Optics Communications 300 (2013) 27–32

Contents lists available at SciVerse ScienceDirect

Optics Communications

0030-40

http://d

n Corr

E-m

snsarka

journal homepage: www.elsevier.com/locate/optcom

Discussion

Effect of variation of core gap radius on the performance of dualconcentric core Raman fiber amplifier

Pratap Kumar Bandyopadhyay a,b, Somenath Sarkar b,n

a Dream Institute of Technology, P.O. Bishnupur, Samali, 24 Parganas(S), Kolkata, Indiab Department of Electronic Science, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India

a r t i c l e i n f o

Article history:

Received 15 June 2012

Received in revised form

13 February 2013

Accepted 14 February 2013Available online 6 March 2013

Keywords:

Fiber Raman amplifier

Effective Raman gain

Effective area

Dispersion coefficient

18/$ - see front matter & 2013 Published by

x.doi.org/10.1016/j.optcom.2013.02.029

esponding author. Tel.: þ91 332 447 3412;

ail addresses: gubli2003@yahoo.co.in (P.K. Ba

r_50@yahoo.com (S. Sarkar).

a b s t r a c t

We vary the core gap radius, the distance between two concentric cores of a single mode Fiber Raman

Amplifier (FRA), having a step profile. Based on well known scalar analysis, we study crucially the

effects of this variation on the performance characteristics of FRA for the first time. For practically

important frequency shift bands of 20–700 cm�1, our interesting observation is that the values of core

gap radii in a certain higher range show better flattening of effective Raman gain in comparison to those

in lower range with single pump; side by side, a larger negative coefficient of dispersion is achieved in

the operating range of wavelength with the increase of core gap radius. These promising criteria should

attract the attention of system designers engaged in fabrication of FRA.

& 2013 Published by Elsevier B.V.

1. Introduction

Today single mode Fiber Raman Amplifier (FRA), constructedby dual concentric cores meets up the high speed need of internettraffic in WDM and DWDM because attenuated signals areoptically amplified by the fiber based amplifier [1]. Here ampli-fication of signals does not require any doping in the narrowregion of fiber core as in erbium doped fiber amplifiers (EDFA).In case of FRA, Stimulated Raman Scattering (SRS), a nonlinearoptical phenomenon, is applied and the photon energy is utilizedfrom one optical domain of higher frequency, known as a pump,to another domain at lower frequencies, known as the signal, foramplification [2]. Also the gain spectrum of FRA depends solely onthe pump wavelength; hence it becomes simpler and easier toaccess inaccessible bands like S-band by EDFAs [3]. Thus FRAwhich can provide 3 dB bandwidth of 90–100 nm, has emerged asthe potential solution to be used as an optical amplifier. Since theeffective Raman gain coefficient [4], which is the ratio of Ramangain coefficient and effective area, assumes almost a uniformvalue with frequency shift in the said wavelength band, there isno restriction of selecting signal frequency band in FRA in whichonly we have to choose proper pump wavelength. We concentratesignals of wavelength to be amplified around a particular value inS-band.

Elsevier B.V.

fax: þ91 332 351 9755.

ndyopadhyay),

Very recently, comparative studies of performance criteria ofFRA with various refractive index profile distributions in coreincluding step, parabolic, and triangular [5] ones have receivedkeen attention in relation to the variation of the effective Ramangain, effective area and dispersion coefficient with frequency shiftkeeping the phase matching condition fixed. Similar work has alsoincluded trapezoidal index profiles of practical interest [6]. Alsoinvestigation on FRA involving photonic crystal structures hasstarted attracting interest in the context of signal amplificationand dispersion compensation [7]. It is relevant to mention thatinvestigation on zero dispersion wavelength and its shiftingtogether with extremely large birefringence have received world-wide attention in case of micro-structured and photonic crystalfibers [8,9].

