+1
6(120596 minus 1205960)
31205733
(2)
4 Advances in Optical Technologies
where
120573119899 = [120597119899120573
120597120596119899]
120596=1205960
(3)
Here 120573(119901)
1= 120591(119901) is mode delay and 120573
(119901)
2is group velocity
dispersion (GVD) associated with chromatic dispersion of119901th guided mode 119871119875119897119898
Therefore according to introduced piecewise regular rep-resentation of irregular multimode fiber link the frequencyresponse of a few-mode optical signal transferred by 119872
guided modes over irregular multimode fiber with length 119911
under given particular length of regular span Δ119911 by takinginto account expression (1) can be written in the followingform
119867119877119909 (120596 119911)
= 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(119901)
(119873119911+1)exp [minus120572(119901)
(119873119911+1)(119911 minus 119873119911Δ119911)]
times exp [minus119895 (120596 minus 1205960) 120591(119901119873119911+1) (119911 minus 119873119911Δ119911)]
times exp [minus11989512(120596 minus 1205960)
2120573(119901119873119911+1)
2(119911 minus 119873119911Δ119911)]
times
119873119911
prod
119902=1
119860(119901)
119902exp (minus120572(119901)
119902Δ119911)
times exp (minus119895 (120596 minus 1205960) 120591(119901119902)
Δ119911)
times exp(minus11989512(120596 minus 1205960)
2120573(119901119902)
2Δ119911)
(4)
where119873119911 = 119864(119911Δ119911) 119864(119909) is the integer part of real number119909
The resulting pulse response at the receiver end ofirregularmultimode link is computed by the following simpleexpression
ℎ119877119909 (119905) = 119865minus1(119867119877119909 (120596)) sdot [119865
minus1(119867119877119909 (120596))]
lowast
(5)
where 119865minus1 is inverse Fourier transform [119909]lowast is the complexconjugate of 119909
It is obvious that the proposed model requires fast andsimple method for the analysis of the optical fibers withcomplicated refractive index profile form close to profiles ofreal fibers which provides evaluation of parameters not onlyfor the fundamental but also higher-order guidedmodes Forthis purpose we propose to apply an extension of modifiedGaussian approximation (EMGA) generalized for estimationof any order guided mode parameters propagating oversilica weakly guiding optical fibers with an arbitrary axial-symmetric refractive index profile [102] EMGA is based oncombination of Gaussian approximation and stratificationmethod It permits amuch reduce computing time especiallyfor the calculations of higher-order mode parameters underthe low error [103] by making a passage to analytical expres-sions both for the variational expression and characteristic
equation Below a detailed description of EMGA and itsapplication for the proposed model is presented
4 Extension of ModifiedGaussian Approximation
EMGA is based on classical Gaussian approximation [25] ofradial mode field distribution 119865
(119897)
119898(119877) in the weakly guiding
optical waveguidewith an arbitrary refractive index profile bythe well-known Laguerre-Gauss function expression [4 25100] describing a mode field distribution in weakly guidingoptical waveguide with ideal infinite parabolic index profile
Ψ(119897)
119898(119877) = (
119877
1198770
)
119897
119871(119897)
119898minus1(1198772
1198772
0
) exp(minus1198772
21198772
0
) (6)
where 119871(119897)119898minus1
is Laguerre polynomialEMGA leads to equivalent normalized mode field radius
1198770 estimation by solving a characteristic equation whichis derived from propagation constant variational expressionunder following passage to square core mode parameter 1198802variational expression written for analyzed weakly guidingoptical waveguide with given refractive index profile Param-eter 1198770 is basic for this method and completely defines modetransmission parameters
Unlike knownmethods and their modifications [25 104ndash110] based on conventional Gaussian approximation inEMGAoptical fiberwith an arbitrary graded axial-symmetricindex profile is considered as multicladding optical fiberTherefore refractive index profile inside core region can berepresented in the form of the set of 119873 layers in which therefractive stays a constant [102 103]
119899 (119877) =
119899119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
119899119873 1 lt 119877 le +infin
(7)
and any profile function 119891(119877) related to refractive indexprofile 119899(119877) as
1198992(119877) = 119899
2
max [1 minus 2Δ sdot 119891 (119877)] (8)
where Δ = (1198992
max minus 1198992
119873)21198992
max is profile height parameter canbe written in terms of profile parameter ℎ119896
119891 (119877) =
ℎ119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
1 1 lt 119877 le +infin
(9a)
where
ℎ119896 =1198992
max minus 1198992
119896
1198992max minus 1198992
119873
(9b)
where 119899119896 is refractive index of 119896 layer (119896 = 0 119873) 119899max isthe maximal core refractive index 119899119873 is cladding refractiveindex
This refractive index profile representation permits writ-ing the variational expression for core mode parameter 1198802
Advances in Optical Technologies 5
and characteristic equation for normalized equivalent modefield radius 12059711988021205971198770 = 0 in the form of finite nested sums asfollows [15 16]
1198802=
(119898 minus 1)
(119897 + 119898 minus 1)
times 119876
1198772
0
+ 1198812[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
1198830 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198831 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901
1198832 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901
minus119876 + 1198812[1198780 +
119873minus1
sum
119896=0
ℎ119896 (1198781 minus 1198782)] = 0
(10)
1198780 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(1 minus 1199011198772
0)
1198781 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
1198782 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
119876 =(119897 + 119898 minus 1) (3119897 + 2119898 minus 1)
(119898 minus 1)
+ 21198972
2119898minus2
sum
119902=0
119863119902 (119902 + 119897 minus 1) minus 4119897
times
2119898minus2
sum
119902=0
119862119902 (119902 + 119897)
(11)
where
119863119902 =
min(119902119898minus1)
sum
119901=max(0119902minus119898+1)
119887(119897119898minus1)
119901119887(119897119898minus1)
119902minus119901 (12a)
119871(119897)
119898minus1(119909) 119871(119897+1)
119898minus1(119909) =
2119898minus2
sum
119902=0
119862119902119909119902 (12b)
and 119887(119897119898)
119901is coefficient of polynomial representation in the
form of power series [111 112]
119871(119897)
119898(119909) =
119898
sum
119902=0
119887(119897119898)
119902119909119902 (13a)
119887(119897119898)
119902= (minus1)
119902 (119897 + 119898)
(119897 + 119902) (119898 minus 119902)119902 (13b)
119881 = 1198960119886119899maxradic2Δ is normalized frequency 1198960 = 2120587120582 iswavenumber 120582 is wavelength
Therefore analysis of weakly guiding single-claddingoptical fiber with an arbitrary profile leads to the followingRefractive index profile is represented by the profile function(9a) and (9b) in the form of 119873 layers Fiber parameters andmode orders 119897 and 119898 are substituted to the characteristicequation (11) By means of numerical solution (11) thenormalized equivalent mode field radius 1198770 will be obtainedThen 1198770 is substituted to the expression (10) and modecore parameter 119880 is estimated that permits evaluating thepropagation constant 120573 for guided mode 119871119875119897119898 by the well-known expression [4 25ndash27 55 100]
1205732= 1198962
01198992
max minus1198802
1198862 (14)
Solution of the characteristic equation (11) is correctunder normalized frequency 119881 gt 1 and it should satisfy theguided mode cutoff condition [4 25ndash27 55 100]
1198960119899119873 lt 120573 le 1198960119899max (15)
Optical confinement factor 119875core can be considered as thesecond criterion for the identification of the ghost solutions
119875(119897119898)
co ge 05 (16)
If we take into account Gaussian approximation parameter119875core is defined by analytical expression derived in [15] fromthe generalized integral form forweakly guiding optical fiberspresented in [25]
119875(119897119898)
co =(119898 minus 1)
(119897 + 119898 minus 1)
2119898minus2
sum
119902=0
119863119902 (119897 + 119902) sdot 119882119902 (17)
where
119882119902 = 1 minus exp(minus 1
1198772
0
)
119897+119902
sum
119901=0
1
1199011198772119901
0
(18)
5 Mode Group Velocity
According to what is mentioned above realization of intro-duced model of piecewise regular multimode fiber linkrequires a passage to analytical expressions for mode groupvelocity Guidedmode group velocity V119892 is related to propaga-tion constant120573 by thewell-known formula [4 25ndash27 55 100]
V119892 = minus2120587119888
1205822
120597120582
120597120573 (19)
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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International Journal of
2 Advances in Optical Technologies
technique of signal generation conversion and processing intransceivers of multi-Gigabit communication systems
A generalized chromatic dispersion parameter is appliedin some models mentioned above It is based on well-known empiric formula which is cited in the most of fiberspecifications and data sheets [22] However this approachdoes not differ of lower- and higher-order modes and doesnot solve the problem of ldquoindividualrdquo chromatic dispersionestimation for guidedmode with particular order Alsomodecoupling is neglected in most of the listed above It is justifiedby the short length of multimode fiber link In some workscorrection factors or modal noise penalty are applied butonly mode power redistribution at fiber interconnections infiber optic distribution systems consolidation points patch-panels and so forth is recalculated without taking intoaccount both real fiber irregularities and fiber bends
Here we introduce an alternative model of multimodefiber links with conventional silica weakly-guiding graded-index irregular multimode fibers under a few-mode opti-cal signal propagation generated by laser source Proposedmodel is based on the piecewise regular representation Ittakes into account launch conditions differential mode delayboth lower- and higher-order mode chromatic dispersiondifferential mode attenuation and mode mixing and powerdiffusion occurring due to real fiber irregularity and microandmacro-bendsWe also present some results of introducedmodel approbation with following pulse propagation simu-lations and matching with measured pulse responses at theoutput of test fibers
2 Brief Overview of Methods forSimulation of Pulse Propagation overIrregular Optical Fibers
A several methods have been developed for computing elec-tromagnetic wave fields during propagation over irregularwaveguidesMonograph [23] is one of the fundamental worksof this direction Effects of wave interaction in irregularwaveguides are exact described by the coupled mode theoryComplete overview description classification and list ofapplications of coupled mode theory methods can be foundin monograph [24]
Generally coupled mode theory is based on represen-tation of mode fields along longitudinal cross-section 119911
of irregular optical waveguide in the kind of mode fieldsuperposition of complete system of guided and leaky modesin regular optical waveguide This expansion varies along thefiber longitudinal axis and it is described by systemof coupledmode equations The system of equations is formulated bysubstitution of mentioned irregular optical waveguide modefield representation to Maxwell equations [25] or Helmholtzequation [26 27] Result of substitution is infinite system ofdifferential equations of the first or second order (it dependson initial equationsmdashMaxwell or Helmholtz) for each modeamplitude depending on longitudinal coordinate 119911
The system of equations exactly describes mode fields inlongitudinal cross-section of irregular optical waveguide butit has not general analytical solution Problem can be much
simplified for the limited number of modesmdashjust one ortwo [28ndash30] For example coupled mode theory is effectivelyapplied for analysis of the fundamental mode propagationover irregular optical fiber under taking into account thebirefringence effect [25ndash27 31] However few-mode regime ofsignal transmission over multimode fibers requires increaseof mode components that much complicates solutions of thesystem of equations This fact limits coupled mode theoryapplication in the case of random fluctuations of fiber coregeometry parameters along the longitudinal axis 119911
In [32 33] mode components of few-mode signal arecombined in mode groups with roughly equal propagationconstants In works [34ndash36] a passage to infinite number ofmode components (mode continuum) is proposed followedby replacing the sum of discrete mode component fields byintegration By neglecting core geometry variation randomfiber macrobends is considered as the main factor of modecoupling in [37ndash39] Also in [40ndash44] a special curvilinearcoordinates are applied to describe mode coupling via fiberbends In [45ndash47] mode mixing and power diffusion issimulated by combination of coupled mode theory and ray-tracing analysis
The system of incoherently coupled nonlinear Schro-dinger equations is applied for mathematical description ofpulse propagation overmultimode fiber in [48ndash54] Althoughthe generalized system of equations was written for infinitenumber of modes by taking into account fiber nonlinearitythemain application of thismethod is used just for simulationof coupling between two orthogonally polarized modes insingle mode fibers occurring due to birefringence
