Research Article Description of Dispersive Wave Emission ...downloads.hindawi.com/journals/amp/2014/180656.pdfuide structures such as single mode bers [ ], photonic ... fourth-order

Research ArticleDescription of Dispersive Wave Emission andSupercontinuum Generation in Silicon Waveguides UsingSplit-Step Fourier and Runge-Kutta Integration Methods

Xuefeng Li

School of Science Xirsquoan University of Post amp Telecommunications Xirsquoan 710121 China

Correspondence should be addressed to Xuefeng Li lixfpost163com

Received 6 January 2014 Accepted 2 March 2014 Published 27 March 2014

Academic Editor Boris G Konopelchenko

Copyright copy 2014 Xuefeng Li This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on solving numerically the generalized nonlinear Schrodinger equation describing the propagation of high orderfemtosecond soliton in siliconwaveguide under certain parametric conditions by the split-step Fourier andRunge-Kutta integrationmethods dispersive wave emission and supercontinuum generation in silicon waveguides are numerically investigated bypropagating femtosecond solitonsThe numerical results show that the efficient dispersive wave emission can be generated in siliconwaveguide which plays an important role in the process of the supercontinuum generation with the form of Cherenkov radiationand it is also shown that the high order low-energy solitons and short waveguides are efficient for the dispersive wave emission

1 Introduction

Supercontinuum generation has important potential appli-cations in the fields such as optical communication spec-troscopy and optical coherence tomography [1] Manyresearches of supercontinuum generation in different waveg-uide structures such as single mode fibers [2] photoniccrystal fibers (PCFs) [3ndash5] and silica nanowires [6] have beenreported These results indicate that it is possible to achievesupercontinuum generation at relative low optical power andover short propagation distances provided that the mediumexhibits high nonlinear response and tailoring dispersionproperties However it is difficult to implement on-chipintegration applications because of the large propagationlength required by the large spectral broadening

A promising solution to generating supercontinuum isprovided by the silicon waveguide which has the advantageof employing an emerging silicon integrated photonics plat-form Compared with the conventional optical waveguidesthe silicon waveguide has several unique properties thatcan be employed to achieve on-chip scale supercontinuumgeneration The silicon waveguide has a smaller transversedimension governing the dispersion properties so that the

dispersion properties can be tailored flexibly by designing thetransverse dimension properly [7 8] At the same time theoptical intensity used is a moderate input power due to highoptical confinement in waveguide caused by the high indexcontrast therefore the nonlinear optical effects are enhancedintensely [9ndash11] Moreover the nonlinear coefficient of thesilicon waveguide is several orders of magnitude higher thanthat of the PCFsThese propertiesmake the siliconwaveguidesignificant and suitable for the supercontinuum generation[12 13]

In this paper through solving numerically the generalizednonlinear Schrodinger equation describing propagation offemtosecond pulse in silicon waveguide under certain para-metric conditions by the combination of split-step Fourierand Runge-Kutta integration methods the dispersive waveemission and supercontinuum generation in silicon waveg-uide are investigated by propagating femtosecond solitons

2 Theory Model

The theory model that describes the femtosecond pulsepropagation along the silicon waveguide including the effects

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 180656 7 pageshttpdxdoiorg1011552014180656

2 Advances in Mathematical Physics

of nonlinear loss such as the two-photon absorption (TPA)and free-carrier absorption (FCA) is based on the generalizednonlinear Schrodinger equation as follows [14]

120597119860

120597119911=

infin

sum

119898=2

119894119898+1

120573119898

119898

120597119898

119860

120597119905119898minus1

2(120572119897+ 120572119891)119860 + 119894120574

times (1 +119894

1205960

120597

120597119905)119860 (119911 119905) times int

119905

minusinfin

119877 (119905 minus 120591) |119860 (119911 120591)|2

119889120591

(1)

where 119860 = 119860(119911 119905) is the electric field envelope as a functionof distance along the waveguide 119911 and retarded time 119905 120572

119897

and 120572119891account for the linear losses and the free-carrier

absorption (FCA) of the waveguide respectively 120573119898s are

dispersion coefficients at center frequency 1205960 120574 denotes the

nonlinear coefficient of thewaveguidewhich is defined as 120574 =11989921198960119886eff + 119894120573TPA119886eff 1198992 is the Kerr coefficient the parameter