In this paper, our aim is to investigate whether there isa suitable structural parameter for a step FRA in order to achievebetter gain flattening w.r.t. a single pump corresponding toa particular wavelength. We focus our attention to concentricstep profile for its conventionally accepted use and variation ofthe core gap radius, the distance of separation between first andsecond cores in the profile, with other structural parametersremaining constant. Our encouraging observation is that thevalues of core gap radii in a certain higher range show betterflattening of effective Raman gain in comparison to those in lowerrange with single pump. Thus structures with higher core gapradii will exhibit flattening with single pump whereas structureswith the lower ranging core gap radii should show gain flatteningwith multiple pumps requiring cost based technological intrica-cies. With the above proposal, it is therefore possible to realize a

P.K. Bandyopadhyay, S. Sarkar / Optics Communications 300 (2013) 27–3228

suitable FRA in the S band operation. Side by side, anotherimportant aspect is revealed that larger negative dispersioncoefficient is also achieved in the said wavelength band for higherrange of core gap radii. Advantageously, one can, then, choose FRAof certain length with a particular value of core gap radius anddetermine the value of dispersion and attach to the total link andmake the proper amplification and minimize dispersion [10].In subsequent sections, we present our analysis for computationand simulation together with results and discussion.

2. Modeling and analysis

2.1. Profile structure

Our proposed optical fiber, shown in Fig. 1, has a coaxialrefractive index profile with inner and outer cores. We confineour attention to a single mode regime with profile distribution as

n2ðrÞ ¼

n21½1�D1f ðrÞ� for 0rrr1

n23 for 1rrrb0

n22 for b0rrrc0

n23 for c0rr

8>>>><>>>>:

ð1Þ

where n1, n2 and n3 are refractive indices of the inner core,outer core and cladding and b0 ¼b/a, c0 ¼c/a, a¼core radius.D1 ¼ ðn

21�n2

3Þ=n21, r¼ r=a; r is the radial distance in cylindrical

coordinate and a is first core radius; b is the core gap radius; (c–b)is outer core radius. The profile function, f ðrÞ is given as f ðrÞ ¼ rq;here q tends to N for step profile. The profile is sampled intosmall units of rectangle to compute the field distribution in eachrectangular segment and effective index of refraction ðZeff Þ. Thetotal field is the vector sum of all such segment based fields [5,9].The effective refractive index bears the relation with propagationconstant and wave number

b2¼ k2

0n2Zef f

ð2Þ

where bð ¼ 2p=lÞ is the propagation constant in the dielectricmedium of optical fiber with k0 ¼ 2p=l0 being the free space wavevector. Here l and l0 are the wavelengths in the medium and freespace respectively. Thus one may take Zef f as a ratio of l0=l.

2.2. Raman gain model

For small signal regime, one can ignore the pump depletiondue to SRS. The variations of pump power ðPpÞ with pumpwavelength lp and signal power ðPsÞ with signal wavelength

Fig. 1. Refrative index vs. radial distance to indicate the profile.

ls are described by the following couple mode equation [3,5]:

dPp=dz¼�apPP ,

dPs=dz¼ gRPPPS�asPs

)ð3Þ

where ap and as are attenuation coefficients at pump and signalwavelength, respectively and gR being effective Raman gaincoefficient, given as [2,6,12]

gR ¼gRðuÞAef f

ð4Þ

where gRðuÞ is Raman gain coefficient; Aef f is the effective area,defined in terms of pump ðcpÞ and signal modal fields ðcsÞ as

Aef f ¼ 2pRc2

pr drRc2

s r drRc2

pc2s r dr

ð5Þ

and is computed from the overlap integral of two modal fields [6].The modal fields are calculated by the solution of the respectivetranscendental equation by the Runge–Kutta method. In the FRAstructure, the effective area Aef f varies in such a way that theeffective Raman gain coefficient becomes constant over the 90 nmbandwidth as the Raman gain decreases sharply and side by sidethe effective area Aef f increases in accordance with the decrease,eventually maintaining gain flattening.