Piecewise regular representation [25 26 55] of irregularoptical waveguide is an alternative method to coupled modetheory It is basic for direct methods like finite-differencetime-domain (FDTD) methodmdashone of the most accurateuniversal and ldquopopularrdquo numerical methods which is widelyused for analysis of electromagnetic wave propagation overwaveguides with arbitrarily complicated configuration (eg[56ndash62] etc) Also well-known direct rigorous methods aremethod of moments [58 62 63] finite-difference frequency-domain (FDFD) method [55 58 62] finite element method[55 58 62] wavelets [64ndash66] and their modifications andcombinations with elements of coupled mode theory [41ndash44 67ndash73]
Rigorous methods provide accurately computing of thetransversal mode field distribution at the particular distanceof emission source and time interval under the lowesterror The main issue of their application is complicatedformulation of boundary and initial conditions according toindividual problem and waveguide properties and configura-tion They are very effective for the modeling of waveguidestructures with quite short dimensionsmdashnot more than(103 10
4) sdot 120582mdashlike integral optics components However
simulation of pulse propagation over conventional fiberlink (with length of hundreds or even tens meters) by thementioned direct rigorousmethods requires extra computingresources and is almost impossible
In terms of piecewise regular representation of both LANMAN (metropolitan area network) and long-haul fiber links
Advances in Optical Technologies 3
50
25
0
minus25
minus50
0 002 004 006 008 01 012 014 016 018 02Length of regular span Δz
(i minus 1)th regular span
iminus1
i i+1
(i + 1)th regular span
Cor
e dia
met
er
r(120583
m)
L (m)
Figure 1 Piecewise regular representation of 200m irregular mul-timode optical fiber with varying core diameter
matrix methods [62 74ndash80] recurrent methods [81ndash85] andquasi-analytic time-domain models based on quasianalyticmethods [86ndash89] and spatial transformmethods [55 86 90ndash95] are widely used The most of them were developed forsimulation of pulse propagation over single mode fibers
From this point of view method described in [96ndash98]can be considered as an exception It combined coupledmode theory with piecewise regular representation and wasdeveloped formodeling of signal transmission processes overlong irregular multimode fiber links Here a random fiberbends are preset and supposed as the main reason of modecoupling occurring Also it takes into account birefringenceHowever the method neglects core geometry variation alonglongitudinal fiber axis and considers only multimode fiberswith ideal infinite 120572-profiles
Therefore the most considered methods above for mod-eling and analysis of pulse propagation over optical fibersshould be adapted for irregular multimode fiber link oper-ating in a few-mode regime Model has to take into accountthe whole total selected excited guided mode compositionof transferred few-mode optical signal and their ldquoindividualrdquoparameters during propagation over multimode fibers withreal graded-index profiles differing form ideal 120572-profiles bylocal defects and refractive index fluctuation Moreover inthis case division ofmode composition on lower- and higher-order mode groups with averaged propagation constant isunacceptable Also model has to take into account launchconditions and mode coupling due to both variation ofcore geometry parameter along fiber length and micro- andmacrobends of fiber occurring in real cable links
3 Model of Piecewise RegularMultimode Fiber Link
In this work we also apply piecewise regular representationcombined with general approach of split-step method [99]to simulate a processes of mode mixing and power diffusiondue to mode coupling (Figure 1) Here single silica weaklyguiding circular multimode fibers with arbitrary axial sym-metric refractive index profile with single continue outercladdings are considered According to piecewise regularrepresentation the fiber is divided into regular spans withlength Δ119911 Inside the span fiber geometry parameters areconsidered as constant and modes propagate independently
without interaction and mixing It is supposed that eachexcited guided mode with propagation constant varyingfrom one regular span to another span satisfies to cut-offcondition for whole regular spans composing fiber Also it isassumed that at the link transmitter end each excited guidedmode starts transferring single optical pulse in particularform identical to input signal (eg Gaussian) During pulsepropagation over regular span its amplitude decreases dueto mode attenuation The signal is mainly distorted due todifference between group velocities and amplitudes ofmodesthat is DMD effect Also transferred by each mode pulsespreading due to chromatic dispersion for particular modeis taken into account
Boundaries of regular spans can be represented gener-ally as ideal axially alignment splice of two almost similaroptical fibers with mismatching parameters However itis correct only for ldquostraightrdquo fibers Therefore it is pro-pose to simulate fiber bends by representation of bound-aries in the form of splice of two mismatched fibers withsome low angular misalignment Mode power redistribu-tion between the amplitudes of signal components as aresult of mode interaction is estimated by mode couplingcoefficient matrix computing at the joints of regular spansHere only guided modes are considered as the main issueunder pulse dynamics research during propagation overmultimode fiber link in a few-mode regime However in factpower loss due lo component transformation from guidedto leaky mode and reflections are also indirectly taken intoaccount
At the receiver end the resulting pulse envelope is con-sidered as superposition of all mode components of signalHere it is propose to apply a well-known expression for thefrequency response of the signal transferred by 119872 modecomponents 119871119875119897119898 over regular multimode fiber with length119911 [4 8ndash15 18 99ndash101]
119867119877119909119905 (120596 119911) = 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(0)
119901exp (minus120572119901119911)
times exp [minus119895 (120596 minus 1205960) 120573(119901)
1119911]
times exp [minus11989512(120596 minus 1205960)
2120573(119901)
2119911]
(1)
where 119865 is direct Fourier transform ℎ119879119909(119905) is initial pulse atthe transmitter end 119860(0)
119901and 120572119901 are starting amplitude and
mode attenuation of 119901th guided mode 119871119875119897119898 (119901 = 1 119872)120573(119901)
1and 120573
(119901)
2are first- and second-order dispersion param-
eters These dispersion parameters are elements of well-knownTaylor series expansion approximation of propagationconstant frequency dependence 120573(120596) [2 55 99]
120573 (120596) = (120596 minus 1205960) 1205731 +1
2(120596 minus 1205960)
21205732
+1
6(120596 minus 1205960)
31205733
(2)
4 Advances in Optical Technologies
where
120573119899 = [120597119899120573
120597120596119899]
120596=1205960
(3)
Here 120573(119901)
1= 120591(119901) is mode delay and 120573
(119901)
2is group velocity
dispersion (GVD) associated with chromatic dispersion of119901th guided mode 119871119875119897119898
Therefore according to introduced piecewise regular rep-resentation of irregular multimode fiber link the frequencyresponse of a few-mode optical signal transferred by 119872
guided modes over irregular multimode fiber with length 119911
under given particular length of regular span Δ119911 by takinginto account expression (1) can be written in the followingform
119867119877119909 (120596 119911)
= 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(119901)
(119873119911+1)exp [minus120572(119901)
(119873119911+1)(119911 minus 119873119911Δ119911)]
times exp [minus119895 (120596 minus 1205960) 120591(119901119873119911+1) (119911 minus 119873119911Δ119911)]
times exp [minus11989512(120596 minus 1205960)
2120573(119901119873119911+1)
2(119911 minus 119873119911Δ119911)]
times
119873119911
prod
119902=1
119860(119901)
119902exp (minus120572(119901)
119902Δ119911)
times exp (minus119895 (120596 minus 1205960) 120591(119901119902)
Δ119911)
times exp(minus11989512(120596 minus 1205960)
2120573(119901119902)
2Δ119911)
(4)
where119873119911 = 119864(119911Δ119911) 119864(119909) is the integer part of real number119909
The resulting pulse response at the receiver end ofirregularmultimode link is computed by the following simpleexpression
ℎ119877119909 (119905) = 119865minus1(119867119877119909 (120596)) sdot [119865
minus1(119867119877119909 (120596))]
lowast
(5)
where 119865minus1 is inverse Fourier transform [119909]lowast is the complexconjugate of 119909
It is obvious that the proposed model requires fast andsimple method for the analysis of the optical fibers withcomplicated refractive index profile form close to profiles ofreal fibers which provides evaluation of parameters not onlyfor the fundamental but also higher-order guidedmodes Forthis purpose we propose to apply an extension of modifiedGaussian approximation (EMGA) generalized for estimationof any order guided mode parameters propagating oversilica weakly guiding optical fibers with an arbitrary axial-symmetric refractive index profile [102] EMGA is based oncombination of Gaussian approximation and stratificationmethod It permits amuch reduce computing time especiallyfor the calculations of higher-order mode parameters underthe low error [103] by making a passage to analytical expres-sions both for the variational expression and characteristic
equation Below a detailed description of EMGA and itsapplication for the proposed model is presented
4 Extension of ModifiedGaussian Approximation
EMGA is based on classical Gaussian approximation [25] ofradial mode field distribution 119865
(119897)
119898(119877) in the weakly guiding
optical waveguidewith an arbitrary refractive index profile bythe well-known Laguerre-Gauss function expression [4 25100] describing a mode field distribution in weakly guidingoptical waveguide with ideal infinite parabolic index profile
Ψ(119897)
119898(119877) = (
119877
1198770
)
119897
119871(119897)
119898minus1(1198772
1198772
0
) exp(minus1198772
21198772
0
) (6)
where 119871(119897)119898minus1
is Laguerre polynomialEMGA leads to equivalent normalized mode field radius
1198770 estimation by solving a characteristic equation whichis derived from propagation constant variational expressionunder following passage to square core mode parameter 1198802variational expression written for analyzed weakly guidingoptical waveguide with given refractive index profile Param-eter 1198770 is basic for this method and completely defines modetransmission parameters
Unlike knownmethods and their modifications [25 104ndash110] based on conventional Gaussian approximation inEMGAoptical fiberwith an arbitrary graded axial-symmetricindex profile is considered as multicladding optical fiberTherefore refractive index profile inside core region can berepresented in the form of the set of 119873 layers in which therefractive stays a constant [102 103]
119899 (119877) =
119899119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
119899119873 1 lt 119877 le +infin
(7)
and any profile function 119891(119877) related to refractive indexprofile 119899(119877) as
1198992(119877) = 119899
2
max [1 minus 2Δ sdot 119891 (119877)] (8)
where Δ = (1198992
max minus 1198992
119873)21198992
max is profile height parameter canbe written in terms of profile parameter ℎ119896
119891 (119877) =
ℎ119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
1 1 lt 119877 le +infin
(9a)
where
ℎ119896 =1198992
max minus 1198992
119896
1198992max minus 1198992
119873
(9b)
where 119899119896 is refractive index of 119896 layer (119896 = 0 119873) 119899max isthe maximal core refractive index 119899119873 is cladding refractiveindex
This refractive index profile representation permits writ-ing the variational expression for core mode parameter 1198802
Advances in Optical Technologies 5
and characteristic equation for normalized equivalent modefield radius 12059711988021205971198770 = 0 in the form of finite nested sums asfollows [15 16]
1198802=
(119898 minus 1)
(119897 + 119898 minus 1)
times 119876
1198772
0
+ 1198812[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
1198830 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198831 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901
1198832 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901
minus119876 + 1198812[1198780 +
119873minus1
sum
119896=0
ℎ119896 (1198781 minus 1198782)] = 0
(10)
1198780 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(1 minus 1199011198772
0)
1198781 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
1198782 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
119876 =(119897 + 119898 minus 1) (3119897 + 2119898 minus 1)
(119898 minus 1)
+ 21198972
2119898minus2
sum
119902=0
119863119902 (119902 + 119897 minus 1) minus 4119897
times
2119898minus2
sum
119902=0
119862119902 (119902 + 119897)
(11)
where
119863119902 =
min(119902119898minus1)
sum
119901=max(0119902minus119898+1)
119887(119897119898minus1)
119901119887(119897119898minus1)
119902minus119901 (12a)
119871(119897)
119898minus1(119909) 119871(119897+1)
119898minus1(119909) =
2119898minus2
sum
119902=0
119862119902119909119902 (12b)
and 119887(119897119898)
119901is coefficient of polynomial representation in the
form of power series [111 112]
119871(119897)
119898(119909) =
119898
sum
119902=0
119887(119897119898)
119902119909119902 (13a)
119887(119897119898)
119902= (minus1)
119902 (119897 + 119898)
(119897 + 119902) (119898 minus 119902)119902 (13b)
119881 = 1198960119886119899maxradic2Δ is normalized frequency 1198960 = 2120587120582 iswavenumber 120582 is wavelength