1198960is the wave vector 119886eff is the effective area of the silicon

waveguide and 120573TPA is the two-photon absorption (TPA)parameter The influence of the FCA is from the nonlinearloss 120572

119891= 120590119873

119888(119911 119905) where 119873

119888(119911 119905) is the density of free

carriers The density of free-carrier 119873119888(119911 119905) can be obtained

by solving the following equation [7]

120597119873119888(119911 119905)

120597119905=120573TPA2ℎ]0

|119860 (119911 119905)|4

1198862effminus119873119888(119911 119905)

120591 (2)

where 120591 is the effective carrier life time which is estimatedto be about 3 ns [12] ℎ is Plank constant and ]

0is the

central frequency All of the parameters above have beenconsidered in our theory model and the initial injectedsoliton is assumed to have a hyperbolic secant field profilewhich is shown as

119860 (0 119879) = radic1198750sec ℎ( 119879

1198790

) (3)

where 1198750is the peak power of the input soliton The order of

input soliton119873 satisfies the condition of

1198732

=Re (120574) 119875

01198792

0

1205732

(4)

The right-hand side of (1) represents the nonlinear responseof the silicon waveguide with the response function 119877(119905)which can be written as

119877 (119905) = (1 minus 119891119877) 120575 (119905) + 119891

119877sdot ℎ119877(119905) (5)

where the value of 119891119877= 0043 which is smaller than that of

silica fibers The function of ℎ119877can be written as [15]

ℎ119877=1205912

1+ 1205912

2

1205912112059122

exp(minus 1199051205912

) sdot sin( 119905

1205911

) (6)

where 1205911= 10 fs and 120591

2= 303 ps

In order to solve (1) numerically the equation can bedivided to two parts the linear term and nonlinear termThelinear term includes the effects of linear loss nonlinear lossand high order dispersions here we consider the dispersioncoefficient to the sixth orderWe can use the split-step Fouriermethod [11 16ndash18] to solve the linear term of (1) while thenonlinear term can be described as follows

120597119860

120597119911= 119894120574 (1 +

119894

1205960

120597

120597119905) (119860119877 lowast |119860|

2

) (7)

where

119877 lowast |119860|2

= int

119905

minusinfin

119877 (1199051015840

)10038161003816100381610038161003816119860 (119911 119905 minus 119905

1015840

)10038161003816100381610038161003816

2

119889119905 (8)

The time derivative term in (7) can be treated as aperturbation and introduce a new function 119861(119911 119905) as

119861 (119911 119905) = 119860 (119911 119905) sdot exp lfloor119894120574119877 lowast 1003816100381610038161003816119860010038161003816100381610038162

sdot (119911 minus 1199110)rfloor (9)

where 1199110represents the initial position of the propagating

femtosecond soliton considering 1199110= 0 in this simulation

From (7) and (9) it can be found that the new function119861(119911 119905)satisfies the following equation

120597119861

120597119911= 119894120574119861 sdot 119877 lowast (

1003816100381610038161003816119861010038161003816100381610038162

+ |119861|2

) minus120574

1205960

sdot120597

120597119905(119861 sdot 119877 lowast |119861|

2

)

(10)

It is possible to obtain 119861(119911 119905) through solving (9) using thefourth-order Runge-Kutta integration method [19ndash21] Thedetails of calculations using the fourth-order Runge-Kuttamethod satisfy the following relationship

119861119896+1

= 119861119896+ℎ0(1198911+ 21198912+ 21198913+ 1198914)

6 (11)

where ℎ0is the step length of the calculation process 119891

1 1198912

1198913 and 119891

4can be written as follows

1198911= 119891 (119911

119896 119861119896) (12)

1198912= 119891(119911

119896+ℎ0

2 119861119896+ℎ0

21198911) (13)

1198913= 119891(119911

119896+ℎ0

2 119861119896+ℎ0

21198912) (14)

1198914= 119891 (119911

119896+ ℎ0 119861119896+ ℎ01198913) (15)

where 119911119896represents the position of the propagation distance

Then we can obtain 119860(119911 119905) through the relationshipbetween 119861(119911 119905) and 119860(119911 119905) in (8) Consequently using thesplit-step Fourier method (SSFM) and the fourth-orderRunge-Kutta integrationmethod (1) can be solved accurately

3 Results and Discussion

In this simulation the silicon waveguide used is the straightwaveguide The transverse section of the silicon waveguide is