The Raman gain coefficient is expressed by the followingequation:

gRðxGeO2,uÞ ¼

n22

n21

gRðSiO2,uÞþCðuÞxGeO2gRPðSiO2, uÞ

l3s

l3sPeak

" #ð6Þ

where gRPis the peak Raman gain coefficient, xGeO2

is thegermanium concentration; CðuÞ is the linear regression coeffi-cient; lsPeak is the signal wavelength at which the peak Ramangain is obtained. It is well known that the doping of germanium inthe specialized manner modulates the refractive index desirablefor the production of dual concentric cores. Consequently, thedegree of concentration in relation to that of SiO2 presentssuitable value of Raman gain coefficient given in Eq. (6) [4,7,12].

2.3. Dispersion coefficient

The dispersion coefficient meaning group velocity dispersion isgiven as

DðlÞ ¼ �lc

d2Zef f

dl2ð7Þ

where c is the velocity of light [5,10].

2.4. Scalar wave equation

In order to find pump and signal fields and effective refractiveindices, we use the following scalar wave equation under weaklyguiding approximation [8]:

@2c@r2þ

1

r

@c@rþ o2eðrÞm0�b

2h i

cðrÞ ¼ 0 ð8Þ

where eðrÞ is dielectric permittivity of optical fiber at radial distancer, m0 is free space permeability of the medium. The modal field isexpressed by Bessel and modified Bessel equation as

cðrÞ ¼ AJ0ðkrÞþBY0ðkrÞ for nðrÞ4Zef f ð9Þ

cðrÞ ¼ CI0ðorÞþDK0ðorÞ for nðrÞoZef f ð10Þ

with k2 and o2 are given as

k2 ¼ k20½n

2ðrÞ�Z2ef f � ð11Þ

P.K. Bandyopadhyay, S. Sarkar / Optics Communications 300 (2013) 27–32 29

o2 ¼ k20½n

2nef f�n2ðrÞ� ð12Þ

The effective refractive index ðZef f Þ, constants A, B, C and D andthe field, cðrÞ in the above equations are evaluated by the matrixmethod [4,11] where we partition the whole index profile of theproposed optical fiber into smaller rectangular segments andapply boundary condition of continuity with the field cðriÞ andð@cðriÞ=@rÞ for ith and (iþ1) th segments, respectively. It may berelevant to mention that Zef f is computed theoretically only forconventional step index fibers in terms of Bessel function in thecore and modified Bessel function in the cladding. However, forarbitrary index profiles, having complex refractive index struc-tures as in our designed dual core profile, one has to take resort tonumerical methods to simulate the field profiles and compute thefiber parameters of practical interest. Further, for the informationof the wider audience, the calculation based on the matrixmethod in this paper follows the transformation of the columnvector of the field and its derivative by 2�2 transformationmatrix at each segment of the profile according to the formulationpresented by Morishita [11] in detail. In the next section, wepresent the results and discussions based on our simulation workusing MATLAB 7.0 and involving the above theoretical framework.

3. Results and discussion

3.1. Profile and phase matched wavelength

With reference to Fig. 1, the refractive index, n1 of the first coreis 1.47299, refractive index of second core n2 is 1.44871 and therefractive index of the cladding, n3 is 1.444388 at 1.55 mm of puresilica. The core gap radius, b ranges from 7.1 mm to 8.9 mm for thefirst case of evaluation and 9.0–9.5 mm in next case. But a and(c–b) are kept fixed all through the simulation. The parametersD1 and D2 ¼ ðn

22�n2

3Þ=n22 are calculated and presented with other

data in Table 1.

1.5 1.55 1.6 1.651.4462

1.4464

1.4466

1.4468

1.447

1.4472

1.4474

1.4476

1.4478

1.448

wavelength (micrometer)

� eff

a=1e-6mtr,b=7.1e-6mtr,c=14.42e-6mtr,d=16.42e-6mtrpmw=1.517e-6mtr at neff=1.447

nco1=1.47299,nco2=1.44871,ncl=1.444388LP01

LP02

phase matched wavelength

Fig. 2. Effective index ðZef f Þ vs. wavelength (mm). Two super modes LP01 and LP02

and phase matched wavelength are shown.

Table 1Profile structure.