Therefore analysis of weakly guiding single-claddingoptical fiber with an arbitrary profile leads to the followingRefractive index profile is represented by the profile function(9a) and (9b) in the form of 119873 layers Fiber parameters andmode orders 119897 and 119898 are substituted to the characteristicequation (11) By means of numerical solution (11) thenormalized equivalent mode field radius 1198770 will be obtainedThen 1198770 is substituted to the expression (10) and modecore parameter 119880 is estimated that permits evaluating thepropagation constant 120573 for guided mode 119871119875119897119898 by the well-known expression [4 25ndash27 55 100]
1205732= 1198962
01198992
max minus1198802
1198862 (14)
Solution of the characteristic equation (11) is correctunder normalized frequency 119881 gt 1 and it should satisfy theguided mode cutoff condition [4 25ndash27 55 100]
1198960119899119873 lt 120573 le 1198960119899max (15)
Optical confinement factor 119875core can be considered as thesecond criterion for the identification of the ghost solutions
119875(119897119898)
co ge 05 (16)
If we take into account Gaussian approximation parameter119875core is defined by analytical expression derived in [15] fromthe generalized integral form forweakly guiding optical fiberspresented in [25]
119875(119897119898)
co =(119898 minus 1)
(119897 + 119898 minus 1)
2119898minus2
sum
119902=0
119863119902 (119897 + 119902) sdot 119882119902 (17)
where
119882119902 = 1 minus exp(minus 1
1198772
0
)
119897+119902
sum
119901=0
1
1199011198772119901
0
(18)
5 Mode Group Velocity
According to what is mentioned above realization of intro-duced model of piecewise regular multimode fiber linkrequires a passage to analytical expressions for mode groupvelocity Guidedmode group velocity V119892 is related to propaga-tion constant120573 by thewell-known formula [4 25ndash27 55 100]
V119892 = minus2120587119888
1205822
120597120582
120597120573 (19)
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
Advances in Optical Technologies 3
50
25
0
minus25
minus50
0 002 004 006 008 01 012 014 016 018 02Length of regular span Δz
(i minus 1)th regular span
iminus1
i i+1
(i + 1)th regular span
Cor
e dia
met
er
r(120583
m)
L (m)
Figure 1 Piecewise regular representation of 200m irregular mul-timode optical fiber with varying core diameter
matrix methods [62 74ndash80] recurrent methods [81ndash85] andquasi-analytic time-domain models based on quasianalyticmethods [86ndash89] and spatial transformmethods [55 86 90ndash95] are widely used The most of them were developed forsimulation of pulse propagation over single mode fibers
From this point of view method described in [96ndash98]can be considered as an exception It combined coupledmode theory with piecewise regular representation and wasdeveloped formodeling of signal transmission processes overlong irregular multimode fiber links Here a random fiberbends are preset and supposed as the main reason of modecoupling occurring Also it takes into account birefringenceHowever the method neglects core geometry variation alonglongitudinal fiber axis and considers only multimode fiberswith ideal infinite 120572-profiles
Therefore the most considered methods above for mod-eling and analysis of pulse propagation over optical fibersshould be adapted for irregular multimode fiber link oper-ating in a few-mode regime Model has to take into accountthe whole total selected excited guided mode compositionof transferred few-mode optical signal and their ldquoindividualrdquoparameters during propagation over multimode fibers withreal graded-index profiles differing form ideal 120572-profiles bylocal defects and refractive index fluctuation Moreover inthis case division ofmode composition on lower- and higher-order mode groups with averaged propagation constant isunacceptable Also model has to take into account launchconditions and mode coupling due to both variation ofcore geometry parameter along fiber length and micro- andmacrobends of fiber occurring in real cable links
3 Model of Piecewise RegularMultimode Fiber Link
In this work we also apply piecewise regular representationcombined with general approach of split-step method [99]to simulate a processes of mode mixing and power diffusiondue to mode coupling (Figure 1) Here single silica weaklyguiding circular multimode fibers with arbitrary axial sym-metric refractive index profile with single continue outercladdings are considered According to piecewise regularrepresentation the fiber is divided into regular spans withlength Δ119911 Inside the span fiber geometry parameters areconsidered as constant and modes propagate independently
without interaction and mixing It is supposed that eachexcited guided mode with propagation constant varyingfrom one regular span to another span satisfies to cut-offcondition for whole regular spans composing fiber Also it isassumed that at the link transmitter end each excited guidedmode starts transferring single optical pulse in particularform identical to input signal (eg Gaussian) During pulsepropagation over regular span its amplitude decreases dueto mode attenuation The signal is mainly distorted due todifference between group velocities and amplitudes ofmodesthat is DMD effect Also transferred by each mode pulsespreading due to chromatic dispersion for particular modeis taken into account
Boundaries of regular spans can be represented gener-ally as ideal axially alignment splice of two almost similaroptical fibers with mismatching parameters However itis correct only for ldquostraightrdquo fibers Therefore it is pro-pose to simulate fiber bends by representation of bound-aries in the form of splice of two mismatched fibers withsome low angular misalignment Mode power redistribu-tion between the amplitudes of signal components as aresult of mode interaction is estimated by mode couplingcoefficient matrix computing at the joints of regular spansHere only guided modes are considered as the main issueunder pulse dynamics research during propagation overmultimode fiber link in a few-mode regime However in factpower loss due lo component transformation from guidedto leaky mode and reflections are also indirectly taken intoaccount
At the receiver end the resulting pulse envelope is con-sidered as superposition of all mode components of signalHere it is propose to apply a well-known expression for thefrequency response of the signal transferred by 119872 modecomponents 119871119875119897119898 over regular multimode fiber with length119911 [4 8ndash15 18 99ndash101]
119867119877119909119905 (120596 119911) = 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(0)
119901exp (minus120572119901119911)
times exp [minus119895 (120596 minus 1205960) 120573(119901)
1119911]
times exp [minus11989512(120596 minus 1205960)
2120573(119901)
2119911]
(1)
where 119865 is direct Fourier transform ℎ119879119909(119905) is initial pulse atthe transmitter end 119860(0)
119901and 120572119901 are starting amplitude and
mode attenuation of 119901th guided mode 119871119875119897119898 (119901 = 1 119872)120573(119901)
1and 120573
(119901)
2are first- and second-order dispersion param-
eters These dispersion parameters are elements of well-knownTaylor series expansion approximation of propagationconstant frequency dependence 120573(120596) [2 55 99]
120573 (120596) = (120596 minus 1205960) 1205731 +1
2(120596 minus 1205960)
21205732
+1
6(120596 minus 1205960)
31205733
(2)
4 Advances in Optical Technologies
where
120573119899 = [120597119899120573
120597120596119899]
120596=1205960
(3)
Here 120573(119901)
1= 120591(119901) is mode delay and 120573
(119901)
2is group velocity
dispersion (GVD) associated with chromatic dispersion of119901th guided mode 119871119875119897119898
Therefore according to introduced piecewise regular rep-resentation of irregular multimode fiber link the frequencyresponse of a few-mode optical signal transferred by 119872
guided modes over irregular multimode fiber with length 119911
under given particular length of regular span Δ119911 by takinginto account expression (1) can be written in the followingform
119867119877119909 (120596 119911)
= 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(119901)
(119873119911+1)exp [minus120572(119901)
(119873119911+1)(119911 minus 119873119911Δ119911)]
times exp [minus119895 (120596 minus 1205960) 120591(119901119873119911+1) (119911 minus 119873119911Δ119911)]
times exp [minus11989512(120596 minus 1205960)
2120573(119901119873119911+1)
2(119911 minus 119873119911Δ119911)]
times
119873119911
prod
119902=1
119860(119901)
119902exp (minus120572(119901)
119902Δ119911)
times exp (minus119895 (120596 minus 1205960) 120591(119901119902)
Δ119911)
times exp(minus11989512(120596 minus 1205960)
2120573(119901119902)
2Δ119911)
(4)
where119873119911 = 119864(119911Δ119911) 119864(119909) is the integer part of real number119909
The resulting pulse response at the receiver end ofirregularmultimode link is computed by the following simpleexpression
ℎ119877119909 (119905) = 119865minus1(119867119877119909 (120596)) sdot [119865
minus1(119867119877119909 (120596))]
lowast
(5)
where 119865minus1 is inverse Fourier transform [119909]lowast is the complexconjugate of 119909
It is obvious that the proposed model requires fast andsimple method for the analysis of the optical fibers withcomplicated refractive index profile form close to profiles ofreal fibers which provides evaluation of parameters not onlyfor the fundamental but also higher-order guidedmodes Forthis purpose we propose to apply an extension of modifiedGaussian approximation (EMGA) generalized for estimationof any order guided mode parameters propagating oversilica weakly guiding optical fibers with an arbitrary axial-symmetric refractive index profile [102] EMGA is based oncombination of Gaussian approximation and stratificationmethod It permits amuch reduce computing time especiallyfor the calculations of higher-order mode parameters underthe low error [103] by making a passage to analytical expres-sions both for the variational expression and characteristic
equation Below a detailed description of EMGA and itsapplication for the proposed model is presented
4 Extension of ModifiedGaussian Approximation
EMGA is based on classical Gaussian approximation [25] ofradial mode field distribution 119865
(119897)
119898(119877) in the weakly guiding
optical waveguidewith an arbitrary refractive index profile bythe well-known Laguerre-Gauss function expression [4 25100] describing a mode field distribution in weakly guidingoptical waveguide with ideal infinite parabolic index profile
Ψ(119897)
119898(119877) = (
119877
1198770
)
119897
119871(119897)
119898minus1(1198772
1198772
0
) exp(minus1198772
21198772
0
) (6)
where 119871(119897)119898minus1
is Laguerre polynomialEMGA leads to equivalent normalized mode field radius
1198770 estimation by solving a characteristic equation whichis derived from propagation constant variational expressionunder following passage to square core mode parameter 1198802variational expression written for analyzed weakly guidingoptical waveguide with given refractive index profile Param-eter 1198770 is basic for this method and completely defines modetransmission parameters
Unlike knownmethods and their modifications [25 104ndash110] based on conventional Gaussian approximation inEMGAoptical fiberwith an arbitrary graded axial-symmetricindex profile is considered as multicladding optical fiberTherefore refractive index profile inside core region can berepresented in the form of the set of 119873 layers in which therefractive stays a constant [102 103]
119899 (119877) =
119899119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
119899119873 1 lt 119877 le +infin
(7)
and any profile function 119891(119877) related to refractive indexprofile 119899(119877) as
1198992(119877) = 119899
2
max [1 minus 2Δ sdot 119891 (119877)] (8)
where Δ = (1198992
max minus 1198992
119873)21198992
max is profile height parameter canbe written in terms of profile parameter ℎ119896
119891 (119877) =
ℎ119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
1 1 lt 119877 le +infin
(9a)
where
ℎ119896 =1198992
max minus 1198992
119896
1198992max minus 1198992
119873
(9b)
where 119899119896 is refractive index of 119896 layer (119896 = 0 119873) 119899max isthe maximal core refractive index 119899119873 is cladding refractiveindex
This refractive index profile representation permits writ-ing the variational expression for core mode parameter 1198802
Advances in Optical Technologies 5
and characteristic equation for normalized equivalent modefield radius 12059711988021205971198770 = 0 in the form of finite nested sums asfollows [15 16]
1198802=
(119898 minus 1)
(119897 + 119898 minus 1)
times 119876
1198772
0
+ 1198812[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
1198830 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198831 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901
1198832 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901
minus119876 + 1198812[1198780 +
119873minus1
sum
119896=0
ℎ119896 (1198781 minus 1198782)] = 0
(10)
1198780 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(1 minus 1199011198772
0)
1198781 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
1198782 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
119876 =(119897 + 119898 minus 1) (3119897 + 2119898 minus 1)
(119898 minus 1)
+ 21198972
2119898minus2
sum
119902=0
119863119902 (119902 + 119897 minus 1) minus 4119897
times
2119898minus2
sum
119902=0
119862119902 (119902 + 119897)
(11)
where
119863119902 =
min(119902119898minus1)
sum
119901=max(0119902minus119898+1)
119887(119897119898minus1)
119901119887(119897119898minus1)
119902minus119901 (12a)
119871(119897)
119898minus1(119909) 119871(119897+1)
119898minus1(119909) =