Advances in Mathematical Physics 3

Air

W

H Silicon

SiO2

Figure 1 Geometry properties of the silicon waveguide The lowercladdingmaterial is SiO

2and the upper claddingmaterial is air119882 =

07 120583m denotes the width of the waveguide and 119867 = 05 120583m is theheight of the waveguide

1200 1300 1400 1500 1600 1700 1800 1900 2000 2100minus3minus2minus1

0123

Wavelength (nm)

25

3

35

4

Refr

activ

e ind

ex

Seco

nd o

rder

disp

ersio

n (p

s2m

)

nsi1205732TE

1205732TM

neffTE

neffTE

Figure 2 Wavelength dependence of 119899si 119899effTE 119899effTM 1205732TE and1205732TM for the fundamental TE mode and fundamental TM mode

using the waveguide shown in Figure 1 with119882 = 07 120583m and 119867 =

05 120583m

shown in Figure 1Thedispersion properties of thewaveguideplay an important role in the process of supercontinuumgeneration which can be tailored by choosing the suitableparameters of thewaveguideThe zero-dispersionwavelength(ZDWL) must be designed to be around the central wave-length of the pump (1550 nm)The ZDWL of such waveguidecan be tailored to fall in this regimewith a suitable designTheZDWL of the transverse electric (TE) mode and transversemagnetic (TM) mode can be tailored to below the pumpwavelength for straight waveguide as shown in Figure 1 whenboth the width and height of the waveguide are close to08 120583m The width and height of waveguide are assumed as119882 = 07 120583m and119867 = 05 120583m respectively

The wavelength dependence of effective mode index ofthe TE mode (119899effTE) and TM mode (119899effTM) is calculatedusing the full-vector finite difference mode solver [22] basedon the refractive index of silicon waveguide 119899si The seconddispersion coefficient for TE mode (120573

2TE) and TM mode(1205732TM) is also calculated as shown in Figure 2 It can be found

that 119899effTE changes from 325 to 295 as the wavelength variesfrom 1200 nm to 2000 nmwhile 119899effTM changes from 32 to 26in the same wavelength range The ZDWLs for fundamentalTE mode and TM mode of the silicon waveguide are around1300 nm and 1500 nm respectively In this simulation thefundamental TE mode can be chosen because the impact ofRaman scattering must be considered in the theory modelThe higher order dispersion (to the sixth order) parametersof the waveguide for fundamental TE mode at 1550 nm canalso be calculated from the same method and the dispersion

parameters at 1550 nm are obtained as 1205732= minus018 ps2m

1205733= 40 times 10

minus3 ps3m 1205734= minus075 times 10

minus5 ps4m 1205735=

01 times 10minus6 ps5m and 120573

6= 012 times 10

minus7 ps6mIn order to simulate dispersive wave emission and super-

continuum generation in silicon waveguide we input afemtosecond pulse exciting the fundamental TE mode andpropagating in the form of high order soliton The linear losscoefficient of the waveguide is 120572

119897= 022 dBcm The TPA

coefficient is 120573TPA = 5 times 10minus12mW the effective area is

119886eff = 03 120583m2 and the Kerr coefficient is 119899

2= 6times10

minus18m2Wand 120590 = 145 times 10

minus21m2 for silicon The input pulse is thefemtosecond soliton with the order 119873 = 3 the pulse peakpower is 119875

0= 25W and the pulse width is 119879

0= 284 fs

Correspondingly the full width of half maximum (FWHM)can be written as

119879119901= 2 ln (1 + radic2)119879

0= 1763119879

0= 50 fs (16)

Compared with the silica waveguide the Raman scat-tering plays a minor role in the process of supercontinuumgeneration in the silicon waveguide Moreover the propertiesof nonlinear loss such as TPA and FCA have impact on thesupercontinuum generation in the silicon waveguide whichhave been described in [13]

In our simulations the third order soliton propagatesin a 1 cm long silicon waveguide with the peak power of25W Figure 3 shows the temporal and spectral profiles atthe waveguide output The features seen can be describedin terms of soliton fission and dispersive wave emission intime domain while in terms of self-phase modulation andCherenkov radiation in spectral domain respectively It isobserved from Figure 3(a) that the single third order solitonevolves into three individual fundamental solitonsMoreoverthe dispersive wave emission produces a long tail in timedomain as shown in Figure 3(a) Correspondingly there aresome spectral peak structures in spectral domain as shownin Figure 3(b) The spectral peak located at the wavelength13 120583m is due to the Cherenkov radiation under the conditionof phase matching while other spectral peak features are theresult of the soliton fission

