Anton Bourdine. 1. Introduction. and approximate propagation constants by following simple ratio (Equation (32.22) in [1]):

Transcription

1 Matheatical Probles in Engineering olue 5, Article ID 843, pages Research Article Fast and Siple Method for Evaluation of Polarization Correction to Propagation Constant of Arbitrary Order Guided Modes in Optical Fibers with Arbitrary Refractive Index Profile Anton Bourdine Departent of Counication Lines, Povolzhsiy State University of Telecounications and Inforatics (PSUTI), 77 Moscow Avenue, Saara 4439, Russia Correspondence should be addressed to Anton Bourdine; Received 5 May 5; Accepted 9 June 5 Acadeic Editor: Xiao-Qiao He Copyright 5 Anton Bourdine. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original wor is properly cited. This wor presents fast and siple ethod for evaluation of polarization correction to scalar propagation constant of arbitrary order guided odes propagating over wealy guiding optical fibers. Proposed solution is based on earlier on developed odified Gaussian approxiation extended for analysis of wealy guiding optical fibers with arbitrary refractive index profile in the core region bounded by single solid outer cladding. Soe results are presented that illustrate the decreasing of coputational error during the estiation of propagation constant when polarization corrections are taen into account. Analytical expressions for the first and second derivatives of polarization correction are derived and presented.. Introduction The wealy guiding approxiation based on scalar wave equation (Helholtz equation) is widely used in the ost well-nown approxiate ethods and their various odifications for analysis of optical waveguides. Because the difference between refractive indexes of the core and the cladding of conventional telecounication silica optical fibers is less than %, they are wealy guiding. Therefore this approxiation providing a passage fro the vector wave equation to siplified scalar wave equation is applied as basic assuption in various approxiate ethods for coputing of optical fiber ode paraeters under sufficient accuracy. However these solutions are not exact, because they do not tae into account polarization effects, and there are soe applications requiring ore high accuracy. The proble can be solved by adding polarization correction δβ to scalar propagation constant β that would iprove accuracy of ode propagation constant calculation. Polarization correction is described in detail by Snyder and Love []. It relates to exact and approxiate propagation constants by following siple ratio (Equation (3.) in []): β exact =β δβ. () This wor presents fast and siple ethod for evaluation of polarization correction to scalar propagation constant of arbitrary order guided odes propagating over wealy guiding optical fibers. Proposed solution is based on earlier on developed odified Gaussian approxiation extended foranalysisofwealyguidingopticalfiberswitharbitrary refractive index profile in the core region bounded by single solid outer cladding. Soe results are presented that illustrate the decreasing of coputational error during the estiation of propagation constant when polarization corrections are taen into account. Analytical expressions for the first and second derivatives of polarization correction are derived and presented.