2119898minus2
sum
119902=0
119862119902119909119902 (12b)
and 119887(119897119898)
119901is coefficient of polynomial representation in the
form of power series [111 112]
119871(119897)
119898(119909) =
119898
sum
119902=0
119887(119897119898)
119902119909119902 (13a)
119887(119897119898)
119902= (minus1)
119902 (119897 + 119898)
(119897 + 119902) (119898 minus 119902)119902 (13b)
119881 = 1198960119886119899maxradic2Δ is normalized frequency 1198960 = 2120587120582 iswavenumber 120582 is wavelength
Therefore analysis of weakly guiding single-claddingoptical fiber with an arbitrary profile leads to the followingRefractive index profile is represented by the profile function(9a) and (9b) in the form of 119873 layers Fiber parameters andmode orders 119897 and 119898 are substituted to the characteristicequation (11) By means of numerical solution (11) thenormalized equivalent mode field radius 1198770 will be obtainedThen 1198770 is substituted to the expression (10) and modecore parameter 119880 is estimated that permits evaluating thepropagation constant 120573 for guided mode 119871119875119897119898 by the well-known expression [4 25ndash27 55 100]
1205732= 1198962
01198992
max minus1198802
1198862 (14)
Solution of the characteristic equation (11) is correctunder normalized frequency 119881 gt 1 and it should satisfy theguided mode cutoff condition [4 25ndash27 55 100]
1198960119899119873 lt 120573 le 1198960119899max (15)
Optical confinement factor 119875core can be considered as thesecond criterion for the identification of the ghost solutions
119875(119897119898)
co ge 05 (16)
If we take into account Gaussian approximation parameter119875core is defined by analytical expression derived in [15] fromthe generalized integral form forweakly guiding optical fiberspresented in [25]
119875(119897119898)
co =(119898 minus 1)
(119897 + 119898 minus 1)
2119898minus2
sum
119902=0
119863119902 (119897 + 119902) sdot 119882119902 (17)
where
119882119902 = 1 minus exp(minus 1
1198772
0
)
119897+119902
sum
119901=0
1
1199011198772119901
0
(18)
5 Mode Group Velocity
According to what is mentioned above realization of intro-duced model of piecewise regular multimode fiber linkrequires a passage to analytical expressions for mode groupvelocity Guidedmode group velocity V119892 is related to propaga-tion constant120573 by thewell-known formula [4 25ndash27 55 100]
V119892 = minus2120587119888
1205822
120597120582
120597120573 (19)
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
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DistributedSensor Networks
International Journal of
4 Advances in Optical Technologies
where
120573119899 = [120597119899120573
120597120596119899]
120596=1205960
(3)
Here 120573(119901)
1= 120591(119901) is mode delay and 120573
(119901)
2is group velocity
dispersion (GVD) associated with chromatic dispersion of119901th guided mode 119871119875119897119898
Therefore according to introduced piecewise regular rep-resentation of irregular multimode fiber link the frequencyresponse of a few-mode optical signal transferred by 119872
guided modes over irregular multimode fiber with length 119911
under given particular length of regular span Δ119911 by takinginto account expression (1) can be written in the followingform
119867119877119909 (120596 119911)
= 119865 [ℎ119879119909 (119905)]
119872
sum
119901
119860(119901)
(119873119911+1)exp [minus120572(119901)
(119873119911+1)(119911 minus 119873119911Δ119911)]
times exp [minus119895 (120596 minus 1205960) 120591(119901119873119911+1) (119911 minus 119873119911Δ119911)]
times exp [minus11989512(120596 minus 1205960)
2120573(119901119873119911+1)
2(119911 minus 119873119911Δ119911)]
times
119873119911
prod
119902=1
119860(119901)
119902exp (minus120572(119901)
119902Δ119911)
times exp (minus119895 (120596 minus 1205960) 120591(119901119902)
Δ119911)
times exp(minus11989512(120596 minus 1205960)
2120573(119901119902)
2Δ119911)
(4)
where119873119911 = 119864(119911Δ119911) 119864(119909) is the integer part of real number119909
The resulting pulse response at the receiver end ofirregularmultimode link is computed by the following simpleexpression
ℎ119877119909 (119905) = 119865minus1(119867119877119909 (120596)) sdot [119865
minus1(119867119877119909 (120596))]
lowast
(5)
where 119865minus1 is inverse Fourier transform [119909]lowast is the complexconjugate of 119909
It is obvious that the proposed model requires fast andsimple method for the analysis of the optical fibers withcomplicated refractive index profile form close to profiles ofreal fibers which provides evaluation of parameters not onlyfor the fundamental but also higher-order guidedmodes Forthis purpose we propose to apply an extension of modifiedGaussian approximation (EMGA) generalized for estimationof any order guided mode parameters propagating oversilica weakly guiding optical fibers with an arbitrary axial-symmetric refractive index profile [102] EMGA is based oncombination of Gaussian approximation and stratificationmethod It permits amuch reduce computing time especiallyfor the calculations of higher-order mode parameters underthe low error [103] by making a passage to analytical expres-sions both for the variational expression and characteristic
equation Below a detailed description of EMGA and itsapplication for the proposed model is presented
4 Extension of ModifiedGaussian Approximation
EMGA is based on classical Gaussian approximation [25] ofradial mode field distribution 119865
(119897)
119898(119877) in the weakly guiding
optical waveguidewith an arbitrary refractive index profile bythe well-known Laguerre-Gauss function expression [4 25100] describing a mode field distribution in weakly guidingoptical waveguide with ideal infinite parabolic index profile
Ψ(119897)
119898(119877) = (
119877
1198770
)
119897
119871(119897)
119898minus1(1198772
1198772
0
) exp(minus1198772
21198772
0
) (6)
where 119871(119897)119898minus1
is Laguerre polynomialEMGA leads to equivalent normalized mode field radius
1198770 estimation by solving a characteristic equation whichis derived from propagation constant variational expressionunder following passage to square core mode parameter 1198802variational expression written for analyzed weakly guidingoptical waveguide with given refractive index profile Param-eter 1198770 is basic for this method and completely defines modetransmission parameters
Unlike knownmethods and their modifications [25 104ndash110] based on conventional Gaussian approximation inEMGAoptical fiberwith an arbitrary graded axial-symmetricindex profile is considered as multicladding optical fiberTherefore refractive index profile inside core region can berepresented in the form of the set of 119873 layers in which therefractive stays a constant [102 103]
119899 (119877) =
119899119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
119899119873 1 lt 119877 le +infin
(7)
and any profile function 119891(119877) related to refractive indexprofile 119899(119877) as
1198992(119877) = 119899
2
max [1 minus 2Δ sdot 119891 (119877)] (8)
where Δ = (1198992
max minus 1198992
119873)21198992
max is profile height parameter canbe written in terms of profile parameter ℎ119896
119891 (119877) =
ℎ119896 119877119896 =119896
119873 0 le 119896 le 119873 minus 1
1 1 lt 119877 le +infin
(9a)
where
ℎ119896 =1198992
max minus 1198992
119896
1198992max minus 1198992
119873
(9b)
where 119899119896 is refractive index of 119896 layer (119896 = 0 119873) 119899max isthe maximal core refractive index 119899119873 is cladding refractiveindex
This refractive index profile representation permits writ-ing the variational expression for core mode parameter 1198802
Advances in Optical Technologies 5
and characteristic equation for normalized equivalent modefield radius 12059711988021205971198770 = 0 in the form of finite nested sums asfollows [15 16]
1198802=
(119898 minus 1)
(119897 + 119898 minus 1)
times 119876
1198772
0
+ 1198812[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
1198830 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198831 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901
1198832 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901
minus119876 + 1198812[1198780 +
119873minus1
sum
119896=0
ℎ119896 (1198781 minus 1198782)] = 0
(10)
1198780 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(1 minus 1199011198772
0)
1198781 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
1198782 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
119876 =(119897 + 119898 minus 1) (3119897 + 2119898 minus 1)
(119898 minus 1)
+ 21198972
2119898minus2
sum
119902=0
119863119902 (119902 + 119897 minus 1) minus 4119897
times
2119898minus2
sum
119902=0
119862119902 (119902 + 119897)
(11)
where
119863119902 =
min(119902119898minus1)
sum
119901=max(0119902minus119898+1)
119887(119897119898minus1)
119901119887(119897119898minus1)
119902minus119901 (12a)
119871(119897)
119898minus1(119909) 119871(119897+1)
119898minus1(119909) =
2119898minus2
sum
119902=0
119862119902119909119902 (12b)
and 119887(119897119898)
119901is coefficient of polynomial representation in the
form of power series [111 112]
119871(119897)
119898(119909) =
119898
sum
119902=0
119887(119897119898)
119902119909119902 (13a)
119887(119897119898)
119902= (minus1)
119902 (119897 + 119898)
(119897 + 119902) (119898 minus 119902)119902 (13b)
119881 = 1198960119886119899maxradic2Δ is normalized frequency 1198960 = 2120587120582 iswavenumber 120582 is wavelength
Therefore analysis of weakly guiding single-claddingoptical fiber with an arbitrary profile leads to the followingRefractive index profile is represented by the profile function(9a) and (9b) in the form of 119873 layers Fiber parameters andmode orders 119897 and 119898 are substituted to the characteristicequation (11) By means of numerical solution (11) thenormalized equivalent mode field radius 1198770 will be obtainedThen 1198770 is substituted to the expression (10) and modecore parameter 119880 is estimated that permits evaluating thepropagation constant 120573 for guided mode 119871119875119897119898 by the well-known expression [4 25ndash27 55 100]
1205732= 1198962
01198992
max minus1198802
1198862 (14)
Solution of the characteristic equation (11) is correctunder normalized frequency 119881 gt 1 and it should satisfy theguided mode cutoff condition [4 25ndash27 55 100]
1198960119899119873 lt 120573 le 1198960119899max (15)
Optical confinement factor 119875core can be considered as thesecond criterion for the identification of the ghost solutions
119875(119897119898)
co ge 05 (16)
If we take into account Gaussian approximation parameter119875core is defined by analytical expression derived in [15] fromthe generalized integral form forweakly guiding optical fiberspresented in [25]
119875(119897119898)
co =(119898 minus 1)
(119897 + 119898 minus 1)
2119898minus2
sum
119902=0
119863119902 (119897 + 119902) sdot 119882119902 (17)
where
119882119902 = 1 minus exp(minus 1
1198772
0
)
119897+119902
sum
119901=0
1
1199011198772119901
0
(18)
5 Mode Group Velocity
According to what is mentioned above realization of intro-duced model of piecewise regular multimode fiber linkrequires a passage to analytical expressions for mode groupvelocity Guidedmode group velocity V119892 is related to propaga-tion constant120573 by thewell-known formula [4 25ndash27 55 100]
V119892 = minus2120587119888
1205822
120597120582
120597120573 (19)
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
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International Journal of
Advances in Optical Technologies 5
and characteristic equation for normalized equivalent modefield radius 12059711988021205971198770 = 0 in the form of finite nested sums asfollows [15 16]
1198802=
(119898 minus 1)
(119897 + 119898 minus 1)
times 119876
1198772
0
+ 1198812[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
1198830 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198831 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901
1198832 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901
minus119876 + 1198812[1198780 +
119873minus1
sum
119896=0
ℎ119896 (1198781 minus 1198782)] = 0
(10)
1198780 = exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(1 minus 1199011198772
0)
1198781 = exp(minus 1198962
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
1198782 = exp(minus(119896 + 1)2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
119876 =(119897 + 119898 minus 1) (3119897 + 2119898 minus 1)
(119898 minus 1)
+ 21198972
2119898minus2
sum
119902=0
119863119902 (119902 + 119897 minus 1) minus 4119897
times
2119898minus2
sum
119902=0
119862119902 (119902 + 119897)
(11)
where
119863119902 =
min(119902119898minus1)
sum
119901=max(0119902minus119898+1)
119887(119897119898minus1)
119901119887(119897119898minus1)
119902minus119901 (12a)
119871(119897)
119898minus1(119909) 119871(119897+1)
119898minus1(119909) =
2119898minus2
sum
119902=0
119862119902119909119902 (12b)
and 119887(119897119898)
119901is coefficient of polynomial representation in the
form of power series [111 112]
119871(119897)
119898(119909) =
119898
sum
119902=0
119887(119897119898)
119902119909119902 (13a)
119887(119897119898)
119902= (minus1)
119902 (119897 + 119898)
(119897 + 119902) (119898 minus 119902)119902 (13b)
119881 = 1198960119886119899maxradic2Δ is normalized frequency 1198960 = 2120587120582 iswavenumber 120582 is wavelength
Therefore analysis of weakly guiding single-claddingoptical fiber with an arbitrary profile leads to the followingRefractive index profile is represented by the profile function(9a) and (9b) in the form of 119873 layers Fiber parameters andmode orders 119897 and 119898 are substituted to the characteristicequation (11) By means of numerical solution (11) thenormalized equivalent mode field radius 1198770 will be obtainedThen 1198770 is substituted to the expression (10) and modecore parameter 119880 is estimated that permits evaluating thepropagation constant 120573 for guided mode 119871119875119897119898 by the well-known expression [4 25ndash27 55 100]
