Propagation characteristics of single-mode graded-index elliptical core linear and nonlinear fiber using

super-Gaussian approximation

Sunil K. Khijwania,1,* Veena M. Nair,1 and Somenath Sarkar2

1Department of Physics, Indian institute of Technology Guwahati, Guwahati 781039, India2Department of Electronic Science, University of Calcutta, Kolkata 700009, India

*Corresponding author: skhijwania@iitg.ernet.in

Received 16 June 2009; revised 8 October 2009; accepted 9 October 2009;posted 9 October 2009 (Doc. ID 112894); published 28 October 2009

A rigorous, much simplified, and accurate analysis of the modal field characteristics such as propagationconstants, mode power confinement, delay and dispersion characteristics of a single-mode graded-indexelliptical core fiber is presented applying a variational method and super-Gaussian approximation of thefundamental modal field. Normalized propagation constants, a fundamental parameter to evaluate othermodal characteristics, obtained through this method showed a greater accuracy over the entire range ofpractical interest in comparison with other reportedmethods. The effects of various aspect ratios on thesecharacteristics are analyzed. In addition, the effect of Kerr nonlinearity on these characteristics is in-vestigated using the reportedmethod, and a comparison is made with the linear results. © 2009 OpticalSociety of America

OCIS codes: 060.2310, 060.2270, 060.2400.

1. Introduction

The past two decades have witnessed a rapid andcontinuous advancement in polarization-maintain-ing optical fiber technologies. In particular, ellipticalcore fibers (ECFs) have generated much research in-terest because of their excellent capability of control-ling and maintaining the required birefringence orphase and polarization modulation in an optical fiberlink. This has made ECFs a meritorious candidatefor potential applications in the areas such as coher-ent optical communication systems and polarimetricand interferometric sensor systems [1,2]. With thegrowing demand for complex and multifaceted appli-cations of these fibers and the need for ultrasensitivedevices, an accurate estimation of modal field char-acteristics (e.g., propagation characteristics) of step-index as well as graded-index ECFs is important.Direct computation of the modal field characteristics

and their fundamental properties in such ECFs withvarious aspect ratios (ellipticity) is extremely diffi-cult. One can solve analytically the eigenvalue equa-tion to evaluate the modal field characteristics forECFs with various aspect ratios. However, the ana-lytical method is applicable to a step-index profileonly. Even for a step-index profile, these characteris-tics are not easily computable because the eigen-value equations in terms of Mathieu functions in-volves infinite determinants [3]. For arbitrary-indexprofiles, the studies of propagation characteristicsare even more complicated. Hence, one must resortto approximation and numerical methods since ana-lytical methods are not available for graded-indexECFs. Various numerical [4–6] as well as approxi-mate methods [7,8] have been instituted to studythe propagation characteristics of ECF. The approx-imate method, e.g., the perturbation method [7], ap-proximates an elliptical core to a rectangular corewith the same aspect ratio and involves intricatemathematics in the calculation of modal fields forECF. On the other hand, the numerical method, e.g.,

0003-6935/09/31G156-07$15.00/0© 2009 Optical Society of America

G156 APPLIED OPTICS / Vol. 48, No. 31 / 1 November 2009

the finite element method [4,6], requires complexmathematics and computation. Moreover, the accu-racy of these methods is observed to be unsatis-factory over the entire normalized waveguide para-meter (V) range of practical interest. A similar obser-vation is made for the accuracy of the propagationconstant calculated for the fundamental modethrough the variational method that involves aGaussian approximation [8]. In the 1990s, the non-Gaussian approximation, like the super-Gaussianapproximation (SGA), was used as the approximatesolution for the radial mode of the optical wave thatpropagates in the combined influence of the non-linear Kerr effect and parabolic variation of the re-fractive index in circular core fiber, and Karlssonet al. studied the effect of nonlinear self-action ofthe optical pulse [9] in the medium. However, tothe best of our knowledge, no studies have been re-ported that use SGA on the ECF with arbitrary-index profiles.In the past decade, research has focused mainly on

the study of dual-mode ECF [10–12]. To the best ofour knowledge, no research attempt has been madeto improve the accuracy of propagation characteris-tics of single-mode ECF over the desired V range forstep-index as well as graded-index profiles and thusto estimate other modal characteristics of practicalinterest. Recently, the propagation constant for step-index ECF for the fundamental mode was computedaccurately using SGA in variational analysis and hasshown excellent agreement with the available analy-tical approximation and numerical methods [13]. Butpropagation characteristics and other fiber relatedparameters, such as core power, modal field intensityalong semimajor and semiminor axes with differentaspect ratios, delay and dispersion using SGA for thefundamental mode of graded-index ECF, have notbeen reported to date.We propose a scalar variational method with SGA