In order to accurately explain the supercontinuum gen-eration in silicon waveguide the temporal and spectralevolution along the waveguide is given in Figure 4 with thesame parameters as shown in Figure 3 We first note that theinitial stage of propagation is dominated by approximatelysymmetrical spectral broadening most of which occurs inthe first 15mm In this process the effect of self-phasemodulation plays a dominated role Meanwhile strong tem-poral compression also occurs over this range However withthe propagation distance increasing the spectral broadeningbecomes greatly asymmetric with the development of distinctspectral peaks on both the short- and long-wavelength sidesof the input pump which is clearly shown in Figure 4(b)As shown in Figure 4(a) when the propagation distanceis over 2mm the soliton fission effect begins to occurThe full width at half maximum (FWHM) of the spectrumat the distance of 2mm is over 100 nm The dispersivewave emission generates at the distance of 5mm which isassociated with the development of stable distinct spectral


minus02 minus01 0 01 02 03 04 05 06 07 080

005

01

Time (ps)

Nor

mal

ized

inte

nsity

Input pulseOutput pulse

(a)

1200 1300 1400 1500 1600 1700 1800 1900minus40minus35minus30minus25minus20minus15minus10

minus50

Nor

mal

ized

inte

nsity

(dB)

Input spectrumOutput spectrum

120582 (nm)

(b)

Figure 3 (a) Input and output temporal properties obtained from propagating the 50 fs (FWHM) 25W peak power input hyperbolic secantsoliton (b) Input and output spectral properties obtained from propagating the 50 fs (FWHM) 25W peak power input hyperbolic secantsoliton The soliton order is119873 = 3 and waveguide length is 1 cm

minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1D

istan

ce (c

m)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)

Figure 4 Results from numerical simulations showing (a) temporal and (b) spectral evolution for 1 cm long silicon waveguide The inputsoliton is 50 fs (FWHM) 25W peak power hyperbolic secant soliton with order119873 = 3

peak in spectral domain called theCherenkov radiationHerewe define the full width at half maximum (FWHM) of thepulse and spectrum as the pulse width and spectral widthrespectively The pulse width and spectral width at differentdistance of the waveguide can be seen clearly in Figure 5

The Cherenkov radiation is emitted in the form ofdispersive wave at a frequency underlying phase matchingconditionThe frequency detuning of Cherenkov radiation isgiven by [12]

Ω119889asymp minus

31205732

1205733

+1205741198751199041205733

312057322

(17)

where 119875119904is the peak power of the fundamental soliton

perturbed by the third order dispersion coefficient Generallyin the absence of losses 119875

119904= (53)

2

1198750 where 119875

0is the

peak power of the input soliton However the linear loss andnonlinear loss reduce the value of 119875

119904along the waveguide

We assume that 119875119904asymp 1198750in this simulation The frequency

detuning of Cherenkov radiation is determined by the highorder dispersions and input peak power of solitons Therelationship between the frequency of Cherenkov radiationand input peak power is presented in Figure 6 when theinput peak power of soliton changes from 10W to 70W thefrequency of dispersive wave emission varies from 170 THzto more than 370 THz It is also known that the frequency ofdispersive wave emission can be changedwhen pumping nor-mal dispersion zone and abnormal dispersion zone which isdescribed by (17)

The results of temporal and spectral evolution byinputting high order soliton with119873 = 6 and the peak powerof soliton which is 50W are obtained as shown in Figure 7 Itis observed that soliton fission occurs in the process of pulseevolution and the broadband dispersive wave is generated asshown in Figure 7(a) In this process the number of solitonsgenerated by the soliton fission is 6 which is the samewith the order of soliton Figure 7(b) shows the bandwidthof the supercontinuum is becoming to over 500 nm which


004006008

01012

Pulse

wid

th (p

s)

0 1Distance (cm)

05

(a)

0 1406080

100120

Distance (cm)Spec

tral

wid

th (n

m)

05

(b)

Figure 5 Results from numerical simulations showing (a) variation of pulse width along the silicon waveguide and (b) variation of spectralwidth along the silicon waveguide

10 20 30 40 50 60 70150

200

250

300

350

400

Peak power of input soliton (W)

Freq

uenc

y of

disp

ersiv

e wav

e (TH

z)