1205732= 1198962
01198992
max minus1198802
1198862 (14)
Solution of the characteristic equation (11) is correctunder normalized frequency 119881 gt 1 and it should satisfy theguided mode cutoff condition [4 25ndash27 55 100]
1198960119899119873 lt 120573 le 1198960119899max (15)
Optical confinement factor 119875core can be considered as thesecond criterion for the identification of the ghost solutions
119875(119897119898)
co ge 05 (16)
If we take into account Gaussian approximation parameter119875core is defined by analytical expression derived in [15] fromthe generalized integral form forweakly guiding optical fiberspresented in [25]
119875(119897119898)
co =(119898 minus 1)
(119897 + 119898 minus 1)
2119898minus2
sum
119902=0
119863119902 (119897 + 119902) sdot 119882119902 (17)
where
119882119902 = 1 minus exp(minus 1
1198772
0
)
119897+119902
sum
119901=0
1
1199011198772119901
0
(18)
5 Mode Group Velocity
According to what is mentioned above realization of intro-duced model of piecewise regular multimode fiber linkrequires a passage to analytical expressions for mode groupvelocity Guidedmode group velocity V119892 is related to propaga-tion constant120573 by thewell-known formula [4 25ndash27 55 100]
V119892 = minus2120587119888
1205822
120597120582
120597120573 (19)
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
6 Advances in Optical Technologies
By squaring and following differentiation of the expres-sion (14) with respect to wavelength the first partial deriva-tive of the square of propagation constant will be obtained
1205971205732
120597120582= minus
21198962
01198992
max120582
+ 1198962
0
1205971198992
max120597120582
minus1
1198862
1205971198802
120597120582 (20)
After substitution of (20) to (19) an expression for themode group velocity can be rewritten in the following form
V119892 = minus4120587120573119888
1205822
120597120582
1205971205732= minus
1198962
0120573119888
120587
120597120582
1205971205732 (21)
Then by differentiating expressions (10) and (11) withrespect to wavelength first derivatives of square core modeparameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
1205971198802
120597120582=
(119898 minus 1)
(119897 + 119898 minus 1)
times minus2119882
1198773
0
1205971198770
120597120582+1205971198812
120597120582[1198830 +
119873minus1
sum
119896=0
ℎ119896 (1198831 minus 1198832)]
+ 21198812119883(1)
0
1205971198770
120597120582+ 1198812
119873minus1
sum
119896=0
120597ℎ119896
120597120582(1198831 minus 1198832)
+ 21198812 1205971198770
120597120582
119873minus1
sum
119896=0
ℎ119896 (119883(1)
1minus 119883(1)
2)
119883(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(1 minus 1199011198772
0)
119883(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
1198962119901
1198732119901(1198962
1198732minus 1199011198772
0)
119883(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
(119896 + 1)2119901
1198732119901((119896 + 1)
2
1198732minus 1199011198772
0)
1205971198770
120597120582= (1198780
1205971198812
120597120582
+
119873minus1
sum
119896=0
[(1198780 minus 1198781) (ℎ119896
1205971198812
120597120582+ 1198812 120597ℎ119896
120597120582)])
times (minus21198812sdot [119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)])
minus1
119878(1)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+3
0
[(1 minus 1199011198772
0)2
minus 1199011198774
0]
119878(1)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+3
0
[(1198962
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
119878(1)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+3
0
times [((119896 + 1)
2
1198732minus 1199011198772
0)
2
minus 1199011198774
0]
(22)
6 Mode Chromatic Dispersion Parameter
The same technique is applied to derive the analytical expres-sions for mode chromatic dispersion parameter119863 associatedwith GVD Chromatic dispersion parameter 119863 is related topropagation constant 120573 by the well-known formula [4 25ndash27 55 100]
119863 = minus120582
2120587119888(2
120597120573
120597120582+ 120582
1205972120573
1205971205822) (23)
By differentiating (20) a second partial derivative of thesquare of propagation constant will be obtained
12059721205732
1205971205822= 1198962
0(12059721198992
max1205971205822
+61198992
max1205822
minus4
120582
1205971198992
max120597120582
) minus1
1198862
12059721198802
1205971205822 (24)
Therefore expression for119863 leads to following form
119863 = minus120587
1198962
0120573119888
[2
120582
1205971205732
120597120582+12059721205732
1205971205822minus
1
21205732(1205971205732
120597120582)
2
] (25)
By differentiating expressions (22) with respect to wave-length required second derivatives of square core mode
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
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International Journal of
Advances in Optical Technologies 7
parameter 1198802 and normalized equivalent mode field radius1198770 will be obtained
12059721198802
1205971205822
=(119898 minus 1)
(119897 + 119898 minus 1)
times 2119872
1198773
0
[3
1198770
(1205971198770
120597120582)
2
minus12059721198770
1205971205822]
+
119873minus1
sum
119896=0
(1198831 minus 1198832)
times (ℎ119896
12059721198812
1205971205822+ 2
120597ℎ119896
120597120582
1205971198812
120597120582+ 1198812 1205972ℎ119896
1205971205822)
+ 2
119873minus1
sum
119896=0
(119883(1)
1minus 119883(1)
2)
times (2ℎ119896
1205971198812
120597120582
1205971198770
120597120582+ 21198812 120597ℎ119896
120597120582
1205971198770
120597120582+ 1198812ℎ119896
12059721198770
1205971205822)
+ 41198812(1205971198770
120597120582)
2
[119883(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119883(2)
1minus 119883(2)
2)]
+ 1198830
12059721198812
1205971205822+ 2119883(1)
0(2
1205971198812
120597120582
1205971198770
120597120582+ 1198812 12059721198770
1205971205822)
119883(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
[(1 minus 1199011198772
0)2
+1198772
0
2(1199011198772
0minus 3)]
119883(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
1198962119901
1198732119901
times [(1198962
1198732minus 1199011198772
0)
2
+1198772
0
2(1199011198772
0minus31198962
1198732)]
119883(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
(119896 + 1)2119901
1198732119901
times [(119896 + 1)
2
1198732minus 1199011198772
0]
2
+1198772
0
2[1199011198772
0minus3(119896 + 1)
2
1198732]
120597119883(1)
0
120597120582= 2119883(2)
0
1205971198770
120597120582
120597119883(1)
1
120597120582= 2119883(2)
1
1205971198770
120597120582
120597119883(1)
2
120597120582= 2119883(2)
2
1205971198770
120597120582
12059721198770
1205971205822= minus [119881
2119878(1)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(1)
1minus 119878(1)
2)]
minus1
times 21205971198770
120597120582
119873minus1
sum
119896=0
(119878(1)
1minus 119878(1)
2)
times (1198812 120597ℎ119896
120597120582+ ℎ119896
1205971198812
120597120582)
+
119873minus1
sum
119896=0
(1198781 minus 1198782)
times (ℎ119896
2
12059721198812
1205971205822+120597ℎ119896
120597120582
1205971198812
120597120582+1198812
2
1205972ℎ119896
1205971205822)
+ 21198812(1205971198770
120597120582)
2
times [119878(2)
0+
119873minus1
sum
119896=0
ℎ119896 (119878(2)
1minus 119878(2)
2)]
+1198780
2
12059721198812
1205971205822+ 2119878(1)
0
1205971198770
120597120582
1205971198812
120597120582
119878(2)
0= exp(minus 1
1198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)
1199011198772119901+6
0
times (1 minus 1199011198772
0)3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus 3]
119878(2)
1= exp(minus 119896
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)1198962119901
11990111987321199011198772119901+6
0
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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DistributedSensor Networks
International Journal of
8 Advances in Optical Technologies
times (1198962
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
31198964
1198734]
119878(2)
2= exp(minus(119896 + 1)
2
11987321198772
0
)
2119898minus2
sum
119902=0
119863119902
times
119897+119902
sum
119901=0
(119897 + 119902)(119896 + 1)2119901
11990111987321199011198772119901+6
0
times ((119896 + 1)
2
1198732minus 1199011198772
0)
3
+1198772
0
2[1199011198774
0(3119901 minus 1) minus
3(119896 + 1)4
1198734]
120597119878(1)
0
120597120582= 2119878(2)
0
1205971198770
120597120582
120597119878(1)
1
120597120582= 2119878(2)
1
1205971198770
120597120582
120597119878(1)
2
120597120582= 2119878(2)
2
1205971198770
120597120582
(26)
7 Material Dispersion and Profile Parameters
We shall apply a well-known Sellmeier equation to take intoaccount the refractive index dependence on wavelength [425ndash27 55 100]
119899 (120582) = radic1 +
3
sum
119894=1
119860 1198941205822
1205822 minus 1198612
119894
(27)
where119860 119894 and119861119894 are Sellmeierrsquos coefficients (119861119894 is also denotedas the resonance wavelength) which has been empiricallymeasured for GeO2-SiO2 glasses [113 114] under the severalparticular dopant concentration Here we shall apply themethod described in [115] to estimate Sellmeier coefficientsat the graded-index profile points
A passage from the differentiation operator to its squareis used by analogously to propagation constant derivativesThis passage will much simplify expressions for the first andsecond derivatives of the refractive index
120597119899
120597120582=
1
2119899
1205971198992
120597120582
1205971198992
120597120582= minus2120582
3
sum
119894=1
119860 1198941198612
119894
(1205822 minus 1198612
119894)2
(28)
1205972119899
1205971205822=
1
2119899[12059721198992
1205971205822minus
1
21198992(1205971198992
120597120582)
2
] (29a)
12059721198992
1205971205822=
3
sum
119894=1
2119860 1198941198612
119894(1198612
119894+ 31205822)
(1205822 minus 1198612
119894)3
(29b)
Therefore the first and second derivatives of the profileheight parameter Δ can be expressed as follows
120597Δ
120597120582=
1
21198992max[(1 minus 2Δ)
1205971198992
max120597120582
minus1205971198992
119873
120597120582]
1205972Δ
1205971205822=
1
21198992max[(1 minus 2Δ)
12059721198992
max1205971205822
minus12059721198992
119873
1205971205822minus 4
120597Δ
120597120582
1205971198992
max120597120582
]
(30)
Then derivatives of profile parameter ℎ119896 defined byformula (9b) are determined by the following expressions
120597ℎ119896
120597120582=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)1205971198992
max120597120582
minus1205971198992
119896
120597120582+ ℎ119896
1205971198992
119873
120597120582]
1205972ℎ119896
1205971205822=
1
1198992max minus 1198992
119873
[(1 minus ℎ119896)12059721198992
max1205971205822
minus12059721198992
119896
1205971205822+ ℎ119896
12059721198992
119873
1205971205822
minus 2120597ℎ119896
120597120582(1205971198992
max120597120582
minus1205971198992
119873
120597120582)]
(31)
Finally the first and second derivatives of the normalizedfrequency can be written with a help of formulas (28)ndash(30) inthe following form
1205971198812
120597120582= 11988621198962
0[1205971198992
max120597120582
minus1205971198992
119873
120597120582minus2
120582(1198992
max minus 1198992
119873)]
12059721198812
1205971205822= minus
2
120582
1205971198812
120597120582+ 11988621198962
0
times 12059721198992
max1205971205822
minus12059721198992
119873
1205971205822
+2
120582[1
120582(1198992
max minus 1198992
119873) minus
1205971198992
max120597120582
+1205971198992
119873
120597120582]
(32)
8 Differential Mode Attenuation
We shall apply a simple empirical relation proposed inwork [116] using experimental data from [117] to estimatedifferential mode attenuation depending on mode order
120572120583 (120582) = 1205720 (120582) + 1205720 (120582) 1198689 [7 35(120583 minus 1
1198720
)
2119892(119892+2)
] (33a)
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Optical Technologies 9
where 120583 = 2119898 + 119897 + 1 is principal mode number 1205720(120582) is theattenuation of lower-order modes (it is supposed equal to theattenuation at the correspondence wavelength mentioned infiber specification) 1198720 is total number of modes satisfyingthe cutoff condition for analyzed fiber
1198720 (120582) = 2120587119899max (120582)
120582radic119892Δ (120582)
119892 + 2 (33b)
where 119892 is gradient factor of smoothed 120572-profile
9 Mode Coupling
Nowadays calculation of mode coupling coefficients is stillone of the most well-known and simple methods for opti-cal waveguide junction analysis It is widely applied forestimation of insertion loss and reflection at the splices ofthe same type optical fibers with nonidentical technologicparameters [117ndash121] modeling and research of the influenceof launch conditions on optical waveguidemode compositionexcitation [25 122ndash124] including the modeling of a few-mode optical signal propagation over multimode fibers [4 810]
Generally the coupling coefficient 120578119898119899 of injected mode119898 with excited mode 119899 is determined by the overlap integralof interacting mode fields [4 8 10 117ndash124]
120578119898119899 =
100381610038161003816100381610038161003816intinfin
0int2120587
0Ψ(119897119898)
119898Ψ(119897119899)
119899119903119889119903 119889120593
100381610038161003816100381610038161003816
2
intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119898)
119898
10038161003816100381610038161003816
2
119903119889119903 119889120593intinfin
0int2120587
0
10038161003816100381610038161003816Ψ(119897119899)
119899
10038161003816100381610038161003816
2
119903119889119903 119889120593
(34)