Core 1 and 2 D1 (%) D2 (%) b (mm) a (mm) c–b (mm)

Step 3.0 0.29 Varying (from 7.1 to 9.5) 1 7.32

Here, we find the fundamental modal field (c) and effectiverefractive index ðZef f Þ for the structure corresponding to thewavelength range from 1.46 to 1.62 mm for a particular value ofb, by solving scalar wave equation and solution thereof in (9) and(10) using the matrix method [11]. Likewise, we repeat the aboveprocess to find effective indices and modal field for differentvalues of b ranging from 7.1 mm to 9.5 mm at step of 0.1 mm.One such plot, Zef f vs. wavelength (l) is depicted for a particularvalue of b at 7.1 mm where the Phase Matched Wavelength(PMW) coming at 1.517 mm is shown in Fig. 2. In Fig. 2, the solidand dotted lines represent the effective indices for two LP01 andLP02 super-modes respectively at different wavelengths. We findthe similar nature of plot for different values of b and see thatPMW lies closely within 1.52 mm and 1.53 mm and does notchange appreciably with b. The choice of pump wavelength iskept at 1.465 mm much below PMW.

3.2. Effective area and effective Raman gain

The values of effective area are obtained by applying Eq. (5)for different frequency shifts and are plotted as the curve shownin Fig. 3 for a particular value of b, 7.1 mm; this is repeated for therange of b from 7.1 mm to 9.5 mm and plot is shown for two cases,one from 7.1 to 8.9 mm and other set from 9.0 mm to 9.5 mm.The graphical presentation of these results is depicted inFigs. 3 and 4. In Fig. 3, we concentrate on the first range of b

whereas Fig. 4 shows other range of b. It shows the effective areais increasing with the increase of b.

The Raman gain is obtained from the formula given by Eq. (6)for a step profile distributed amplifier. For our proposed amplifierwe choose a single pump system at wavelength 1.465 mm muchbelow PMW (1.517 mm). The frequency corresponding to peakRaman gain coefficient is found at 13.2 THz and gain is1.046�10�13 m/W at lp equal to 1.465 mm [3, 4]. The operatingregion of wavelength is chosen for frequency shift 20–700 cm�1

and the value of xGeO2is taken as 19.83. The effective Raman gain

coefficient vs. b is plotted and presented in Figs. 5 and 6 for twosets of values of b0s. In the lower set of values of b, we find anovershoot occuring in the value of effective gain near thefrequency shift of 450 cm�1. This overshoot in the values ofeffective Raman gain is much reduced with other set of values

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

x 10-10

frequency shift (1/cm)

b=7.5

two concentric cores(step)

b=7.1 micromtr

b=7.9 micomtre

b=8.3mromtr.

b=8.7micromtr.

Aef

f(sq

mtr.

)

b=9micromtr.

Fig. 3. Effective area (sq m) vs. wavelength (mm) for different values of core gap

radius (b: below 9 mm) indicated in the plot.

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3x 10-3

Frequency Shift (1/cm)

Effe

ctiv

e R

aman

Gai

nC

oeffi

cien

t(1/(m

-W))

b=9 micromtr.b=9.2 micromtr

b=9.3micromtr.

b=9.4micromtr.

b=9.5 micromtr.9.1micromtr

Fig. 6. Effective Raman gain (m/W) vs. frequency shift (cm�1) for different values

of core gap radii (range of b: above 9 to 9.5 mm).

1.44

1.46

1.48 1.5 1.5

21.5

41.5

61.5

8 1.6 1.62

1.64

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

wavelength (micrometre)

Cof

f. of

Dis

pers

ion(

ps/(n

m.k

m))

b=7.5micromtre.

b=7.1 micrometre

b=7.9micromtrb=8.3micromtr.

a=1micromtr.b is varyingtwoconcentriccores

Fig. 7. Coefficient of dispersion (ps/(nm km)) vs. wavelength (mm) for range of

outer gap radius from 7.1 mm to 8.3 mm.

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5

4x 10-3

Frequency Shift (1/cm)

Effe

ctiv

e ra

man

gai

n co

effic

ient

(1/(m

-W))

b=8.3 micromtr.

b=7.1 micromtr.

b=8.7micromtr

b=7.5micromtr.