2 Matheatical Probles in Engineering. Polarization Correction: General Inforation Generally polarization correction to approxiate scalar propagation constant for odes propagating over optical waveguide with arbitrary cross section and refractive index profile can be found by following well-nown perturbation theory forula (Equation (3.4)in[]): δβ = a (Δ)3/ A ( t E) E t f(x,y)da, () A E da a is core radius, Δ = (n ax n N )/n ax is profile height paraeter, n ax is axial value of refractive index in the core region, n N is outer cladding refractive index, = an ax Δ is noralized frequency, = π/λ is wavenuber, λ is wavelength, and E is transverse electric field [ 6]: E= (r) [{cos (lφ) sin (lφ) (lφ) } x {sin } y], (3) cos (lφ) (r) is radial ode field distribution, l is aziuthal ode nuber (l=,,,...), is radial ode nuber ( =,, 3,...), x and y are unit vectors in the directions of the xand y-axis, A is infinite cross section square, and t is the transverse coponent of nabla differential operator: t = x x + y y ; (4) f(r) is profile function related with refractive index profile n(r) by well-nown expression [ 5]: n (R) =n [ Δ f (R)]. (5) Passage to optical fibers will split expression () into following integrals (Table (4.) in []): I = (Δ)3/ 4a I = l (Δ)3/ 4a ( F(l) [ / R) ( f/ R) R dr [F (l) ] ; (6a) RdR ] ( f/ R) dr [F (l) ], (6b) RdR R = r/a is noralized radius. Here polarization correction to scalar propagation constant of particular order guided ode in optical fiber is represented in the for of entioned two integrals cobination. Table copares the relations between LP odes and conventional odes and contains corresponding polarization corrections defined by (6a) and (6b) integrals (Table (4.) in []). By substituting radial ode field distribution and profile function f(r)into integrals (6a) and (6b) the final expression for polarization correction can be obtained. For exaple, below there is polarization correction δβ to the Table LP ode Conventional ode δβ LP (l=) HE (l=) I TE TM LP (l=) (I +I ) HE I I EH I +I HE LP l (l>) l+, I I EH l+, I +I ode HE (LP ) propagation constant for wealy guiding step index optical fiber (Table (4.3) in []): δβ = (Δ)3/ a U W K (W) K (W), (7) U and W are core and cladding ode paraeters: U + W = ; K l is second ind odified Bessel function. And polarization correction to the fundaental ode propagation constant for the graded optical fiber with ideal infinite parabolic index profile is described by following expression(table(4.)in[]): δβ = (Δ)3/ a. (8) Both ideal step and ideal parabolic refractive index profiles correspond to exact solutions of scalar wave equation that copletely describe ode field distribution. However unlie ideal profiles, analysis of real optical fibers requires a passage to ore coplicated refractive index profiles differing fro ideal fors by local defects and refractive index fluctuations [3, 7]. Also a passage to higher order guided odes should be required in the case of a few or ultiode optical fibers. In this wor, the entioned proble is solved by use of earlier on developed extension of odified Gaussian approxiation (EMGA). EMGA provides coputing of transission paraeters of arbitrary order guided odes. It was developed for analysis of wealy guiding silica optical fibers with axialsyetric arbitrary refractive index profile in the core region bounded by one solid outer cladding. 3. Extension of Modified Gaussian Approxiation EMGA is based on conventional well-nown Gaussian approxiation [] ofradialodefielddistribution (R) in the wealy guiding optical waveguide with an arbitrary refractive index profile by the well-nown Laguerre-Gauss function expression [ 6], describing a ode field distribution in wealy guiding optical waveguide with ideal infinite parabolic index profile: (R) = ( R l ) R (R R ) exp ( R R ), (9)