Therefore substitution of radial mode field distributionexpressions to integrals (34) is required for estimation ofmode coupling coefficientsThe simplestmode field structurecorresponds to the fundamental mode 11987111987501 As a result itpermits obtaining analytical formulas for the overlap integral(34) by applying simple approximation expressions or bya particular function series expansion for the fundamentalmode field representation These algorithms are widely usedfor analysis of the single mode fiber splices [25 117ndash123]
Contrariwise higher-order mode field structure is morecomplicated For this reason in certain papers concerningmultimode fiber analysis [4 8 10] it is propose to integrateexpression (34) numerically or to substitute simple approxi-mation expressions which are solutions of wave equation forthe fibers with ideal step or infinite parabolic refractive indexprofile However this substitution is not correct entirely asit was mentioned above real refractive index profiles differmuch from the model ideal smoothed 120572-profiles [100] evenfor the last generation of commercial silica optical fibers bothsingle mode and multimode
Here we propose to apply the overlap integral methodcombined with introduced EMGA Under the EMPG appli-cation the use of the algorithm mentioned above for thefollowing mode coupling estimation becomes acceptablehere local features of real silica optical fiber refractive indexprofile are taken into account that provides decreasing theerror of calculations Thereby the fiber splice analysis willled to substitution of simple approximation expression (6)
(a)
(b)
Figure 2 Multimode fiber samples (a) spools of modern mul-timode fibers Corning of cat OM2+OM3 (b) coil of the first-generation multimode fiber of cat OM2
to the overlap integral (34) under preliminary estimation ofequivalent mode field radiuses of injected and excited modeswith given azimuthal and radial orders by EMPG In termsof introduced model of piecewise regular multimode fiberlink this approach provides both simulation of mode mixingand power diffusion by recalculation of the matrix of modecoupling coefficients at the junctions of regular span bound-aries and modeling of initial mode composition excitation inmultimode fiber by laser source at the transmitter end undertaking into account launch condition (single mode pigtailcentral launching offset air gap special matching fiber etc)
Another advantage of proposed method combining theoverlap integral method and EMGA is the ability of passageto analytical expressions both for mode coupling coefficientsat the ideal centralized splice and splice with offset or tiltfor modes with arbitrary order In recent work [125] thederivation of them is described in detail Therefore here onlyfinal heavy analytical expressions are presented
First of all let us consider the idealized model of diverseoptical fiber splice jointed mismatched type fibers have idealsmooth ends cleaved under 90∘ to the core axes and there areno any longitudinal axial or angular misalignments betweencore centers On the one hand it is very simplified fiber splicedescription on the other hand this ideal splice representationcorresponds to central launching during themodeling of fiberexcitation by laser source
A passage from the generalized form of the overlapintegral (1) to analytical expression for the arbitrary ordermode coupling coefficient estimation at the central splice ofthe optical fibers with mismatched parameters and withoutany misalignments was proposed in work [121] Analyticalexpression deriving is described in detail in [125] Finally the
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
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International Journal of
10 Advances in Optical Technologies
0 10 20 30 400
02
04
06
08
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(a)
0 10 20 30 400
01
02
03
04
05
06
07
08
09
1
minus40 minus30 minus20 minus10
Δ(
)
r (120583m)
(b)
Figure 3 Refractive index profiles measured by EXFO NR-9200 (a) sample no 01 multimode fiber Corning of cat OM2+OM3 length is373m sample no 02 first-generation multimode fiber of cat OM2 length is 456m
Pulsegenerator
1 2
SM FCPCFCPC
3 4
MM FCPC
5
Receiver10
MM FCPCFCPC
Micro-processor
116
9 8127
MM FCPC
SM pigtail
MM pigtail
MM pigtail
MM pigtail
Tested MMFcoilspool
Fusion splice
Fusion splice
SM LD 1310HM
Figure 4 Simplified block scheme if pulse response workstation R2D2 and experimental setup
Figure 5 Experimental setup Figure 6 Software screenshot
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Advances in Optical Technologies 11
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
t05 = 340ps
Figure 7 Input pulse
0 005 01 015 02 025 03
0
005
01
015
02
025
minus02
minus015
minus01
minus005
z (km)
Δa
(120583m
)
Figure 8 Core diameter variations over fiber length 340m
formula for arbitrary order mode coupling coefficient at thecentral splice is written as follows
120578119898119899 =Γ (119898) Γ (119899)
Γ (119897 + 119898) Γ (119897 + 119899)(2120588119898120588119899)
2119897+2[
(120588119898 minus 120588119899)119898+119899minus2
(1205882119898+ 1205882119899)119898+119899+119897minus1
]
2
times
min(119898minus1119899minus1)
sum
119896=0
(minus1)119896 Γ (119898 + 119899 + 119897 minus 119896 minus 1)
Γ (119898 minus 119896) Γ (119899 minus 119896) 119896
times (1205882
119898+ 1205882
119899
1205882119898minus 1205882119899
)
119896
(35)
Here mode coupling occurs only for modes with the sameazimuthal order 119897 Γ is Gamma function 120588119898 and 120588119899 areinjected 119871119875119897119898 and excited 119871119875119897119899 mode field radiuses
The analytical expression for arbitrary order mode cou-pling coefficient at the optical fiber splice under low anglemisalignment 120579 lt 10
∘ has the following form [125]
120578(120579)
119898119899= 4
(119898 minus 1)
(119897119898 + 119898 minus 1)
(119899 minus 1)
(119897119899 + 119899 minus 1)
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898 119898minus1)
119902119887(119897119899 119899minus1)
119901minus119902
times2(1198971198982)+119901(1198960119899120579120579)
119897119899Γ ((1198971198982) + 119897119899 + 119901 + 1)
(1205882119898+ 1205882119899)(1198971198982)+119897119899+119901+1
(119897119899)
times 1205882(119897119899+119901minus119902)+1
119898120588119897119898+119897119899+2119902+1
119899
sdot11198651 [(
119897119898
2+ 119897119899 + 119901 + 1)
(119897119899 + 1) minus(1198960119899120579120579)2 1205882
1198981205882
119899
(1205882119898+ 1205882119899)]
2
(36)
where11198651 is confluent hypergeometric function of the 1st
kind [111 112]
11198651 (119886 119887 119909) = 1 +
infin
sum
119901=1
[
119901minus1
prod
119902=0
(119886 + 119902) 119909
(1 + 119902) (119887 + 119902)] (37)
where 119887(119897119898 119898minus1)
119902and 119887
(119897119899 119899minus1)
119902are Laguerre polynomial expan-
sion factors of (13a) (13b) 119899120579 is refractive index of launchingmedium (air gap core of adjustingexiting fiber etc)
The final expression for mode coupling coefficients for0th azimuth order modes 1198711198750119898 and 1198711198750119899 at the splice withtransverse offset 119889 has the form [125]
0120578(119889)
119898119899= 1198720 sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(0119898minus1)
119902119887(0119899minus1)
119901minus119902
times
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)
sum
119896=0
(minus1)119906(2119904 minus 1) [2 (119902 minus 119905)]119906
[2 (119902 minus 119905) minus 119896]119896
times
119864(1199062)
sum
119908=0
1
2119908 (119906 minus 2119908)119908sdot
1198892(119901minus119904minus119908)
(1205882119898+ 1205882119899)119904+119906minus119908
times 1205882(119904minus119902+119908)
1198981205882(119904minus119901+119902+119906minus119908)
119899
2
(38)
where the lower and upper limits of second and forth sumsare 119902 = max(0 119901 minus 119899 + 1) min(119901119898minus 1) and 119905 = max(0 119904 minus
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
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DistributedSensor Networks
International Journal of
12 Advances in Optical Technologies
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
MeasuredSimulated
t (ns)
AA
max
MMF N08 120579 = 42∘
(a)
MeasuredSimulated
0 05 1 15 2 250
01
02
03
04
05
06
07
08
09
1
t (ns)
AA
max
MMF N12 120579 = 42∘
(b)
Figure 9 Measured and simulated pulse responses (a) multimode fiber of the OM2+OM3 category (profile is shown in Figure 3(a)) (b) (a)multimode fiber of the OM2 category (profile is shown in Figure 3(b))
119901 + 119902) min(119904 119902) and the ldquooddrdquo factorial is (2119904 minus 1) = 1 sdot
3 (2119904 minus 1)
1198720 =41205882
1198981205882
119899
(1205882119898+ 1205882119899)2exp[minus 119889
2
1205882119898
(1 minus1205882
119899
1205882119898+ 1205882119899
)]
119876(119902119901minus119902)
119905=
119902
(119902 minus 119905)119905sdot
(119901 minus 119902)
(119901 minus 119902 minus 119904 + 119905) (119904 minus 119905)
119906 = 119896 + 2 (119901 minus 119902 minus 119904 + 119905)
(39)
The analytic formula mode coupling coefficients for the1st azimuth order modes 1198711198751119898 and 1198711198751119899 at the splice withtransverse offset 119889 is written in the following form [125]
1120578(119889)119898119899
=41198720
119898119899sdot
119898+119899minus2
sum
119901=0
sum
119902
119887(1119898minus1)
119902119887(1119899minus1)
119901minus119902
119901
sum
119904=0
sum
119905
119876(119902119901minus119902)
119905
times
2(119902minus119905)+1
sum
119896=0
((minus1)119906+1
(2119904 minus 1) [2 (119902 minus 119905) + 1]
times (119906 + 1))
times ([2 (119902 minus 119905) + 1 minus 119896]119896)minus1
times
119864((119906+1)2)
sum
119908=0
1
2119908 (119906 + 1 minus 2119908)119908
sdot1198892(119901minus119904minus119908+1)
(1205882119898+ 1205882119899)119904+119906+1minus119908
times 1205882(119904minus119902+119908)minus1
1198981205882(119904minus119901+119902+119906minus119908)+1
119899
2
(40)
Finally the expression for mode coupling coefficients forthe arbitrary higher-order (119897 ge 2) modes 119871119875119897119898 and 119871119875119897119899 at thesplice with transverse offset 119889 leads to the following form
1120578(119889)119898119899
= 1198720
(119898 minus 1)
(119897 + 119898 minus 1)
(119899 minus 1)
(119897 + 119899 minus 1)
times 22119897[119872(119897)
1+119872(119897)
2+119872(119897)
3+119872(119897)
4]2
(41)
where
119872(119897)
1=
1
2
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902)
119876(119902119901minus119902)
119905
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
2=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902+120583+1)
sum
119905=max(0119904minus119901+119902)
119876(119902+120583+1119901minus119902)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
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Hindawi Publishing Corporation httpwwwhindawicom
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Advances in Optical Technologies 13
2
SM FCPC
3
10
MM FCPC
9
R2D2
FCPC
4
FCPC
8
5
SM FCPC
7
MM FCPC 6
MM pigtail
Ericsson FSU-975
13
Receiver
Tested MMFcoilspool
Fusion splice
Offset fusion splice
SM LD 1310HM SM pigtail
Figure 10 Experimental setup modified for pulse response measurements under control of launching conditions
0 10 20 30
0
005
01
015
02
025
03
035
04
minus30
minus005
minus20 minus10
Δ(
)
r (120583m)
Figure 11 Refractive index profile of the single mode fiber
Figure 12 Screenshot of the Ericsson FSU-975 display with offsetfusion splice
119872(119897)
3=
119864((119897minus2)2)
sum
120583=0
119875(119897)
120583
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+120583+1
sum
119904=0
min(119904119902)
sum
119905=max(0119904minus119901+119902minus120583minus1)
119876(119902119901minus119902+120583+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119872(119897)
4= 2
2119864((119897minus2)2)
sum
119894=0
min[119894119864((119897minus2)2)]sum
119895=max[0119864((119897minus2)2)]119875(119897)
119895119875(119897)
119894minus119895
times
119898+119899minus2
sum
119901=0
min(119901119898minus1)
sum
119902=max(0119901minus119899+1)
119887(119897119898minus1)
119902119887(119897119899minus1)
119901minus119902
times
119901+119894+2
sum
119904=0
min(119904119902+119895+1)
sum
119905=max(0119904minus119901+119902minus119894+119895minus1)
119876(119902+119895+1119901minus119902+119894minus119895+1)
119905
times
119897+2(119902minus119905)
sum
119896=0
Ω(119897)
119896
119864((119906+119897)2)
sum
119908=0
Ψ(119897)
119908
119875(119897)
120583=(minus1)120583+1
22120583+3
119897
120583 + 1
(119897 minus 120583 minus 2)
[119897 minus 2 (120583 + 1)]120583
Ω(119897)
119896=(minus1)119906+119897
(2119904 minus 1) [119897 + 2 (119902 minus 119905)] (119906 + 119897)
[119897 + 2 (119902 minus 119905) minus 119896]119896
Ψ(119897)
119908=
1
2119908 (119906 + 119897 minus 2119908)119908sdot
1198892(119897+119901minus119904minus119905minus119908)
(1205882119898+ 1205882119899)119897+119904+119906minus119908
times 1205882(119904minus119902+119908)minus119897
1198981205882(119904minus119901+119902+119906minus119908)+119897
119899
(42)
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
14 Advances in Optical Technologies
0 05 1 15 2 250
0203
01
040506070809
1
MeasuredSimulated
t (ns)
AA
max
OM2+OM3MMF N02 centr launch d = 0120583m
(a) Central launching
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 18120583m
(b) Offset 119889 = 18 120583m
MeasuredSimulated
0 05 1 15 2 25t (ns)
AA
max
OM2+OM3
0
0203
01
040506070809
1MMF N02 d = 235 120583m
(c) Offset 119889 = 23 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 centr launch d = 0120583m
(d) Central launching
MeasuredSimulated
OM2
0 05 1 15 2 25t (ns)
AA
max
0
0203
01
040506070809
1MMF N12 d = 50 120583m
(e) Offset 119889 = 5 120583m
MeasuredSimulated
OM2
05 1 15 2 250t (ns)A
Am
ax
0
0203
01
040506070809
1MMF N12 d = 235 120583m
