Analysis of graded-index optical fibers by the spectral parameter power series method
Raul Castillo Perez, Vladislav V. Kravchenko, Sergii M. Torba
aa r X i v : . [ phy s i c s . op ti c s ] A ug Analysis of graded-index optical fibers by the spectral parameterpower series method
Ra´ul Castillo-P´erez , Vladislav V. Kravchenko and Sergii M. Torba SEPI, ESIME Zacatenco, Instituto Polit´ecnico Nacional, Av. IPN S/N, C.P. 07738, D.F. Mexico Departamento de Matem´aticas, CINVESTAV del IPN, Unidad Quer´etaro, Libramiento Norponiente e-mail: [email protected], [email protected],[email protected] 12, 2018
Abstract
Spectral parameter power series (SPPS) method is a recently introduced technique [18],[19] for solving linear differential equations and related spectral problems. In the present workwe develop an approach based on the SPPS for analysis of graded-index optical fibers. Thecharacteristic equation of the eigenvalue problem for calculation of guided modes is obtainedin an analytical form in terms of SPPS. Truncation of the series and consideration in this wayof the approximate characteristic equation gives us a simple and efficient numerical method forsolving the problem. Comparison with the results obtained by other available techniques revealsclear advantages of the SPPS approach, in particular, with regards to accuracy. Based on thesolution of the eigenvalue problem, parameters describing the dispersion are analyzed as well.
Analysis of graded-index cylindrical waveguides, in particular of optical fibers, presents considerablemathematical and computational difficulties. The main differential equation involved is singular,the fact which restricts application of purely numerical techniques. The typically used approach isbased on the asymptotic WKB approximation (Wentzel–Kramers–Brillouin). It can offer analyticalrelations between the mode-propagation parameters (see, e.g., [21], [23, Sect. 3.7]). However, it iswell known (see, e.g., the discussion in [20]) that they are not accurate enough for the practicalstudy of optical fibers.We develop a completely new approach for analysis of the computationally difficult spectralproblems involved. It is based on the spectral parameter power series (SPPS) representation forsolutions of second-order linear differential equations of the Sturm-Liouville type. The SPPS repre-sentation for regular Sturm-Liouville equations was first obtained in [18] and applied to numericalstudy of spectral problems in [19]. It was used in a number of applications (we refer to the review[16]). In particular in [7] the SPPS method was applied to the study of wave propagation in layeredmedia. In [8] the SPPS representation was obtained for solutions of the singular differential equa-tions belonging to the class of perturbed Bessel equations. The main differential equation of thegraded-index cylindrical waveguides is of that type. In the present paper we use the main resultof [8] to develop a numerical method for analysis of graded-index optical fibers. The characteristicequation equivalent to the spectral problem for calculation of guided modes is obtained in an ex-1licit analytical form. The numerical method consists in approximating this equation and findingzeros of the approximate characteristic function.The paper contains several numerical tests. In the case of the well studied parabolic profilewe compare our results with the available exact solution, with the WKB approach as well as withthe results from [20] where the finite element method was applied with the results of the WKBapproximation used as the initial guess. Our method gives substantially more accurate values.We show that the SPPS method presented here allows one a quick and accurate analysis ofgraded-index optical fibers and can be used as an efficient tool for their design.