3 Matheatical Probles in Engineering 3 is Laguerre polynoial, R =ρ /a is equivalent noralized ode field radius, and ρ is equivalent ode field radius. EMGA leads to equivalent noralized ode field radius R estiation by solving a characteristic equation, which is derived fro propagation constant variational expression under following passage to square core ode paraeter U variational expression, written for analyzed wealy guiding optical waveguide with given refractive index profile. R is variational paraeter. It is basic for Gaussian approxiation and copletely defines ode transission paraeters. The ost well-nown ethods utilized variational approach with conventional Gaussian approxiation and their various odifications [, 8 4] are based on representation of optical fiber refractive index profile by siple power functions, while the real coercial fiber profiles have ore coplex for with local defects and refractive index fluctuations [3, 7]. Soe ethods solve this proble by using representation of profile or/and ode fields by series expansions together with adding other variational paraeters [ 5]. This leads to increasing the nuber of equations in characteristic set or nuber of series ters for coplex profiles and higher order odes. That is the ain reasonwhyostethodsforanalysisofopticalfibersare concerned with coputing of transission paraeters only for the fundaental ode. Even in the case of generalization of conventional Gaussian approxiation proposed by Snyder andlove(section5.6andtable(5.4)in[]) analytical variational expressions were obtained only for odes LP l with the radial order =due to eliination of Laguerre polynoial. Unlie the aboveentioned nown ethods, in EMGA the stratification ethod approach [3] isappliedforrepresentation of coplicated profile for corresponding to real coercial telecounication optical fibers. Here optical fiber with an arbitrary axial-syetric refractive index profile is considered as ulticladding optical fiber. Therefore, refractive index profile inside core region can be represented in the for of the set of N layers in which the refractive index stays a constant: n (R) = { n, { { n N, R = N, N <R +, () and any profile function f(r) can be written in ters of profile paraeter h : f (R) = { h, R = { N, N {, < R +, (a) h = n ax n nax, (b) n N n is refractive index of layer ( =,...,N), n ax is the axial core refractive index, and n N is cladding refractive index. This approach based on stratification ethod aes EMGAtobeversatilethatprovidesabilityofanalysisofreal coercial optical fibers with nonideal disturbed refractive index profile. Another proble occurring during a passage fro wellnown integral variational expressions for propagation constant or core ode paraeter (Equations (5.4) and (5.8) in []) to the analytical forulas for higher order odes is a presence of nonstandard integrals with product of Laguerre polynoials of nonequal degree and order in the nuerator. Here it is solved by representation of Laguerre polynoial in the for of finite power series according to its definition [6]. Proposed approach perits writing the variational expression for core ode paraeter U and characteristic equation for noralized equivalent ode field radius U / R =in the for of finite nested sus for any order ode LP l as follows: U = ( )! (l+ )! { Q R + N [X + = X = exp ( R ) X = exp ( N R h (X X )]} ; q= ) D q p= q= X = exp ( (+) N R ) Q+ N [S + = S = exp ( R ) S = exp ( N R p= (l + q)! p!r p q= q= ()! ; p!r p (l + q)! D q p= D q p= p!r p ()! p!r p p N p ; (+) p N p ; () (3) h (S S )] =; (4) ) (l + q)! D q p= p!r p ( pr ); q= D q p ( Np N pr ); S = exp ( (+) N R ) (+)p N p q= ( (+) N pr ); (l + q)! D q p= p!r p (l+ )! (3l + ) Q= ( )! +l D q (q+l )! 4l C q (q + l)!, q= q= (5)

4 4 Matheatical Probles in Engineering D q = C q = in(q, ) p=ax(,q +) in(q, ) p=ax(,q +) b (l, ) p b (l, ) q p ; (6a) b (l, ) p b (l+, ) q p ; (6b) and b (l,) p is coefficient of polynoial representation in the for of power series [5, 6]: (x) = q= b (l,) q x q ; (7a) b (l,) q = ( ) q (l+)! (l + q)! ( q)!q!. (7b) Therefore analysis of wealy guiding single-cladding optical fiber with an arbitrary profile leads to the following. Refractive index profile is represented by profile function () in the for of N layers. Fiber paraeters and ode orders l and are substituted to characteristic equation (4). By eans of nuerical solution (4), the noralized equivalent ode field radius R will be obtained. Then R is substituted into expression (), and ode core paraeter U is estiated, which perits evaluating the propagation constant β for guided ode LP l by the well-nown expression [ 6]: β = n ax U a. (8) Solution of characteristic equation (4) is correct under noralized frequency >,anditshouldsatisfytheguided ode cutoff condition [ 6]: n N <β n ax. (9) Optical confineent factor P core can be considered as the second criterion for identification of the ghost solutions: P (l) co.5. () By taing into account Gaussian approxiation paraeter P core is defined by analytical expression derived fro the generalized integral for for wealy guiding optical fibers presented in []: P (l) co = ( )! (l+ )! q= W q = exp ( R ) D q ()!W q, () p= p!r p. () Therefore EMGA provides coputation of transission paraeters of arbitrary order guided odes propagating over wealy guiding optical fibers with arbitrary axial-syetric refractive index profile under high accuracy deonstrated in [7] also due to taing into account researched fiber profile local distortions. 4. Polarization Correction to the Propagation Constant of Guided Modes in Wealy Guiding Optical Fiber with an Arbitrary Refractive Index Profile According to the aboveentioned, it is supposed that the optical fiber is analyzed by EMGA based on Gaussian approxiation. Therefore substitution of (9) into integral expressions (6a) and (6b) will lead the denoinator integral to the siple ratio of aziuthal and radial ode nubers l and [6]: ( R l ) [ R = (l+ )!. ( )! (R R )] exp ( R R )RdR (3) By applying Laguerre polynoial recurrence and derivation forulas [6] and carrying out awward algebraic transforations, the analytical expression for radial ode field distribution derivation (R)/ R in the nuerator of integral (6a) can be written in the for d dr = Rl R l [( R R +l) exp ( R R ) (R R =( l R + R R ) F (l+) R. ) R R L (l+) (R R )] (4) According to the aboveentioned, EMGA uses stratification ethod approach for refractive index profile representation inside the fiber core region by finite nuber of N layers. As a result profile function f(r) is defined by expression ().Bytaingintoaccountthegreatnuberoflayers(N> [7]) required for detailed description of real optical fiber refractive index profile, the derivative of profile function fro the nuerator integral of expression (6a) can be written in the following for: f = f(r +) f(r ) R R=R R + R = { N(h + h ), R = { N, N {, < R +. (5) As a result the nuerator integral of (6a) is led to finite su, and expression (6a) by tainginto account (3), (4),and(5) will be rewritten as follows: I = ( )! (Δ) 3/ (l+ )! 4a N N = (h + h )[ R R ]. R =/N (6)