Figure 6 Frequency of dispersive wave emission as a function of inputting peak power of the soliton The relative parameters are the sameas used in Figure 4

minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1

Dist

ance

(cm

)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)


is broader than that generated by the third order solitonIt can be also found that the wavelength of Cherenkovradiation is around 1250 nm This value agrees with thepredictions of (12) Compared with the supercontinuumgeneration in silica fibers the effect of stimulated Ramanscattering in silicon waveguide plays a minor role The effectof TPA and FCA must be considered in the process ofsupercontinuum generation in silicon waveguide Figure 8(a)shows the temporal evolution along the waveguide withoutconsidering the effect of TPA the generated dispersive waveemission is more continuous and the efficiency is higherthan that generated considering TPA The contrast between

the input and output pulse without the effect of TPA isshown in Figure 8(b) in which the intensity of solitonfission is reduced and the intensity of fundamental solitonis nearly 80 of input intensity The effect of TPA is notdetrimental to the supercontinuum generation process eventhough it reduces the supercontinuum bandwidth to someextent The numerical results of the dispersive wave emissionwithout considering the effect of stimulatedRaman scattering(SRS) were also presented Figure 9(a) shows the temporalevolution along the waveguide without considering the effectof SRS The contrast between the input and output pulsewithout the effect of SRS is shown in Figure 9(b) The soliton


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity

Input pulseOutput pulse without TPA

(b)

Figure 8 (a) Temporal evolution for the 1 cm long silicon waveguide without considering the effect of TPA (b) Contrast of the input pulseand output pulse without considering the effect of TPAThe input soliton is 50 fs (FWHM) 25W peak power hyperbolic secant soliton withorder119873 = 3

minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)

Input pulseOutput pulse with SRS

(b)

Figure 9 (a) Temporal evolution for the 1 cm long silicon waveguide without considering the effect of stimulated Raman scattering (b)Contrast of the input pulse and output pulse without considering the effect of stimulated Raman scatteringThe input soliton is 50 fs (FWHM)25W peak power hyperbolic secant soliton with order119873 = 3

fission effect is weak without the role of SRS which gives riseto the reduction of the number of the peak in time domain

4 Conclusions

Based on solving the generalized nonlinear Schrodingerequation in the silicon waveguide through the SSFM andfourth-order Runge-Kutta integrationmethod the dispersivewave emission and supercontinuum generation are inves-tigated in silicon waveguide The numerical results showthat the efficient dispersive wave emission can be generatedby propagating femtosecond solitons in silicon waveguidewhich plays an important role in the process of the super-continuum generation with the form of Cherenkov radiationand it is also shown that the high order low-energy solitons

and short waveguides are efficient for the dispersive waveemission and the dominated impacts on the generation ofthe dispersive wave emission are the high order dispersionsand the peak power of input soliton

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant no 61275134


References

[1] J M Dudley and J R Taylor ldquoTen years of nonlinear optics inphotonic crystal fibrerdquoNature Photonics vol 3 no 2 pp 85ndash902009

[2] P L Baldeck and R R Alfano ldquoIntensity effects on thestimulated four photon spectra generated by picosecond pulsesin optical fibersrdquo Journal of Lightwave Technology vol 5 no 12pp 1712ndash1715 1987

[3] J HerrmannU Griebner N Zhavoronkov et al ldquoExperimentalevidence for supercontinuum generation by fission of higher-order solitons in photonic fibersrdquo Physical Review Letters vol88 no 17 Article ID 173901 2002

[4] A L Gaeta ldquoNonlinear propagation and continuum generationin microstructured optical fibersrdquo Optics Letters vol 27 no 11pp 924ndash926 2002

[5] A Demircan and U Bandelow ldquoAnalysis of the interplaybetween soliton fission andmodulation instability in supercon-tinuum generationrdquo Applied Physics B Lasers and Optics vol86 no 1 pp 31ndash39 2007

[6] M A Foster J M Dudley B Kibler et al ldquoNonlinearpulse propagation and supercontinuum generation in photonicnanowires experiment and simulationrdquo Applied Physics BLasers and Optics vol 81 no 2-3 pp 363ndash367 2005

[7] Q Lin O J Painter and G P Agrawal ldquoNonlinear opticalphenomena in silicon waveguides modeling and applicationsrdquoOptics Express vol 15 no 25 pp 16604ndash16644 2007