The basic wave equation governing the wave propagation in graded-index fibers has the form (see,e.g., [23, Sect. 3.7]) d dr ψ + 1 r ddr ψ + (cid:18) k n ( r ) − β − m r (cid:19) ψ = 0 , r ∈ (0 , a ] , (2.1)where the knowledge of the solution ψ allows one to compute the corresponding electromagneticfield, n ( r ) is the radial refractive index profile, k = 2 π/λ is the vacuum wave number, β is thepropagation constant, and m is a mode parameter given by m = ℓ = 0) ℓ + 1 EH mode ( ℓ ≥ ℓ − ℓ ≥ . As in most practical fibers the refractive index varies in the core but is constant in the cladding, itis usually convenient to solve the wave equation in the core and cladding separately and to matchthose solutions at the core-cladding boundary. If the solutions in the core and cladding are denotedby ψ ( r ) and ψ clad ( r ), respectively, the boundary conditions at the interface r = a under the weaklyguiding approximation (see, e.g., [23, p. 94]) are given by ψ ( a ) = ψ clad ( a ) and dψdr (cid:12)(cid:12)(cid:12)(cid:12) r = a = dψ clad dr (cid:12)(cid:12)(cid:12)(cid:12) r = a . (2.2)We note that for the method presented here it is not essential that the boundary conditions bethat simple. For example, they could be dependent on k .Introducing the function U ( r ) = √ rψ ( r ) we write equation (2.1) in the form − U ′′ + (cid:18) β + m − / r − k n ( r ) (cid:19) U = 0 . Finally, introducing the new independent variable x = r/a we obtain an equation for the function u ( x ) = U ( ax ), − u ′′ + (cid:18) a β + m − / x − a k n ( ax ) (cid:19) u = 0 . (2.3)The boundary conditions (2.2) take the form u (1) = u clad (1) and dudx (cid:12)(cid:12)(cid:12)(cid:12) x =1 = du clad dx (cid:12)(cid:12)(cid:12)(cid:12) x =1 . (2.4)2he solution for the cladding ( x >
1) has the form u clad ( x ) = C √ xK m (cid:16) a q β − k n x (cid:17) where K m is the modified Bessel function of the second kind, the constant n is the value of therefractive index in the cladding, and C is an arbitrary constant.The bounded solution u in the core is unique up to a multiplicative constant. Thus, from theboundary conditions (2.4) the following characteristic equation of the problem is obtained u ′ (1) u (1) = u ′ clad (1) u clad (1) , which then admits the form2 K m (cid:16) a q β − k n (cid:17) u ′ (1) − (cid:18) (1 + 2 m ) K m (cid:16) a q β − k n (cid:17) − a q β − k n K m +1 (cid:16) a q β − k n (cid:17)(cid:19) u (1) = 0 . (2.5)Pairs of values of β and k which satisfy this characteristic equation together with the light propa-gation condition k n < β < k n := k max ≤ x ≤ n ( ax ) (2.6)correspond to guided modes in the fiber and are the main object of computation. As we showbelow, the SPPS approach allows one to approximate directly the characteristic equation and tosolve it with a considerable accuracy and speed. Remark . Note that the characteristic equation (2.5) remains valid also for the case of absorbingmedia, i.e., when the refractive index n ( r ) has non-zero imaginary part. The only change is in thepropagation condition (2.6), it should be written asRe n < β k < max ≤ x ≤ Re n ( ax ) . The solution of the characteristic equation (2.5) gives us the dependence of k on β (or, since k = 2 π/λ = ω/c , the dependencies of λ and ω on β ). A way to employ the obtained informationis in the analysis of dispersion. The modal description of dispersion is related to the differentmode indices (or group velocities) associated with different modes. The group velocity associatedwith the fundamental mode is frequency dependent because of chromatic dispersion. As a result,different spectral components of the pulse travel at slightly different group velocities, a phenomenonreferred to as group velocity dispersion (GVD) which has two contributions: material dispersionand waveguide dispersion [2]. Group velocity v g is defined as [5] v g = dω/dβ (3.1)where ω stands for the frequency. If ∆ ω is the spectral width of the pulse, the extent of pulsebroadening for a fiber of length L is governed by [2]∆ T = ddω (cid:18) Lv g (cid:19) ∆ ω = L d βdω ∆ ω =: Lβ ∆ ω. (3.2)3here ∆ T stands for the differential group delay and the parameter β = d β/dω is known as theGVD parameter. In terms of the range of wavelengths ∆ λ emitted by the optical source, and byusing ω = 2 πc/λ and ∆ ω = ( − πc/λ )∆ λ , (3.2) can be written as∆ T = ddλ (cid:18) Lv g (cid:19) ∆ λ = DL ∆ λ, where the dispersion parameter D, expressed in units of ps/(km-nm), is given by D = ddλ (cid:18) v g (cid:19) = − πcλ β = − λ πc (cid:18) dβdλ + λ d βdλ (cid:19) . (3.3)Interest in studying fibers with different refractive index profiles comes from the fact that tayloringsuch profiles the dispersion, which is one of the main impairments for optical communications, canbe manipulated. Fibers where the dispersion