Fig. 5. Effective Raman gain (m/W) vs. frequency shift (cm�1) for different values

of core gap radii (range of b: below 9 mm).

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10-10

Aef

f (sq

. met

er) b=9.5micromtr.

b=9.4micrmtr.

b=9.3 micro mtr.

b=9 micromtr.

b=9.1micromtr

Frequency shift (1/cm)

Fig. 4. Effective area (sq m) vs. wavelength (mm) for different values of core gap

radius (b: above 9 mm ) indicated in the plot.

1.44

1.46

1.48 1.5 1.5

21.5

41.5

61.5

8 1.6 1.62

1.64

-700

-600

-500

-400

-300

-200

-100

0

wavelength (micrometer)

Coe

ffici

ent o

f Dis

pers

ion

(ps/

(nm

.km

))step,a=1,b=9.1pmw= -620 at 1.53 e-6mtr

b=9.3e-6mtr.

b=9.5e-6mtr

Fig. 8. Coefficient of dispersion(ps/(nm km)) vs. wavelength (mm ) for range of

outer gap radius from 9.1 mm to 9.5 mm.

P.K. Bandyopadhyay, S. Sarkar / Optics Communications 300 (2013) 27–3230

of b0s as we increase b, shown in Fig. 6. When signal wavelengthsare below 1.517 mm, the pump and signal are tightly confined tothe inner core for which we get much overlapping between thetwo, pump and signal, resulting in smaller effective area, Aef f .As the signal wavelength approaches towards PMW, the fractionalpower spreads to the outer core and overlapping between pumpand signal is less, resulting in larger effective area, Aef f . Howeveralthough we get better flattening of effective Raman gain inhigher range of b without any appreciable overshooting of values,the highest value of effective Raman gain coefficient is more incase of lower range of b for DuE400 cm�1. Therefore whereasone can get better flattening by using a high power single pumpat a suitable wavelength for FRA structure corresponding tohigher range of b, one should have to use multiple pumps forsuch structure relating to lower range of b.

3.3. Dispersion coefficient

Now although gR is more crucial in studies on FRA, one cannotavoid the importance of wide knowledge of dispersion coefficient

7 7.5 8 8.5 9 9.5-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

-70

core gap radius (micromtr.)

dDm

in/d

b(ps

/(nm

.km

.mic

rom

tr.))

1st core: step2nd core: step

Fig. 10. Minimum dispersion coefficient rate, dDmin=db (ps/(nm km mm)) with

respect to core gap radius vs. core gap radius (mm).

7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9-650

-600

-550

-500

-450

-400

-350

-300

Core gap radius (micrometer)

min

imum

dis

pers

ion

coef

ficie

nt(p

s/(n

m.k

m))

Fig. 9. Minimum dispersion coefficient, Dmin ((ps/(nm km)) vs. core gap radius (mm).

Table 2Dispersion coefficient for higher core gap radius.

Sl. no. b-value (mm) Min. value of dispersion coefficient (ps/(nm km))

1 9.1 �620

2 9.2 �636.58

3 9.3 �653

4 9.4 �669.67

5 9.5 �685.9

P.K. Bandyopadhyay, S. Sarkar / Optics Communications 300 (2013) 27–32 31

in choosing a dispersion-compensated module. Fig. 7 shows theplot of dispersion coefficient vs. wavelength for b values rangingfrom 7.1 mm to 8.9 mm. Similarly, Fig. 8 shows the same variationfor the other range of b from 9.0 mm to 9.5 mm. The minimumcoefficient of dispersion, Dmin goes down below towards moreminimum, with the increase of values of b. The minimumdispersion for five such values is presented in Table 2.