for single-mode graded-index ECF to study variousmodal characteristics of the fiber. A precise anal-ysis for the computation of such characteristicsfor graded-index ECF [e.g., parabolic-index ECF(PIECF)] with different aspect ratios using theproposed formalism is carried out. For this, the nor-malized propagation constant and variational para-meters are optimized by solving the linear scalarwave equation with the variational method andSGA. Results obtained with the SGA-based varia-tional technique are found to be extremely consistentwith the values obtained by analytical [14] and exactnumerical methods [5]. In addition, these resultsshow greater improvement over the existing resultsobtained from other approximation techniques. It isalso worth mentioning that the method we used forour study is much simpler in comparison with otherreported methods to evaluate the propagation char-acteristics of ECF with various index profiles. Thisoptimized propagation constant is used to calculatethe intensity distributions along the major and min-or axes and delay and dispersion characteristics for

linear graded-index profile with different aspectratios. SGA-based variational formalism is furtherextended to precisely calculate the propagation con-stant, core power, and modal field intensity of non-linear graded-index ECF having Kerr nonlinearityin both core and cladding. To this end, the normalizedpropagation constant and three variational para-meters are optimized by solving the nonlinear scalarwave equation that uses the SGA of the fundamentalmode. Finally, a comparison is made between linearand nonlinear characteristics of the PIECF.

2. Analysis

The refractive-index profile for linear graded-indexECF is given as

n2ðx; yÞ ¼ n12

�1 − 2δ

��x2

a2

�þ�y2

b2

��q=2

�

forx2

a2 þy2

b2≤ 1

¼ n12f1 − 2δg ¼ n2

2 forx2

a2 þy2

b2≥ 1; ð1Þ

where n1 is the refractive index at the core axis andn2 is the refractive index of the cladding of the fiber, aand b are the semimajor and semiminor axes of theelliptical fiber core, q is the profile exponent that de-termines the profile shape, and δ is the grading para-meter. In this study the fundamental modal field ofECF is approximated by the following normalizedsuper-Gaussian function:

ψðξ;ηÞ ¼ 21=2ppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiabρ01ρ02

p

×1

Γ�

12p

exp�−

�� ξρ01

�2p

þ� ηρ02

�2p��

; ð2Þ

where ξ ¼ x=a and η ¼ y=b are normalized coordi-nates and ρ01 and ρ02 are modal spot sizes alongthe semimajor and semiminor axes of the fiber core.The SGA, employing as many as three optimizationparameters (namely, ρ01, ρ02, and p), is used in thewell-known variational technique; Γ is the gammafunction [15]. The expression for the propagation con-stant is given as [8]

β2 ¼�k02

Z∞

0

dξZ∞

0

dηn2ðξ; ηÞjψðξ; ηÞj2

−1

a2

Z∞

0

dξZ∞

0

dηj∂ψ=∂ξj2

−1

b2

Z∞

0

dξZ∞

0

dηj∂ψ=∂ηj2��Z∞

0

dξZ∞

0

dηjψðξ; ηÞj2�−1;

ð3Þ

1 November 2009 / Vol. 48, No. 31 / APPLIED OPTICS G157

where k0 is the free-space wave vector. Substitutingn2 and ψðξ; ηÞ in Eq. (3), we obtain the ex-pression for Uð¼ bðk02n1

2 − β2Þ1=2Þ and normalizedpropagation constant P2 as

U2 ¼ ðVb2ðT1 þ T2Þ þ Ω2T3 þ T4Þ=T5;

P2 ¼ ½1 − ðU2=Vb2Þ�; ð4Þ

where

Vb ¼ k0bðn12 − n2

2Þ1=2;

Ω is the aspect ratio a=b, and

T1 ¼Z1

0

dξZffiffiffiffiffiffiffi1−ξ2

p

0

dηðξ2 þ η2Þq=2jψðξ; ηÞj2;

T2 ¼Z1

0

dξZ∞ffiffiffiffiffiffiffi1−ξ2

pdηjψðξ; ηÞj2 þ

Z∞

1

dξZ∞

0

dηjψðξ; ηÞj2;

T3 ¼Z∞

0

dξZ∞

0

dη∂ψ∂ξ

2;

T4 ¼Z∞

0

dξZ∞

0

dη∂ψ∂η

2;