5 Matheatical Probles in Engineering 5 δ b (%) HE λ = 3 n HE λ = 3 n Gauss Gauss + δβ (a) (b) Figure : Error reducing of calculation of the fundaental ode HE noralized propagation constant produced by EMGA by polarization correction under low value of noralized frequency: (a) relative error, with and without polarization correction; (b) difference of relative error curves. After substituting (4) into (6) the final analytic expression for integral I will be obtained: I = ( )! N (Δ) 3/ N (l+ )! ar [l+ R R R R = (h + h ){[ ] L(l+) (R /R ) (R /R ) ]}. R =/N (7a) By applying the sae approach to I,aftersubstituting(3) and (4) into (6b) the expression for integral I will be written in the following analytical for: I = l ( )! (Δ) 3/ (l+ )! 4a N N = 5. Results and Discussion (h + h ){[ ] }. R =/N (7b) The worst casefrothepointofviewofemgaapplication for analysis of optical fibers, ideal step index refractive index profile, will be considered for estiation of polarization correction influence on error reducing. According to [3, 4, 8], it corresponds to coputing of fundaental ode propagation constant β under the low values of noralized frequency near =.45. Therefore conventional silica step index optical fiber with pure silica cladding and 3.% Geraniu doped core was considered. Noralized frequency range =.5.7 was researched under wavelength λ = 3 n. Soe results of error coparison between exact solution of wave equation and approxiate values of noralized propagation constant obtained for the fundaental ode HE by EMGA without and by taing into account polarization correction are presented in Figure. Here polarization correction provides error reducing on.5.5%over researched low value noralized frequency range. The influence of polarization correction on the error reducing decreases under noralized frequency enhanceent. Further results of the higher order ode propagation constant estiation produced by exact solution and EMGA without and with polarization correction near the particular value of noralized frequency corresponding to ode cutoff condition were also copared. Noralized propagation constant relative error curves δ b () obtained for guided ode HE and their difference are presented in Figure. Here polarization correction provides reducing of relative error on.5.5% under the researched range of = 35. Following increasing value of noralized frequency also reduces influence of polarization correction: it decreases relative error not ore than..%. Following passage to higher order guided odes also deonstrates decreasing of polarization correction influence on propagation constant error reducing. Differences of relative error curves for guided higher order odes HE 3 and HE 4 are shown in Figure 3. Here it is less than.% for ode HE 3 and.4% for ode HE 4 even under values of near the cutoff condition corresponding to the entioned guided odes. Further the sae analysis was produced for step index ultiode optical fiber 5/5 also with pure silica cladding and 3.% Geraniu doped core over wavelength range λ = 8 35 n. Relative error curves calculated for the