[8] M Zhu H Liu X Li et al ldquoUltrabroadband flat dispersiontailoring of dual-slot siliconwaveguidesrdquoOptics Express vol 20pp 15899ndash15907 2012

[9] L Yin and G P Agrawal ldquoImpact of two-photon absorptionon self-phase modulation in silicon waveguidesrdquoOptics Lettersvol 32 no 14 pp 2031ndash2033 2007

[10] Z Wang H Liu N Huang Q Sun J Wen and X LildquoInfluence of three-photon absorption on Mid-infrared cross-phase modulation in silicon-on-sapphire waveguidesrdquo OpticsExpress vol 21 pp 1840ndash1848 2013

[11] X Li ZWang andH Liu ldquoOptimizing initial chirp for efficientfemtosecond wavelength conversion in silicon waveguide bysplit-step Fourier methodrdquo Applied Mathematics and Compu-tation vol 218 pp 11970ndash11975 2012

[12] L Yin Q Lin and G P Agrawal ldquoSoliton fission and supercon-tinuumgeneration in siliconwaveguidesrdquoOptics Letters vol 32no 4 pp 391ndash393 2007

[13] I-W Hsieh X Chen X Liu et al ldquoSupercontinuum generationin silicon photonic wiresrdquo Optics Express vol 15 no 23 pp15242ndash15249 2007

[14] J Wen H Liu N Huang Q Sun and W Zhao ldquoInfluence ofthe initial chirp on the supercontinuum generation in silicon-on-insulator waveguiderdquo Applied Physics B Lasers and Opticsvol 104 no 4 pp 867ndash871 2011

[15] K J Blow and D Wood ldquoTheoretical description of transientstimulated Raman scattering in optical fibersrdquo IEEE Journal ofQuantum Electronics vol 25 no 12 pp 2665ndash2673 1989

[16] H Wang ldquoNumerical studies on the split-step finite differencemethod for nonlinear Schrodinger equationsrdquo Applied Mathe-matics and Computation vol 170 no 1 pp 17ndash35 2005

[17] X Xiangming and T R Taha ldquoParallel split-step fouriermethods for nonlinear Schrodinger-type equationsrdquo Journal ofMathematical Modelling and Algorithms vol 2 pp 185ndash2012003

[18] G M Muslu and H A Erbay ldquoHigher-order split-step Fourierschemes for the generalized nonlinear Schrodinger equationrdquoMathematics and Computers in Simulation vol 67 no 6 pp581ndash595 2005

[19] S Zhang Z Deng and W Li ldquoA precise Runge-Kutta inte-gration and its application for solving nonlinear dynamicalsystemsrdquo Applied Mathematics and Computation vol 184 no2 pp 496ndash502 2007

[20] M Z Liu S F Ma and ZW Yang ldquoStability analysis of Runge-Kutta methods for unbounded retarded differential equationswith piecewise continuous argumentsrdquo Applied Mathematicsand Computation vol 191 no 1 pp 57ndash66 2007

[21] B S Attili K Furati and M I Syam ldquoAn efficient implicitRunge-Kutta method for second order systemsrdquo Applied Math-ematics and Computation vol 178 no 2 pp 229ndash238 2006

[22] T E Murphy software httpwwwphotonicsumdedu

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of nonlinear loss such as the two-photon absorption (TPA)and free-carrier absorption (FCA) is based on the generalizednonlinear Schrodinger equation as follows [14]

120597119860

120597119911=

infin

sum

119898=2

119894119898+1

120573119898

119898

120597119898

119860

120597119905119898minus1

2(120572119897+ 120572119891)119860 + 119894120574

times (1 +119894

1205960

120597

120597119905)119860 (119911 119905) times int

119905

minusinfin

119877 (119905 minus 120591) |119860 (119911 120591)|2

119889120591

(1)

where 119860 = 119860(119911 119905) is the electric field envelope as a functionof distance along the waveguide 119911 and retarded time 119905 120572

119897

and 120572119891account for the linear losses and the free-carrier

absorption (FCA) of the waveguide respectively 120573119898s are

dispersion coefficients at center frequency 1205960 120574 denotes the

nonlinear coefficient of thewaveguidewhich is defined as 120574 =11989921198960119886eff + 119894120573TPA119886eff 1198992 is the Kerr coefficient the parameter