(f) Offset 119889 = 23 5 120583m
Figure 13
10 Results and Discussion
Commercial GeO2-doped graded-index multimode fibersamples of different generations of categories OM2 andOM2+OM3 with length about 300ndash500m packed in coilsand conventional spools (Figure 2) were selected for exper-imental approbations of introduced model
The refractive index profiles of selected samples werepreliminary measured [126] by optical fiber analyzer EXFONR-9200 via refracted near field scan technique [127] Pro-files reconstructed from the protocol data are presentedin Figure 3 Also each fiber sample passed an OTDR testwith following fiber length measurement by backscatteringmethod The OTDR Hewlett Packard E6000A with singlemode plug-in unit 13101550 was used and test was producedat the wavelength 120582 = 1310 nm [128 129]
We applied a typical optical pulse response workstationR2D2 for DMD diagnostics [128 129] Simplified blockscheme of R2D2 is presented in Figure 4 The experimentalsetup and software R2D2 screenshot are presented in Figures5 and 6
The FP LD emits light with a wavelength 120582 = 1310 nmThe input pulse is shown in Figure 7 It has the full width
at a half maximum (FWHM) of about 340 ps The firstseries of measurements did not apply special launchingconditions Here only conventional FCPC adaptors wereapplied Therefore an angular misalignment was taking intoaccount 120579 = 42
∘ according to [8] during the modeling oftransmitter end of link
We simulated fiber irregularity by core diameter varia-tion It was set from the data of fiber diameter measurementsbeen produced during fiber drawing [130] (Figure 8) Micro-and macrobends were modeled by equivalent low 120579 = 2 3
∘
angularmisalignment at the boundaries of regular spans withlength Δ119911 = 479m [130] Measured and simulated pulseresponses of fibers OM2+OM3 and OM2 with refractiveindex profiles shown in Figure 3 are presented in Figures9(a) and 9(b) Here a close matching can be noticed Thusfor OM2+OM3 fiber the error of the FWHM is 116 andoutput pulse dispersion error is 742 For OM2 fiber theerrors of second peak amplitude and time scale position is163 and 043 and DMD error is 139
Another series of tests were produced under the control oflaunching conditions Simplified block-scheme of the exper-imental setup is shown in Figure 10 Here emission from theLD launched to testedmultimode fiber via singlemode pigtail
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Advances in Optical Technologies 15
(ITU-T Rec G652) which was spliced with multimode fiberwith precision offset Refractive index profile of the singlemode fiber is presented in Figure 11The offset was realized byEricsson FSU-975 programnumber 8 ldquoAttenuatorrdquo [131] Hereparameter ldquoattenuationrdquo was set to 0 ldquooffsetrdquomdashto particularvalue and ldquoECF factorrdquo (surface tension compensation fac-tor) also was set to 0 to avoid deformations in the heat zoneof the offset fusion splice (Figure 12) Some measured andsimulated pulse responses of fibers OM2+OM3 and OM2under particular offset launching conditions are presented inFigure 13 A close matching also can be noticed
11 Conclusion
An alternative model of multimode fiber links with con-ventional silica weakly-guiding graded-index irregular mul-timode fibers under a few-mode optical signal propagationgenerated by laser source is introduced The model is basedon the piecewise regular representation It takes into accountlaunch conditions differential mode delay both lower- andhigher-order mode chromatic dispersion differential modeattenuation andmodemixing and power diffusion occurringdue to real fiber irregularity and micro- and macrobendsResults of theoretical approbation are presented A goodagreement between developed model and measurements ofpulse response were obtained
References
[1] T Koonen ldquoTrends in optical access and in-building networksrdquoin Proceedings of the 34th European Conference on OpticalCommunication (ECOC rsquo08) pp 61ndash92 Brussels BelgiumSeptember 2008
[2] ldquoBSRIA update structured cabling market data centers IIMSand potential new studiesrdquo BSRIA Presentation MaterialsJanuary 2009
[3] Sh Bois ldquoNext generation fibers and standardsrdquo FOLS Presen-tation Materials October 2009
[4] S Bottacchi Multi-Gigabit Transmission over Multimode Opti-cal Fibre Theory and Design Methods for 10GbE Systems JohnWiley amp Sons 2006
[5] C Degen I Fischer and W Elsaszliger ldquoTransverse modes inoxide confined VCSELs influence of pump profile spatial holeburning and thermal effectsrdquo Optics Express vol 5 no 3 pp38ndash47 1999
[6] L-G Zei S Ebers J-R Kropp and K Petermann ldquoNoiseperformance ofmultimode VCSELsrdquo IEEE Journal of LightwaveTechnology vol 19 no 6 pp 884ndash892 2001
[7] A Gholami Z Toffano A Destrez M Pez and F QuentelldquoSpatiotemporal and thermal analysis of VCSEL for short-rangegigabit optical linksrdquo Optical and Quantum Electronics vol 38no 4ndash6 pp 479ndash493 2006
[8] L Raddatz I H White D G Cunningham and M C NowellldquoAn experimental and theoretical study of the offset launchtechnique for the enhancement of the bandwidth of multimodefiber linksrdquo IEEE Journal of Lightwave Technology vol 16 no 3pp 324ndash331 1998
[9] J S AbbotM JHackert D EHarshbargerDGCunninghamT Ch di Minico and I H White ldquoAnalysis of multimode fiberbehavior with laser sources in the development of the Gigabit
Ethernet fiber optic specificationsrdquo in Proceedings of the 47thInternationalWireampCable Symposium (IWCS rsquo98) pp 897ndash907Philadelphia Pa USA 1998
[10] M Webster L Raddatz I H White and D G CunninghamldquoStatistical analysis of conditioned launch for Gigabit Ethernetlinks using multimode fiberrdquo IEEE Journal of Lightwave Tech-nology vol 17 no 9 pp 1532ndash1541 1999
[11] P Pepeljugoski M J Hackert J S Abbott et al ldquoDevelopmentof system specification for laser-optimized 50-120583m multimodefiber for multigigabit short-wavelength LANsrdquo IEEE Journal ofLightwave Technology vol 21 no 5 pp 1256ndash1275 2003
[12] P Pepeljugoski S E Golowich A J Ritger P Kolesar and ARisteski ldquoModeling and simulation of next-generation multi-mode fiber linksrdquo IEEE Journal of Lightwave Technology vol 21no 5 pp 1242ndash1255 2003
[13] P K Pepeljugoski andDMKuchta ldquoDesign of optical commu-nications data linksrdquo IBM Journal of Research and Developmentvol 47 no 2-3 pp 223ndash237 2003
[14] AGholamiDMolin PMatthijsse GKuyt andPA Sillard ldquoAcomplete physical model for Gigabit Ethernet optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 289ndash294 Providence RIUSA 2008
[15] A Gholami D Molin and P Sillard ldquoPhysical modelingof 10 gbe optical communication systemsrdquo IEEE Journal ofLightwave Technology vol 29 no 1 Article ID 5645645 pp 115ndash123 2011
[16] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoSimulation-based prediction ofmultimodefiber bandwidth for10Gbs systemsrdquo in Proceedings of the 15th Annual Meeting ofthe IEEE Laser and Electro-Optics Society (LEOS rsquo02) vol 2 pp604ndash605 Glasgow Scotland November 2002
[17] J Morikuni P Mena B K Whitlock and R ScarmozzinoldquoLink-level design analysis and simulation of multimode datacommunication systemsrdquo in Proceedings of the 19th AnnualNational Fiber Optic Engineers Conference (NFOEC rsquo03) pp858ndash867 Orlando Fla USA 2003
[18] G Shaulov and B Whitlock ldquoMultimode fiber communicationsystem simulationrdquo IEEE 8023aq 10GBase-LRM Task ForcePresentation Materials July 2004
[19] F J Achten M Bingle and B P de Hon ldquoDMD measure-ment and simulation on short length 850-nm laser-optimized50mm core diameter GIMM optical fibresrdquo in Proceedings ofthe 12th Symposium on Optical FiberMeasurements pp 101ndash104Boulder Colo USA 2002
[20] A Sengupta ldquoSimulation of 10GbEmultimode optical commu-nication systemsrdquo in Proceedings of the 57th International Wireamp Cable Symposium (IWCS rsquo08) pp 320ndash326 Providence RIUSA 2008
[21] N Sharma A S Chitambar H Ch Bianchi and K SivaprasadldquoModel for differential mode delay distribution in multimodefibersrdquo Progress in Electromagnetic Research vol 1 no 4 pp469ndash472 2005
[22] A Listvin V Listvin and D Shvyrkov Fibers for Telecommuni-cation Lines LESARart Moscow Russia 2003
[23] B Katsenelenbaum Theory of Irregular Waveguides with SlowVarying Parameters Mir Moscow Russia 1961
[24] A Baribin Electrodynamics of Waveguide Structures Theoryof Perturbation and Coupled Mode Theory Fizmalit MoscowRussia 2007
[25] A Snyder and J Love Optical Waveguide Theory Chapman ampHall London UK 1983
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
16 Advances in Optical Technologies
[26] H G Unger Planar Optical Waveguides and Fibres ClarendonPress Oxford UK 1977
[27] D Marcuse Light Transmission Optics Van Nostrand ReinholdCompany London UK 1972
[28] S Geckeler ldquoPulse broadening in optical fibers with modemixingrdquo Applied Optics vol 18 no 13 pp 2192ndash2198 1979
[29] T A Eftimov and W J Bock ldquoPolarization analysis of LP01and LP11 intermodal interference in highly birefringent bow-tieoptical fibersrdquo IEEE Journal of Lightwave Technology vol 11 no12 pp 1925ndash1936 1993
[30] I O Zolotovski and D I Sementsev ldquoDispersion influence toGaussian pulse transformation in periodically irregular fiberrdquoOptics and Spectroscopy vol 85 no 2 pp 304ndash308 1998
[31] F Tian Y-Z Wu and P-D Ye ldquoAnalysis of polarization fluc-tuation in single-mode optical fibers with continuous randomcouplingrdquo IEEE Journal of Lightwave Technology vol 5 no 9pp 1165ndash1168 1987
[32] L Jeunhomme and J P Pocholle ldquoMode coupling in a multi-mode optical fiber withmicrobendsrdquoApplied Optics vol 14 no10 pp 2400ndash2405 1975
[33] M Miyagi S Kawakami M Ohashi and S Nishida ldquoMea-surement of mode conversion coefficients an mode dependentlosses in a multimode fibersrdquo Applied Optics vol 17 no 20 pp3238ndash3244 1978
[34] D Swierk and A Heimrath ldquoCoupled power equations in mul-timode waveguides several random correlated imperfectionsrdquoJournal of Modern Optics vol 43 no 11 pp 2301ndash2310 1996
[35] R Vacek ldquoElectromagnetic wave propagation in general mul-timode waveguide structures exhibiting simultaneously at-tenuation dispersion and coupling phenomenardquo Journal ofMicrowaves and Optoelectronics vol 1 no 3 pp 42ndash52 1998
[36] E Perrey-Debain I D Abrahams and J D Love ldquoA continuousmodel for mode mixing in graded-index multimode fibres withrandom imperfectionsrdquo Proceedings of the Royal Society A vol464 no 2092 pp 987ndash1007 2008
[37] R Olshansky ldquoMode coupling effects in graded-index opticalfibersrdquo Applied Optics vol 14 no 4 pp 935ndash945 1975
[38] G Yabre ldquoComprehensive theory of dispersion in graded-indexoptical fibersrdquo IEEE Journal of Lightwave Technology vol 18 no2 pp 166ndash177 2000
[39] G Yabre ldquoInfluence of core diameter on the 3-dB bandwidthof graded-index optical fibersrdquo IEEE Journal of LightwaveTechnology vol 18 no 5 pp 668ndash676 2000
[40] J N Kutz J A Cox and D Smith ldquoMode mixing andpower diffusion in multimode optical fibersrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1195ndash1202 1998
[41] M Skorobogatiy M Ibanescu S G Johnson et al ldquoAnalysisof general geometric scaling perturbations in a transmittingwaveguide fundamental connection between polarization-mode dispersion and group-velocity dispersionrdquo Journal of theOptical Society of America B vol 19 no 12 pp 2867ndash2875 2002
[42] M Skorobogatiy S A Jacobs S G Johnson and Y Fink ldquoGeo-metric variations in high index-contrast waveguides coupledmode theory in curvilinear coordinatesrdquoOptics Express vol 10no 21 pp 1227ndash1243 2002
[43] M Skorobogatiy S G Johnson S A Jacobs and Y FinkldquoDielectric profile variations in high-index-contrast waveg-uides coupled mode theory and perturbation expansionsrdquoPhysical Review E vol 67 no 4 Article ID 046613 11 pages2003
[44] M Skorobogatiy K Saitoh and M Koshiba ldquoFull-vectorialcoupled mode theory for the evaluation of macro-bending lossin multimode fibers application to the hollow-core photonicbandgap fibersrdquo Optics Express vol 16 no 19 pp 14945ndash149532008
[45] D Gloge ldquoBending loss in multimode fibers with graded andungraded core indexrdquo Applied Optics vol 11 no 11 pp 2506ndash2513 1972
[46] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibre waveguide IrdquoOptical and QuantumElectronics vol 8 no 6 pp 503ndash508 1976
[47] M Eve and J H Hannay ldquoRay theory and random modecoupling in an optical fibrewaveguide IIrdquoOptical andQuantumElectronics vol 8 no 6 pp 509ndash512 1976
[48] B Crosignani and P di Porto ldquoSoliton propagation in multi-mode optical fibersrdquo Optics Letters vol 6 no 7 pp 329ndash3301981
[49] B Crosignani A Cutolo and P di Porto ldquoCoupled-modetheory of nonlinear propagation in multimode and single-mode fibers envelope solitons and self-confinementrdquo Journalof Optical Society of America vol 72 no 9 pp 1136ndash1141 1982
[50] S Solimeno B Crosignani and P di PortoGuiding Diffractionand Confinement of Optical Radiation Academic Press SanDiego Calif USA 1986
[51] Y S Kivshar and G P Agrawal Optical Solitons From Fibers toPhotonic Crystals Academic Press SanDiego Calif USA 2003
[52] C RMenyuk ldquoStability of solitons in birefringent optical fibersI Equal propagation amplitudesrdquo Optics Letters vol 12 no 8pp 614ndash616 1987
[53] C R Menyuk ldquoPulse propagation in an elliptically birefringentKerrmediumrdquo IEEE Journal of QuantumElectronics vol 25 no12 pp 2674ndash2682 1989
[54] F Poletti and P Horak ldquoDescription of ultrashort pulse propa-gation inmultimode optical fibersrdquo Journal of theOptical Societyof America B vol 25 no 10 pp 1645ndash1654 2008
[55] K Okamoto Fundamentals of Optical Waveguides AcademicPress San Diego Calif USA 2000