6 6 Matheatical Probles in Engineering δ b (%) HE λ = 3 n δ b (%) Gauss Gauss + δβ HE λ = 3 n HE λ = 3 n Gauss Gauss + δβ (a) (b) Figure : Error reducing of calculation of the higher order ode HE noralized propagation constant produced by EMGA by polarization correction under low value of noralized frequency: (a) relative error, with and without polarization correction; (b) difference of relative error curves δ b (%) HE 3 λ = 3 n δ b (%) HE 4 9 λ = 3 n (a) (b) Figure 3: Difference of relative error curves of calculation of the higher order odes noralized propagation constant produced by EMGA with and without polarization correction: (a) HE 3 ode; (b) HE 4 ode. fundaental ode HE and their difference are represented in Figure 4.Herenoralizedfrequency reaches high values over researched wavelength range. That is why the error is so sall and is less than.3%, and polarization correction ensures its decrease to only.5.35%, while for higher order odes it helps to iprove accuracy up to..% depending on ode order. For exaple, differences of relative error curves for guided higher order odes HE 3 and HE 3 are shown in Figure 5.ForodeHE 3 the error reduces by..% duringwavelengthgrowing and corresponding noralized frequency decreasing under fixed core diaeter value, while for ode HE 3 it reaches.5.4%. According to obtained results, the ain influence of polarization correction δβ on reducing of noralized propagation constant calculation error corresponds to lower order guided odes under critical values of noralized frequency near cutoff condition (low core diaeter or wavelength

7 Matheatical Probles in Engineering HE 3.5 δ b (%) HE λ () λ () Gauss Gauss + δβ (a) (b) Figure 4: Error reducing of calculation of the fundaental ode HE noralized propagation constant produced by EMGA by polarization correction: (a) relative error, with and without polarization correction; (b) difference of relative error curves HE HE λ () (a) (b) λ () Figure 5: Difference of relative error curves of calculation of the higher order odes noralized propagation constant produced by EMGA with and without polarization correction: (a) HE 3 ode; (b) HE 3 ode. near ode cutoff wavelength). Under described conditions, relative error can be reduced by up to %. By taing into account the following passage to the first and second propagation constant derivatives β/ and β/ required for estiation of ode dispersion paraeters, it would uch reduce the errors of coputing of ode delay and chroatic dispersion of guided odes with particular order. Derived analytical expressions for polarization correction to the scalar propagation constant estiated by EMGA have analytical for and do not require high-perforance coputers. For exaple, coputing tie of any order guided ode exact propagation constant β exact by taing into account polarization correction δβ at the particular wavelength λ and the nuber of layers for refractive index profile representation in the fiber core region N > 3 is less than s under the following coputing resource paraeters: AMD Pheno II x4 965, 3.4 HHz, RAM 4 Gb, Microsoft Windows 7. Professional, and Matlab 6.. Derived analytical forulas for polarization correction δβ and its derivatives represented in Appendix provide