T5 ¼Z∞

0

dξZ∞

0

dηjψðξ; ηÞj2: ð5Þ

Case 1: For step-index profile q → ∞. In this caseT1 ¼ 0 and Eq. (4) becomes

U2 ¼ V2 þ p221p ×

1Γð1=2pÞΓ

�2 −

12p

�� Ω2

ρ012þ 1

ρ022�

−2p2

12pV2

ρ01×

1

Γ2ð1=2pÞZ1

0

dξ expð−2ðξ=ρ01Þ2pÞγ

�

12p

; 2

� ffiffiffiffiffiffiffiffiffiffiffiffi1 − ξ2

pρ02

�2p�; ð6Þ

where γ is the incomplete gamma function [15].Case 2: For parabolic-index profile q ¼ 2. In this

case T1 ≠ 0 and Eq. (4) becomes

U2 ¼V2þp221p

1Γð1=2pÞΓ

�2−

12p

�� Ω2

ρ012þ 1

ρ022�

þ2pV2ρ012

ρ01212p

1

Γ2ð1=2pÞZ1

0

dξexpð−2ðξ=ρ01Þ2pÞγ

�

32p

;2

� ffiffiffiffiffiffiffiffiffiffiffiffi1− ξ2

pρ02

�2p�

þ2pV2212p

ρ011

Γ2ð1=2pÞZ1

0

dξexpð−2ðξ=ρ01Þ2pÞðξ2 −1Þγ

�

12p

;2

� ffiffiffiffiffiffiffiffiffiffiffiffi1− ξ2

pρ02

�2p�:

ð7Þ

In the limit of variational parameter p ¼ 1, Eq. (6) aswell as Eq. (7) reduces to a simple Gaussian approx-imation [8]. The modal spot sizes can now be ob-tained by optimizing U2 with respect to ρ01, ρ02,and p. The optimized value of P2 is then used forthe calculation of normalized group delay [DðVbÞ]and normalized dispersion [GðVbÞ] using the follow-ing relations:

DðVbÞ ¼dðVbP2ÞdVb

; GðVbÞ ¼ Vbd2ðVbP2ÞdVb

2 : ð8Þ

Case 3: The nonlinear stationary wave equationfor the fundamental mode in an ECF is given by

1

a2

∂2ψ∂2ξ2þ

1

b2∂2ψ∂2η2þfk02ðn2ðξ;ηÞþn2nNLψ2Þ−β2gψ¼0;

ð9Þ

where nðξ; ηÞ represents the refractive-index profile[Eq. (1)], nNL represents the nonlinear Kerr coeffi-cient, and β denotes the propagation constant. Equa-tion (9) can be rewritten as a variational expression:

Vb2

Z∞

0

Z∞

0

½Qðξ; ηÞ þ αψ2ðξ; ηÞ − P2�ψ2ðξ; ηÞdξdη

− Ω2

Z∞

0

Z∞

0

∂ψ∂ξ2dξdη −

Z∞

0

Z∞

0

∂ψ∂η2dξdη ¼ 0; ð10Þ

where αð¼ 4nNLF=ca2ðn12 − n2

2ÞÞ is the normalizednonlinear coefficient, F is the total power flow in thefiber, c is the free-space light velocity, and Qðξ; ηÞ isthe linear normalized profile, which is given as

Qðξ; ηÞ ¼ n2ðξ; ηÞ − n22

n12 − n2

2 : ð11Þ

Substituting Qðξ; ηÞ and ψðξ; ηÞ into Eq. (10), thefinal expression for the normalized propagation

G158 APPLIED OPTICS / Vol. 48, No. 31 / 1 November 2009

constant for PIECF with a nonlinear optical Kerr ef-fect results as

P2 ¼ 2p21=2p

ρ01Γð1=2pÞÞZ1

0

dξð1 − ξ2Þe−2ðξ=ρ01Þ2pγ

�1=2p;

�2

ffiffiffiffiffiffiffiffiffiffiffiffi1 − ξ2

pρ02

�2p�− 2pρ022

ρ0121=2pΓ2ð1=2pÞÞ

×Z1

0

dξe−2ðξ=ρ01Þ2pγ�3=2p;

�2

ffiffiffiffiffiffiffiffiffiffiffiffi1 − ξ2

pρ02

�2p�

−Ω2p221=2pΓð2 − 1=2pÞ

V2bρ012Γð1=2pÞ

�1

ρ012−

1

ρ022�

þ αp2

Ωρ01ρ02Γ2ð1=2pÞ : ð12Þ

The simple three-parameter optimization yields op-timized values of three variational parameters ρ01,ρ02, and p and hence P2 for the fundamental modeof PIECF. The normalized intensity (πabS) [2] alongthe semimajor and semiminor axes and core power(Pcore) [16] were then calculated from this optimizedvalue of P2.