[56] S T Chu and S K Chaudhuri ldquoFinite-difference time-domainmethod for optical waveguide analysisrdquo Progress in Electromag-netic Research vol 11 pp 255ndash300 1995
[57] K Sawaya ldquoNumerical techniques for analysis of electromag-netic problemsrdquo IEICE Transactions on Communications vol83 no 3 pp 444ndash452 2000
[58] G Bao L Cowsar and W Masters Mathematical Modeling inOptical Science SIAM Philadelphia Pa USA 2001
[59] A B Shvartsburg Time Domain Optics of Ultrashort Wave-forms Clarendon Press Oxford UK 1996
[60] A B Shvartsburg Impulse Time-Domain Electromagnetics ofContinues Media Birkhaauser Boston Mass USA 1999
[61] D M Sullivan Electromagnetic Simulation Using the FDTDMethod IEEE Press New York NY USA 2000
[62] V A Neganov O V Osipov S B Rayevskiy and G PYarovoy Electrodynamics and Wave Propagation Radiotech-nika Moscow Russia 2007
[63] R Mitra Computational Methods for Electrodynamics MirMoscow Russia 1977
[64] GWWornell Signal ProcessingWith Fractals AWavelet-BasedApproach Prentice-Hall Upper Saddle River NJ USA 1996
[65] T K Sarkar C Su R Adve M Salazar-Palma L Garcia-Castillo and R R Boix ldquoA tutorial on wavelets from an electri-cal engineering perspective part 1 discrete wavelet techniquesrdquo
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Optical Technologies 17
IEEEAntennas and PropagationMagazine vol 40 no 5 pp 49ndash68 1998
[66] T K Sarkar and C Su ldquoA tutorial on wavelets from an electricalengineering perspective part 2 the continuous caserdquo IEEEAntennas and Propagation Magazine vol 40 no 6 pp 36ndash481998
[67] A Vukovic P Sewell T M Benson and P C Kendall ldquoSpectralmethod applied to design of spotsize convertersrdquo ElectronicsLetters vol 33 no 25 pp 2121ndash2123 1997
[68] A Vukovic P Sewell T M Benson and P C KendallldquoNovel half space radiationmodemethod for buried waveguideanalysisrdquoOptical and Quantum Electronics vol 31 no 1 pp 43ndash51 1999
[69] M Reed TM Benson P Sewell P C Kendall GM Berry andS V Dewar ldquoFree space radiation mode analysis of rectangulardielectric waveguidesrdquo Optical and Quantum Electronics vol28 no 9 pp 1175ndash1179 1996
[70] A Niiyama and M Koshiba ldquo3-dimensional eam propagationanalysis of nonlinear optical fibersrdquo IEICE Transactions onCommunications vol E80-B no 4 pp 522ndash527 1997
[71] D Schulz C Glingener M Bludszuweit and E Voges ldquoMixedfinite element beam propagation methodrdquo IEEE Journal ofLightwave Technology vol 16 no 7 pp 1336ndash1341 1998
[72] R Magnanini and F Santosa ldquoWave propagation in a 2-Doptical waveguiderdquo SIAM Journal on Applied Mathematics vol61 no 4 pp 1237ndash1252 2000
[73] E Silvestre M V Andres and P Andres ldquoBiorthonormal-basismethod for the vector description of optical-fiber modesrdquo IEEEJournal of Lightwave Technology vol 16 no 5 pp 923ndash928 1998
[74] P Sarrafi and L Qian ldquoModeling of pulse propagation inlayered structures with resonant nonlinearities using a gener-alized time-domain transfer matrix methodrdquo IEEE Journal ofQuantum Electronics vol 48 no 5 pp 559ndash567 2012
[75] M Hammer ldquoHybrid analyticalnumerical coupled-modemodeling of guided-wave devicesrdquo IEEE Journal of LightwaveTechnology vol 25 no 9 pp 2287ndash2298 2007
[76] G-W Chern J-F Chang and L A Wang ldquoModeling ofnonlinear pulse propagation in periodic and quasi-periodicbinary long-period fiber gratingsrdquo Journal of the Optical Societyof America B vol 19 no 7 pp 1497ndash1508 2002
[77] G B Malykin V I Pozdnyakova and I A ShereshevskiildquoRandom groups in the optical waveguides theoryrdquo Journal ofNonlinear Mathematical Physics vol 8 no 4 pp 491ndash517 2001
[78] N A Galchenko ldquoMatrix theory of excitation of electro-magnetic waves in irregular waveguidesrdquo Radiophysics andQuantum Electronics vol 40 no 6 pp 744ndash751 1997
[79] O O Silitchev ldquoMatrix method for computing of coherent laserpulse propagationrdquo Quantum Electronics vol 20 no 10 pp983ndash990 1993
[80] T A Martynova ldquoOn the computing of irregular waveguideswith local layered inhomogeneityrdquo Radiophysics and QuantumElectronics vol 18 no 5 pp 1178ndash1188 1975
[81] T A Ramadan ldquoA recurrence technique for computing theeffective indexes of the guided modes of coupled single-modewaveguidesrdquoProgress in Electromagnetic Research vol 4 pp 33ndash46 2008
[82] V V Merkoulov and I S Sineva ldquoOn the repeated reflectionsin the non-homogeneous transmission linerdquo Journal of RadioElectronics no 5 paper 8 2000
[83] G A Gesell and I R Ciric ldquoRecurrence modal analysisfor multiple waveguide discontinuities and its application to
circular structuresrdquo IEEE Transactions on Microwave Theoryand Techniques vol 41 no 3 pp 484ndash490 1993
[84] S G Akopov V N Korshunov B S Solovrsquoev and B NFomichev ldquoEnergetic characteristics of distribution functionsof irregularities of geometric structure of optic fiberrdquo Elek-trosvyaz no 9 pp 9ndash11 1992
[85] S G Akopov V N Korshunov B S Solovyov and V NFomichev ldquoEstimation of characteristics of random spatial pro-cess of irregularity geometry structure distribution in opticalfiberrdquo Electrosvyaz no 10 pp 20ndash22 1992
[86] Y Y Lu ldquoSome techniques for computing wave propagation inoptical waveguidesrdquoCommunications in Computational Physicsvol 1 no 6 pp 1056ndash1075 2006
[87] C-L Bai H Zhao and W-T Wang ldquoExact solutions toextended nonlinear Schrodinger equation in monomode opti-cal fiberrdquo Communications in Theoretical Physics vol 45 no 1pp 131ndash134 2006
[88] B Li and Y Chen ldquoOn exact solutions of the nonlinearSchrodinger equations in optical fiberrdquo Chaos Solitons ampFractals vol 21 no 1 pp 241ndash247 2004
[89] V Radisic D R Hjelme A Horrigan Z B Popovic and AR Mickelson ldquoExperimentally verifiable modeling of coplanarwaveguide discontinuitiesrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 41 no 9 pp 1524ndash1533 1993
[90] U Osterberg ldquoSignal processing in optical fibersrdquo ModernSignal Processing vol 46 pp 301ndash316 2003
[91] P-O Persson and O Runborg ldquoSimulation of a waveguidefilter using wavelet-based numerical homogenizationrdquo Journalof Computational Physics vol 166 no 2 pp 361ndash382 2001
[92] M Fujii andW J R Hoefer ldquoAwavelet formulation of the finite-difference method full-vector analysis of optical waveguidejunctionsrdquo IEEE Journal of Quantum Electronics vol 37 no 8pp 1015ndash1029 2001
[93] M Fujii and W J R Hoefer ldquoTime-domain wavelet Galerkinmodeling of two-dimensional electrically large dielectricwaveguidesrdquo IEEE Transactions on Microwave Theory andTechniques vol 49 no 5 pp 886ndash892 2001
[94] D Arakaki W Yu and R Mittra ldquoOn the solution of aclass of large body problems with partial circular symmetry(multiple asymmetries) by using a hybrid-dimensional Finite-Difference Time-Domain (FDTD) methodrdquo IEEE Transactionson Antennas and Propagation vol 49 no 3 pp 354ndash360 2001
[95] K A Amaratunga ldquoA wavelet-based approach for compressingkernel data in large-scale simulations of 3D integral problemsrdquoComputing in Science and Engineering no 7-8 pp 34ndash35 2000
[96] M B Shemirani W Mao R A Panicker and J M KahnldquoPrincipal modes in graded-index multimode fiber in presenceof spatial- and polarization-mode couplingrdquo IEEE Journal ofLightwave Technology vol 27 no 10 pp 1248ndash1261 2009
[97] M B Shemirani and J M Kahn ldquoHigher-order modal dis-persion in graded-index multimode fiberrdquo IEEE Journal ofLightwave Technology vol 27 no 23 pp 5461ndash5468 2009
[98] S Fan and J M Kahn ldquoPrincipal modes in multimode waveg-uidesrdquo Optics Letters vol 30 no 2 pp 135ndash137 2005
[99] G P Agrawal Nonlinear Fiber Optics Academic Press SanDiego Calif USA 1989
[100] M J AdamsAn Introduction to OpticalWaveguides JohnWileyamp Sons New York NY USA 1981
[101] J J R Hansen and E Nicolaisen ldquoPropagation in graded-indexfibers comparison between experiment and three theoriesrdquoApplied Optics vol 17 no 17 pp 2831ndash2835 1978
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 Advances in Optical Technologies
[102] A V Bourdine ldquoMethod for chromatic dispersion estimationof high-order guided modes in graded index single-claddingfibersrdquo inOptical Technologies for Telecommunications vol 6605of Proceedings of SPIE pp 660509-1ndash660509-13 November2006
[103] A V Bourdine and O R Delmukhametov ldquoCalculation oftransmission parameters of the launched higher-order modesbased on the combination of a modified Gaussian approxi-mation and a finite element methodrdquo Telecommunications andRadio Engineering vol 72 no 2 pp 111ndash123 2013
[104] H Meher and S I Hosain ldquoVariational approximations forsingle-mode graded-index fibers some interesting applica-tionsrdquo Journal of Optical Communications vol 24 no 1 pp 25ndash30 2003
[105] M-S Wu M-H Lee and W-H Tsai ldquoVariational analysis ofsingle-mode graded-core W-fibersrdquo IEEE Journal of LightwaveTechnology vol 14 no 1 pp 121ndash125 1996
[106] M J Holmes D M Spirit and F P Payne ldquoNew Gaussian-based approximation for modelling non-linear effects in opticalfibersrdquo IEEE Journal of Lightwave Technology vol 12 no 2 pp193ndash201 1994
[107] A Ankiewicz and G-D Peng ldquoGeneralized Gaussian approx-imation for single-mode fibersrdquo IEEE Journal of LightwaveTechnology vol 10 no 1 pp 22ndash27 1992
[108] M I Oksanen and I V Lindell ldquoVariational analysis ofanisotropic graded-index optical fibersrdquo IEEE Journal of Light-wave Technology vol 7 no 1 pp 87ndash91 1989
[109] R Tewari S I Hosain and K Thyagarajan ldquoScalar variationalanalysis of single mode fibers with Gaussian and smoothed-out profilesrdquoOptics Communications vol 48 no 3 pp 176ndash1801983
[110] A Sharma S I Hosain and A K Ghatak ldquoThe fundamentalmode of graded-index fibres simple and accurate variationalmethodsrdquo Optical and Quantum Electronics vol 14 no 1 pp7ndash15 1982
[111] I Gradstein and I Ryjik Tables of Integrals GIFML MoscowRussia 1963
[112] M Abramovitz and I SteganHandbook of Mathematical Func-tions with Formulas Graphs and Mathematical Tables NaukaMoscow Russia 1979
[113] L N Binh ldquoDesign guidelines for ultra-broadmand dispersion-flattened optical fibers with segmented-core index profilerdquoTech Rep MECSE-14-2003 Monash University Clayton Aus-tralia 2003
[114] J W Fleming ldquoDispersion in GeO2-SiO2 glassesrdquo AppliedOptics vol 23 no 24 pp 4886ndash4493 1984
[115] V A Burdin ldquoMethods for computation of Sellmeier coeffi-cients for dispersion analysis of silica optical fibersrdquo Infocom-munication Technologies vol 4 no 2 pp 30ndash34 2006
[116] M Rousseau and L Jeunhomme ldquoOptimum index profile inmultimode optical fiber with respect to mode couplingrdquo OpticsCommunications vol 23 no 2 pp 275ndash278 1977
[117] J P Meunier Z H Wang and S I Hosain ldquoEvaluation ofsplice loss between two nonidentical single-mode graded-indexfibersrdquo IEEE Photonics Technology Letters vol 6 no 8 pp 998ndash1000 1994
[118] J-P Meunier and S I Hosain ldquoAccurate splice loss analysis forsingle-mode graded-index fibers withmismatched parametersrdquoIEEE Journal of Lightwave Technology vol 10 no 11 pp 1521ndash1526 1992
[119] Q Yu P-H Zongo and P Facq ldquoRefractive-index profileinfluences on mode coupling effects at optical fiber splices andconnectorsrdquo IEEE Journal of Lightwave Technology vol 11 no8 pp 1270ndash1273 1993
[120] S P Gurji and V B Katok ldquoLoss calculation at the splices ofsingle mode fibers with complicated refractive index profilerdquoElektrosvyaz no 10 pp 25ndash27 1990
[121] V A Srapionov ldquoMode coupling at the splices of optical fiberswith mismatched parametersrdquo Elektrosvyaz no 10 pp 10ndash121985
[122] S I Hosain J-PMeunier and ZHWang ldquoCoupling efficiencyof butt-joined planar waveguides with simultaneous tilt andtransverse offsetrdquo IEEE Journal of Lightwave Technology vol 14no 5 pp 901ndash907 1996
[123] I A Avrutski V A Sychugov and A V Tishenko ldquoThe studyof excitation radiation and reflection processes in corrugatedwaveguidesrdquo Proceedings of IOFAN no 34 pp 3ndash99 1991
[124] R Chandra K Thyagarajan and A K Ghatak ldquoMode exci-tation by tilted and offset Gaussian beams in W-type fibersrdquoApplied Optics vol 17 no 17 pp 2842ndash2847 1978
[125] A V Bourdine ldquoMode coupling at the splice of diverse opticalfibersrdquo inOptical Technologies for Telecommunications vol 8787of Proceedings of SPIE pp 878706-1ndash878706-12 2012
[126] A V Bourdine and K A Yablochkin ldquoInvestigations of refrac-tive index profile defects of silica graded-index multimodefibers of telecommunication cablesrdquo Infocommunication Tech-nologies vol 8 no 2 pp 22ndash27 2010
[127] ldquoOptical Fiber Analyzer EXFO NR-9200NR-9200HRrdquo EXFO2006
[128] A V Bourdine ldquoDifferential mode delay of different genera-tions of silicamultimode optical fibersrdquoPhoton Express vol 5-6no 69-70 pp 20ndash22 2008
[129] A V Bourdine ldquoOnmultimode optical fiberDMDdiagnosticsrdquoInfocommunication Technologies vol 6 no 4 pp 33ndash38 2008
[130] K J Lyytikainen Control of complex structural geometry inoptical fibre drawing [PhD thesis] School of Physics andOptical Fibre Technology Centre University of Sydney 2004AThesis Submitted for the Degree of Doctor of Philosophy
[131] ldquoEricsson FSU-975 Instruction Manualrdquo Ericsson 2001
International Journal of
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RoboticsJournal of
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Active and Passive Electronic Components
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International Journal of
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Hindawi Publishing Corporation httpwwwhindawicom
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