8 8 Matheatical Probles in Engineering both iproving of EMGA accuracy and ability of taing into account polarization effects and specifically polarization ode dispersion that is especially topical for probles of siulation of Multi-Gigabit data transission over fiber optic lins 6. Conclusion Based on EMGA fast and siple ethod for calculation of polarization correction to any order ode propagation constant in wealy guiding optical fiber with an arbitrary axial-syetric refractive index profile is introduced. Soe results of estiation of propagation constant coputing error decreasing by taing into account polarization correction are deonstrated. The ain influence of polarization correction δβ on reducing of propagation constant calculation error was noticed for lower order guided odes under critical values of noralizedfrequency near cutoff condition (low core diaeter or wavelength near ode cutoff wavelength). Derived analytical forulas for polarization correction δβ and its derivatives provide both iproving of EMGA accuracy and ability of taing into account polarization effects and specifically polarization ode dispersion that is especially topical for probles of siulation of Multi-Gigabit data transission over fiber optic lins. Appendix First and Second Derivatives of Polarization Correction A passage fro the propagation constant to ode delay and chroatic dispersion requires expressions for polarization correction first and second derivatives. First, let us rewrite forula (7a) in the copacted following for: G () I = ( )! N (l+ )! = =P(h + h )[ ] Q ; P= Q =[l+ R R R R N (Δ)3/ ar ; G (), (A.) L(l+) (R /R ) (R /R ) ]. R =/N (A.) It provides representing the first and second derivatives of the first coponent of polarization correction I also in the for of finite su: I = ( )! N G () (l+ )! ; = I = ( )! N G () (l+ )! =. (A.3) After differentiation and carrying out necessary transforations the derivatives of paraeter G () are led to the following for: G () + G () =P[F(l) ] Q( h + h ( P P + Q Q ) )+G() =P[ ] Q[( h + h ) + h ]+ G() +G () + + Q h + Q )+ ( P P + { P { P [ P ( P ) ] { [ [ ( F(l) ) ] ] + Q Q [ ] Q ( Q ) } }. } ( P P + Q Q ) (A.4) The derivatives of paraeter P are expressed as follows: P =P[ 3 Δ Δ R ( R +R )] ; P = P ( P ) +P{ 3 Δ Δ [ Δ ( Δ ) ] + R [ R ( R ) R ] + [ ( ) ]}. (A.5) First- and second-order derivatives of paraeter Q are defined as follows: Q = R R { L(l+) (R /R ) (R /R ) [ R R + (R /R ) (R /R ) L (l+) [L(l) (R R [L(l+) (R R )]] R R )]} ;

9 Matheatical Probles in Engineering 9 Q = R Q R + R R { [ ( R R R ) +F (l+) R + R R ( R ) ]+ (l + R R R ) R ] (R /R ) [L(l+) (R R + (R /R ) [ R R + (R /R ) )] [ R + F(l) R R ( R ) ]. (A.8) [L(l) (R R L(l+) (R /R ) (R /R ) )]] [ [L(l+) [L(l) (R R (R R )]] )] + L(l+) (R /R ) (R /R ) [ [ R R ( R R ) ] + (R /R ) [L(l) (R R )]]}. (A.6) Polarization correction coponent I first- and secondorder derivatives can be expressed analogously to I in the for of finite sus: I = ( )! N (l+ )! = G () ; I = ( )! N G () (l+ )! ; = I = ( )! N G () (l+ )! =, (A.9) By applying recurrent expressions of orthogonal polynoial order and differentiation forulas [5, 6], first and second derivatives of the Laguerre polynoials will be obtained: G () =P(h + h )[ ] ; [L(l) (R R )] = R R 3 L (l+) (R R ) R G () =P[F(l) ] ( h + h )+G() ( P P = R R 3 [L (l+) (R R ) (R R )] R ; + F (l) ); [L(l) (R R )] = R R 3 {L (l+) (R R ) R (A.7) G () =P[ ] [( h + h ) (R R ) R + R (3 + R R R ) R [L(l) [L(l+) (R R (R R )]. )]} Finally, by differentiating the radial ode field distribution, first- and second-order derivatives of are defined by the following expressions: = R R [R F (l+) R R (l + R )] ; ( P P + G() +G () + + ( P P h + )+ h ] + ) { P { P [ P ( P ) ] { [ [ ( F(l) ) } ] }. ]} (A.) = R R [ F(l+) R R F (l+) ( R ) By differentiating characteristic equation (4) the derivatives of the noralized ode field radius will be obtained:

10 Matheatical Probles in Engineering R = S ( /) + N = [(S S )(h ( /) + ( h /))] [S () + N = h (S () S () )], (A.) S () = exp ( R ) pr 4 ]; q= S () = exp ( N R ) S () q= D q p= (l + q)! p p!n p R p+3 = exp ( (+) N R ) q= pr 4 ]. ()! D q p= p!r p+3 [( pr ) [( N pr ) pr 4 ]; ()!(+) p D q p= p!n p R p+3 [( (+) N pr ) R = [ S () N + N = = h (S () S () )] { R (S () S () h )( +h N )+ (S S ) = ( h + h + h )+ ( R ) [S () N + }, = h (S () S () )] + S +S () R (A.) S () q= D q p= (l + q)! p p!n p R p+6 + R [pr4 34 (3p ) N 4 ]} ; = exp ( (+) N R ) q= {( N pr ) 3 (l + q)! (+) p D q p= p!n p R p+6 {( (+) 3 N pr ) + R 3 [pr4 (+)4 (3p ) N 4 ]}. (A.3) Finally the derivatives of profile paraeter h defined by forula (b) are deterined by the following expressions: h = nax [( h ) n ax n N n +h n N ]; h = nax [( h ) n ax n N n +h n N Conflict of Interests h ( n ax n N )]. (A.4) The author declares that there is no conflict of interests regarding the publication of this paper. S () = exp ( R ) q= + R [pr4 (3p ) 3]} ; S () = exp ( N R ) ()! D q p= p!r p+6 {( pr )3 References [] A. Snyder and J. Love, Optical Waveguide Theory, Chapan& Hall, London, UK, 983. [] H.-G. Unger, Planar Optical Waveguides and Fibres,Clarendon Press, Oxford, UK, 977. [3] M. J. Adas, An Introduction to Optical Waveguides,JohnWiley & Sons, New Yor, NY, USA, 98. [4] K. S. Chiang, Review of nuerical and approxiate ethods for the odal analysis of general optical dielectric waveguides, Optical and Quantu Electronics, vol. 6, no. 3, pp. S3 S34, 994.

11 Matheatical Probles in Engineering [5] D. Marcuse, Light Transission Optics, an Nostrand Reinhold, London, UK, 97. [6] K. Oaoto, Fundaentals of Optical Waveguides, Acadeic Press, San Diego, Calif, USA,. [7] A.. Bourdine and K. A. Yablochin, Investigations of refractive index profile defects of silica graded-index ultiode fibers of telecounication cables, Infocounication Technologies,vol.8,no.,pp. 7,. [8] H. Meher and S. I. Hosain, ariational approxiations for single-ode graded-index fibers: soe interesting applications, Optical Counications,vol.4,no.,pp.5 3, 3. [9] M.-S. Wu, M.-H. Lee, and W.-H. Tsai, ariational analysis of single-ode graded-core W-fibers, Lightwave Technology,vol.4,no.,pp. 5,996. []M.J.Holes,D.M.Spirit,andF.P.Payne, NewGaussianbased approxiation for odelling non-linear effects in optical fibers, IEEEJournalofLightwaveTechnology,vol.,no.,pp. 93, 994. [] A. Aniewicz and G.-D. Peng, Generalized Gaussian approxiation for single-ode fibers, Lightwave Technology, vol.,no.,pp. 7,99. [] M. I. Osanen and I.. Lindell, ariational analysis of anisotropic graded-index optical fibers, IEEE Lightwave Technology,vol.7,no.,pp.87 9,989. [3] R. Tewari, S. I. Hosain, and K. Thyagarajan, Scalar variational analysis of single ode fibers with Gaussian and soothedout profiles, Optics Counications, vol. 48, no. 3, pp. 76 8, 983. [4] A.Shara,S.I.Hosain,andA.K.Ghata, Thefundaental ode of graded-index fibres: siple and accurate variational ethods, Optical and Quantu Electronics, vol. 4, no., pp. 7 5, 98. [5]M.C.Kundu,S.Gangopadhyay,T.Basu,andS.N.Sarar, Evaluation of diensionless vector and scalar propagation constants for single-ode graded index fibers by a siple technique, Opti,vol.6,no.,pp.5 54,5. [6] I. Gradstein and I. Ryji, Tables of Integrals, GIFML, Moscow, Russia, 963. [7] A.. Bourdine and O. P. Deluhaetov, Calculation of transission paraeters of the launched higher-order odes based on the cobination of a odified Gaussian approxiation and a finite eleent ethod, Telecounications and Radio Engineering,vol.7,no.,pp. 3,3. [8] T. Begou, B. Bêche,N.Grossardetal., Marcatili sextended approach: coparison to sei-vectorial ethods applied to pedestal waveguide design, Optics A: Pure and Applied Optics, vol., no. 5, Article ID 553, 8.

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