3. Results and Discussions

To optimize P2 we used the classical Broyden method[17] and developed a MATLAB program by use of theBroyden algorithm. Variation of the normalized pro-pagation constant of the fundamental mode for step-index linear ECF with normalized frequency Vb, ascalculated by the SGA-based variational technique,is illustrated in Fig. 1 for Ωð¼ b=aÞ ¼ 0:5. For a rig-orous comparison, we also plotted the variation of P2

obtained with the standard numerical method [1,5]and the available perturbation method [7] with Vbfor Ω ¼ 0:5 in Fig. 1. It is worth noting that the nor-malized propagation constant calculated through theSGA shows spectacular improvement over the resultobtained with the perturbation method and matchesexcellently well over the whole V range with that ofthe standard numerical method. An accurate esti-mate of the normalized propagation constant for the

fundamental mode through SGA-based variationalanalysis is also done in a much simplified manner.As a comparison, in the case of the numerical meth-od, e.g., the finite element method, one must select alarge number of elements to obtain the desired accu-racy of the propagation constant, and this selectioninvolves intricate computation.

Since a Gaussian-type function works better forparabolic-index ECF in comparison with step-indexECF [4], and we have clearly shown excellent perfor-mance of the SGA for step-index ECF, the results ob-tained by the proposed method are expected to behighly accurate for graded-index profiles as well.Hence, we extend our study to graded-index ECFto obtain, for example, the propagation characteris-tics of parabolic-index ECF. The variation of P2 as afunction of normalized frequency Vb for the funda-mental mode of parabolic-index ECF with ellipticityΩ ¼ 0:8 is depicted in Fig. 2. One of the most impor-tant advantages of the proposed method is the factthat it can be applied for any arbitrary-index profile.This is not the case for, say, analytic methods, whichis applicable only to the step-index fiber and cannotbe extended to arbitrary-index profiles. Another sig-nificant advantage of the proposed method is the factthat the highly accurate SGA for the fundamentalmode enables the computation of various other

Fig. 1. (Color online) Normalized dispersion curve (P2 − VB) forstep-index ECF.

Fig. 2. Normalized dispersion curve (b − VB) for parabolic-indexECF.

Fig. 3. Intensity along the semimajor axis for PIECF.

1 November 2009 / Vol. 48, No. 31 / APPLIED OPTICS G159

parameters of interest, e.g., core power, group delay,and dispersion, for an arbitrary-index ECF with agreater accuracy. In the first step we investigatethe effect of ellipticity over the normalized propaga-tion constant. A variation of P2 as a function of nor-malized frequency Vb for the fundamental mode ofparabolic-index ECF with a higher ellipticity Ω ¼0:3 is also plotted in Fig. 2 for this purpose. As canbe observed, a change of ellipticity does not effectthe overall variation of P2 with Vb. However, the ab-solute value of the normalized propagation constantincreases with an increase in the ellipticity. The nor-malized propagation constant is also used to esti-mate modal intensities along the semimajor andsemiminor axes for various Vb values.

Figure 3 depicts one such typical result for the in-tensity (πabS) along the normalized semimajor axis ξfor Vb ¼ 2 and Ω ¼ 0:3. To analyze the effect of ellip-ticity, results for another ellipticity (Ω ¼ 0:8) are alsoplotted in Fig. 3. As is evident, intensity along thesemimajor axis in the central region increases withhigher ellipticity in comparison with the intensity forsmaller ellipticity. The intensity distribution clearlydepicts that the fundamental mode is tightly con-fined within the core along this axis for an ECF hav-ing high ellipticity in comparison with an ECFhaving small ellipticity. Figure 4 depicts the varia-tion of πabS along the normalized semiminor axisη for Ω ¼ 0:3 and 0.8. No considerable change inthe intensity is observed along this axis while the el-lipticity of the fiber changes. Comparing the intensi-ties along the semimajor and semiminor axes for a

fixed ellipticity, it is worth mentioning that, with alower ellipticity, e.g., in the case of Ω ¼ 0:8, thereis not much difference in the intensity pattern alongthe semimajor and semiminor axes. However, forhigher ellipticity, e.g., in the case of Ω ¼ 0:3, there issignificant variation of modal intensities along thesemimajor and semiminor axes. The total power inthe core (Pcore) of the fiber is another important para-meter that can be used to realize an optical fiber com-munication link or fiber-based devices. Hence theSGA analysis is extended to calculate Pcore.