We present the graphical variation of minimum dispersionðDminÞ with different values of b in Fig. 9. It is found that the curve

goes down till 7.8 mm. where a kink is observed. After this kink, itsfall is sharper. This aspect motivates us to compute the rate ofchange of minimum dispersion ðdDmin=dbÞ w.r.t. b and to get theplot of dDmin=db vs. b, shown in Fig. 10. It shows ðdDmin=dbÞ

initially falls with a sharp linearity to b¼7.3 mm. Afterwards itbecomes flat upto 8.1 mm, which indicates change is constant inthis range of b. Then the fall is with slower linearity up tob¼9.0 mm after which it assumes almost a constant value. It isalso observed within the range of b (7.3 mm to 9.5 mm), PMW liesclosely within the two super-modes leading to tolerance of FRA.We get two constant regions, one from 7.3 to 8.1 mm and otherfrom 9.0 to 9.5 mm as shown in Fig. 10 where one does not haveaccelerating fall of dispersion w.r.t. b and thus one will have muchtolerant use of FRA if used as dispersion compensation. Thus weobserve that when we increase the core gap radius in the higherside, we not only get gain flattening in the performance of FRA butalso achieve more negative dispersion coefficient with the PMWclosely lying within two super-modes. The achievement of highnegative dispersion coefficient over C and S bands in high valuesof core gap radii will be able to compensate the accumulateddispersion in conventional single mode fiber [10].

4. Conclusion

We have presented an analysis of dual core single pump FRA inorder to achieve better performance with respect to the core gapradius. This work highlights the gain characteristics as well asgives an advantage in dispersion compensation in the link by thechoice of core gap radius. We demarcate the two ranges of core-gap radii. The higher range of core gap radii needs only a singlepump for gain flattening whereas the lower range requiresmultiple pumps for the same. We observe that when we increasethe core gap radius in the higher side, we not only get gainflattening in the performance of FRA but also achieve morenegative dispersion coefficient with PMW closely lying withintwo super-modes. The achievement of high negative dispersioncoefficient over C and S bands in high values of core gap radii willbe able to compensate the accumulated dispersion in the con-ventional single mode fiber. Hence the high negative dispersioncoefficient along with signal amplification makes FRA, a prospec-tive candidate for Raman amplification and dispersion compensa-tion. The results based on our investigation are in the lowest losswindow of glass and should invite attention of system designersto find wide uses to prescribe and design suitable FRA structure inamplifier optics and dispersion management.

Acknowledgment

The authors acknowledge partial financial support of Centre ofResearch in Nano-science and Technology of Calcutta Universityin this work.

References

[1] G.P. Agarwal, Nonlinear Fiber Optics, 4th. ed., Academic, New York, 2007.[2] Y. Kang, Calculations and measurements of Raman gain coefficients of

different fiber types, M.S. Thesis, Department of Electrical Engineering,Virginia Polytechnic Institute, Blacksburg, VA, 2002, available from:/http://scholar.lib.vt.edu/thesis/available/etd-01102003-020757/S.

[3] K. Thyagrajan, Charu Kakkar, Journal of Lightwave Technology 22 (10) (2004)2279.

[4] K. Thyagrajan, Charu Kakkar, IEEE Photonics Technology Letters 15 (12)(2003) 1701.

[5] A.C.O. Chan, Journal of Lightwave Technology 25 (5) (2007) 1190.[6] Sanchita Prmanik, Gautam Das, Somenath Sarkar, Optical Engineering 49 (5)

(2010) 055001-1.

P.K. Bandyopadhyay, S. Sarkar / Optics Communications 300 (2013) 27–3232

[7] Shailendra K. Vershney, Kunimasa Saitoh, IEEE Photonics Technology Letters17 (10) (2005) 2062.

[8] Swapan Konar, Swapan K. Ghorai, Rakhi Bhattacharya, Fiber and IntegratedOptics 28 (2009) 138.

[9] Rakhi Bhattacharya, S. Konar, Optics and Laser Technology 44 (2012) 2210.

[10] Charu Kakkar, K. Thyagarajan, Applied Optics 44 (12) (2005) 2396.[11] Katsumi Morishita, IEEE Transaction of Microwave Theory and Solutions

MIT-29 (4) (1981) 348.[12] Sanchita Prmanik, Somenath Sarkar, Optics and Laser Technology 48 (2013) 206.