1198960is the wave vector 119886eff is the effective area of the silicon

waveguide and 120573TPA is the two-photon absorption (TPA)parameter The influence of the FCA is from the nonlinearloss 120572

119891= 120590119873

119888(119911 119905) where 119873

119888(119911 119905) is the density of free

carriers The density of free-carrier 119873119888(119911 119905) can be obtained

by solving the following equation [7]

120597119873119888(119911 119905)

120597119905=120573TPA2ℎ]0

|119860 (119911 119905)|4

1198862effminus119873119888(119911 119905)

120591 (2)

where 120591 is the effective carrier life time which is estimatedto be about 3 ns [12] ℎ is Plank constant and ]

0is the

central frequency All of the parameters above have beenconsidered in our theory model and the initial injectedsoliton is assumed to have a hyperbolic secant field profilewhich is shown as

119860 (0 119879) = radic1198750sec ℎ( 119879

1198790

) (3)

where 1198750is the peak power of the input soliton The order of

input soliton119873 satisfies the condition of

1198732

=Re (120574) 119875

01198792

0

1205732

(4)

The right-hand side of (1) represents the nonlinear responseof the silicon waveguide with the response function 119877(119905)which can be written as

119877 (119905) = (1 minus 119891119877) 120575 (119905) + 119891

119877sdot ℎ119877(119905) (5)

where the value of 119891119877= 0043 which is smaller than that of

silica fibers The function of ℎ119877can be written as [15]

ℎ119877=1205912

1+ 1205912

2

1205912112059122

exp(minus 1199051205912

) sdot sin( 119905

1205911

) (6)

where 1205911= 10 fs and 120591

2= 303 ps

In order to solve (1) numerically the equation can bedivided to two parts the linear term and nonlinear termThelinear term includes the effects of linear loss nonlinear lossand high order dispersions here we consider the dispersioncoefficient to the sixth orderWe can use the split-step Fouriermethod [11 16ndash18] to solve the linear term of (1) while thenonlinear term can be described as follows

120597119860

120597119911= 119894120574 (1 +

119894

1205960

120597

120597119905) (119860119877 lowast |119860|

2

) (7)

where

119877 lowast |119860|2

= int

119905

minusinfin

119877 (1199051015840

)10038161003816100381610038161003816119860 (119911 119905 minus 119905

1015840

)10038161003816100381610038161003816

2

119889119905 (8)

The time derivative term in (7) can be treated as aperturbation and introduce a new function 119861(119911 119905) as

119861 (119911 119905) = 119860 (119911 119905) sdot exp lfloor119894120574119877 lowast 1003816100381610038161003816119860010038161003816100381610038162

sdot (119911 minus 1199110)rfloor (9)

where 1199110represents the initial position of the propagating

femtosecond soliton considering 1199110= 0 in this simulation

From (7) and (9) it can be found that the new function119861(119911 119905)satisfies the following equation

120597119861

120597119911= 119894120574119861 sdot 119877 lowast (

1003816100381610038161003816119861010038161003816100381610038162

+ |119861|2

) minus120574

1205960

sdot120597

120597119905(119861 sdot 119877 lowast |119861|

2

)

(10)

It is possible to obtain 119861(119911 119905) through solving (9) using thefourth-order Runge-Kutta integration method [19ndash21] Thedetails of calculations using the fourth-order Runge-Kuttamethod satisfy the following relationship

119861119896+1

= 119861119896+ℎ0(1198911+ 21198912+ 21198913+ 1198914)

6 (11)

where ℎ0is the step length of the calculation process 119891

1 1198912

1198913 and 119891

4can be written as follows

1198911= 119891 (119911

119896 119861119896) (12)

1198912= 119891(119911

119896+ℎ0

2 119861119896+ℎ0

21198911) (13)

1198913= 119891(119911

119896+ℎ0

2 119861119896+ℎ0

21198912) (14)

1198914= 119891 (119911

119896+ ℎ0 119861119896+ ℎ01198913) (15)

where 119911119896represents the position of the propagation distance

Then we can obtain 119860(119911 119905) through the relationshipbetween 119861(119911 119905) and 119860(119911 119905) in (8) Consequently using thesplit-step Fourier method (SSFM) and the fourth-orderRunge-Kutta integrationmethod (1) can be solved accurately

3 Results and Discussion

In this simulation the silicon waveguide used is the straightwaveguide The transverse section of the silicon waveguide is


Air

W

H Silicon

SiO2





0123

Wavelength (nm)