Figure 5 depicts the variation of Pcore, which is cal-culated using normalized P2 obtained by SGA, alongwith Vb for Ω ¼ 0:3 and 0.8. As can be observed, thefractional power in the core is considerably larger forhigher ellipticity in comparison with smaller elliptic-ity for Vb ≤ 2:8. For a Vb value higher than 2.8, powerin the core remains unaffected with the variation ofellipticity of the fiber. SGA analysis is further ex-tended to evaluate the delay and dispersion charac-teristics of parabolic-index ECF. To the best of ourknowledge, this is the first such report of results ob-tained with the SGA. To evaluate delay and disper-sion characteristics, the normalized propagationconstant as obtained through SGA is used withEq. (8). Figure 6 depicts the normalized delay as afunction of Vb for PIECF with ellipticity of 0.5. Tocompare the effect of ellipticity on delay characteris-tics, the normalized group delay with ellipticity of 0.8is also plotted in Fig. 6. It is evident that the normal-ized group delay follows the same trend regardless of

Fig. 4. Intensity along the semiminor axis for PIECF.

Fig. 5. Fractional core power for PIECF.

Fig. 6. (Color online) Normalized delay versus Vb for PIECF.

Fig. 7. Dispersion versus Vb for PIECF.

G160 APPLIED OPTICS / Vol. 48, No. 31 / 1 November 2009

the ellipticity. Also, the group delay is observed to besignificantly higher for larger ellipticity in compari-son with the lower ellipticity in a Vb range that isapproximately ≤3:0. Contrary to this, no considerabledifference is observed for the normalized group delayin the Vb range that is approximately ≥3:0. The nor-malized group delay is then used to calculate dis-persion characteristics of the PIECF. A typical dis-persion characteristic [GðVbÞ] versus Vb of such fiberwith Ω ¼ 0:5 is depicted in Fig. 7.In the next step, we extend the SGA variational

analysis to PIECF with Kerr effect nonlinearity inthe core and cladding. In this case, the normalizednonlinear coefficient α is 0.062, nNL is 1:1 × 10−13

esu (silica fiber) [18], and P is 200 × 1010 erg=s forthe optimization of P2. The variation of P2 with Ω ¼0:3 and 0.8 as a function of Vb is illustrated in Fig. 8.To study the effect of nonlinearity on the propagationconstant, Fig. 2 is also plotted in Fig. 8. As can beseen, the nonlinear P2 follows the same trend as thecase of linear P2. However, nonlinear P2 is slightlyhigher in comparison with the linear P2 for higherV values. This deviation is less for larger ellipticityin comparison with smaller ellipticity. Figure 9 de-picts the variation of nonlinear πabS along the nor-malized axis ξ. These results are also compared withthe linear results (Fig. 3) in Fig. 9. As can be ob-served, the nonlinear πabS follows the same trendas the linear πabS for Ω ¼ 0:3 as well as 0.8. The non-linear πabS is lower in comparison with the linearπabS at the center of the core for both ellipticities in-vestigated in this study. However, for higher elliptic-

ity, this change is more dominant in comparison withlower ellipticity, which shows that the nonlinearityhas a considerably large effect on πabS for high ellip-ticity. The variation of nonlinear πabS along η is illu-strated in Fig. 10; this result is also compared withlinear results (Fig. 4). Linear as well as nonlinearπabS follows the same trend. In addition, there is noeffect when the ellipticity is changed on the non-linear πabS along the normalized semiminor axis.However a significant drop in intensity is observedfor the nonlinear πabS in comparison with the linearπabS at η ¼ 0.

4. Conclusion

Super-Gaussian approximation for the fundamentalmode of elliptical core fiber with an arbitrary-indexprofile has been proposed to study the propagationcharacteristics of such fibers within a simple varia-tional framework. This approximation with threeparameters has been shown to yield highly accurateresults for the normalized propagation constant. Themethod is simple in comparison with available ap-proximate and analytical methods. For the first time,to the best of our knowledge, the propagation charac-teristics such as normalized propagation constant,normalized intensity along semimajor and semi-minor axes, fractional core power, and delay and dis-persion have been computed using the proposedtechnique for the fundamental mode in a graded-in-dex ECF with a greater accuracy. This study was ex-tended to analyze the propagation characteristics ofthe same fiber under the inclusion of Kerr nonlinear-ity in the core and cladding. Finally, a significantcomparison of the linear and nonlinear results hasbeen presented.

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