25

3

35

4

Refr

activ

e ind

ex

Seco

nd o

rder

disp

ersio

n (p

s2m

)

nsi1205732TE

1205732TM

neffTE

neffTE



05 120583m






1205733= 40 times 10




6= 012 times 10






2= 6times10




0= 284 fs


119879119901= 2 ln (1 + radic2)119879

0= 1763119879

0= 50 fs (16)





minus02 minus01 0 01 02 03 04 05 06 07 080

005

01

Time (ps)

Nor

mal

ized

inte

nsity


(a)


minus50

Nor

mal

ized

inte

nsity

(dB)


120582 (nm)

(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1D

istan

ce (c

m)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)




Ω119889asymp minus

31205732

1205733

+1205741198751199041205733

312057322

(17)



119904= (53)

2

1198750 where 119875

0is the







004006008

01012

Pulse

wid

th (p

s)

0 1Distance (cm)

05

(a)

0 1406080

100120

Distance (cm)Spec

tral

wid

th (n

m)

05

(b)


10 20 30 40 50 60 70150

200

250

300

350

400


Freq

uenc

y of

disp

ersiv

e wav

e (TH

z)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1

Dist

ance

(cm

)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)





minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity


(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)


(b)



4 Conclusions





Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Air

W

H Silicon

SiO2





0123

Wavelength (nm)

25

3

35

4

Refr

activ

e ind

ex

Seco

nd o

rder

disp

ersio

n (p

s2m

)

nsi1205732TE

1205732TM

neffTE

neffTE



05 120583m






1205733= 40 times 10




6= 012 times 10






2= 6times10




0= 284 fs


119879119901= 2 ln (1 + radic2)119879

0= 1763119879

0= 50 fs (16)





minus02 minus01 0 01 02 03 04 05 06 07 080

005

01

Time (ps)

Nor

mal

ized

inte

nsity


(a)


minus50

Nor

mal

ized

inte

nsity

(dB)


120582 (nm)

(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1D

istan

ce (c

m)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)




Ω119889asymp minus

31205732

1205733

+1205741198751199041205733

312057322

(17)



119904= (53)

2

1198750 where 119875

0is the







004006008

01012

Pulse

wid

th (p

s)

0 1Distance (cm)

05

(a)

0 1406080

100120

Distance (cm)Spec

tral

wid

th (n

m)

05

(b)


10 20 30 40 50 60 70150

200

250

300

350

400


Freq

uenc

y of

disp

ersiv

e wav

e (TH

z)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1

Dist

ance

(cm

)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)





minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity


(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)


(b)



4 Conclusions





Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus02 minus01 0 01 02 03 04 05 06 07 080

005

01

Time (ps)

Nor

mal

ized

inte

nsity


(a)


minus50

Nor

mal

ized

inte

nsity

(dB)


120582 (nm)

(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1D

istan

ce (c

m)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)




Ω119889asymp minus

31205732

1205733

+1205741198751199041205733

312057322

(17)



119904= (53)

2

1198750 where 119875

0is the







004006008

01012

Pulse

wid

th (p

s)

0 1Distance (cm)

05

(a)

0 1406080

100120

Distance (cm)Spec

tral

wid

th (n

m)

05

(b)


10 20 30 40 50 60 70150

200

250

300

350

400


Freq

uenc

y of

disp

ersiv

e wav

e (TH

z)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1

Dist

ance

(cm

)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)





minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity


(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)


(b)



4 Conclusions





Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










004006008

01012

Pulse

wid

th (p

s)

0 1Distance (cm)

05

(a)

0 1406080

100120

Distance (cm)Spec

tral

wid

th (n

m)

05

(b)


10 20 30 40 50 60 70150

200

250

300

350

400


Freq

uenc

y of

disp

ersiv

e wav

e (TH

z)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

0

05

1

Dist

ance

(cm

)

1200 1300 1400 1500 1600 1700 1800 1900120582 (nm)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(b)





minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity


(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)


(b)



4 Conclusions





Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 140

010203040506070809

1

Time (ps)

Nor

mal

ized

inte

nsity


(b)


minus02 0 02 04 06 080

05

1

Time (ps)

Dist

ance

(cm

)

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

(dB)

(a)

minus02 0 02 04 06 08 1 12 14 16 180

02

04

06

08

1

Nor

mal

ized

inte

nsity

Time (ps)


(b)



4 Conclusions





Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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