parameter can be reduced at a certain wavelenght,flattened for a wide range of wavelenghts or even made highly negative are used for aplications con-cerning long haul communications, wavelenght division multiplexing and dispersion compensation,among others (see, e.g., [3],[22, Chapter 4]). The main equation (2.3) is of the form − u ′′ + (cid:18) m − / x + q ( x ) (cid:19) u = µr ( x ) u, x ∈ (0 , a ] , (4.1)where m ∈ N , q and r are known functions (which in general can be complex valued), µ is a spectralparameter. The equation belongs to the class of perturbed Bessel equations and was studied in aconsiderable number of publications (e.g., [6], [8], [9], [13], [17], [24]). In what follows we alwaysassume that q and r are piecewise continuous functions on the whole segment of interest with atmost a finite number of step discontinuities.The SPPS method applied to (4.1) consists of two steps. On the first step one needs to constructa nonvanishing on (0 , a ] bounded solution u of the equation − u ′′ + (cid:18) m − / x + q ( x ) (cid:19) u = 0 . (4.2)In [8] it was proved that if q ( x ) ≥ x ∈ (0 , a ] then such solution u exists and has the form u ( x ) = x m +1 / ∞ X k =0 e Y (2 k ) ( x ) , (4.3)where the functions e Y ( j ) are defined recursively as follows e Y (0) ≡ , e Y ( j ) ( x ) = Z x e Y ( j − ( t ) t m +1 q ( t ) dt, for odd j, Z x e Y ( j − ( t ) t − (2 m +1) dt, for even j. (4.4)4he series (4.3) converges uniformly on [0 , a ]. When q does not satisfy the nonnegativity conditionthe series (4.3) still defines a bounded solution of (4.2) but in general not necessarily nonvanishingon (0 , a ]. The derivative of u has the form u ′ ( x ) = (cid:18) m + 12 (cid:19) x m − / ∞ X k =0 e Y (2 k ) ( x ) + x − ( m +1 / ∞ X k =1 e Y (2 k − ( x ) . (4.5)On the second step the solution u is used for constructing the unique (up to a multiplicativeconstant) bounded solution of (4.1) for any value of µ . This solution has the form u ( x ) = u ( x ) ∞ X k =0 µ k e X (2 k ) ( x ) (4.6)where the functions e X ( j ) are defined recursively as follows e X (0) ≡ , e X ( j ) ( x ) = Z x u ( t ) r ( t ) e X ( j − ( t ) dt, if j is odd , − Z x e X ( j − ( t ) u ( t ) dt, if j is even . (4.7)The series (4.6) converges uniformly on [0 , a ]. The first derivative of u is given by u ′ = u ′ u u − u ∞ X k =1 µ k e X (2 k − , (4.8)where the series converges uniformly on an arbitrary compact K ⊂ (0 , a ].In (4.6) the solution is represented in the form of a power series with respect to the spectralparameter µ . Such representations were called in [19] the spectral parameter power series (SPPS).The representation (4.6) is based on a particular solution of equation (4.1) for µ = 0. Similarlyto the Taylor series, the approximations given by the truncation of the series (4.6) and (4.8) aremore accurate near the origin, while the accuracy deteriorates with the increase of the parameter µ , see [8, Example 7.7]. In [8, 19] it was mentioned that it is also possible to construct the SPPSrepresentation of a general solution starting from a non-vanishing particular solution for some µ = µ . Such a procedure is called a spectral shift and consists in the following. Equation (4.1)can be written in the form − u ′′ + (cid:18) m − / x + q ( x ) − µ r ( x ) (cid:19) u = ( µ − µ ) r ( x ) u (4.9)where µ is an arbitrary number. This equation is again of the form (4.1) but with the spectralparameter Λ := µ − µ . The SPPS technique applied to this equation leads to a representation ofthe solution in the form of a power series in terms of Λ and allows one to improve the accuracy ofthe approximations given by the truncated series (4.6) and (4.8) near the point µ . The spectralshift has already proven its usefulness for numerical applications [19], [15], [8] and is used below innumerical computations. Remark . Since equation (4.1) for each µ possesses a unique (up to a multiplicative constant)bounded solution, it might look difficult to find a µ providing a non-vanishing particular solution.The sufficient condition q ( x ) − µ ≥ µ . Allowing µ to be complex valued can solve the problem. Assumethat q is real valued and r ( x ) > , a ] (which is sufficient for the scope ofthe paper). Then every particular bounded solution of equation (4.1) having µ such that Im µ = 0is non-vanishing on (0 , a ]. Indeed, the equality u ( x ) = 0, x ∈ (0 , a ] contradicts the well-knownproperty (see, e.g., [26, Chapter 10.4]) of the operator τ u = − u ′′ + (cid:16) m − / x + q ( x ) (cid:17) u with asuitable domain to be self-adjoint in the Hilbert space L ((0 , x ] , r ) and have only real eigenvalues. Let the parameter β be fixed. Writing equation (2.3) as − u ′′ + (cid:18) m − / x + a β (cid:19) u = a k n ( ax ) u we obtain an equation in the form considered in Section 4 with the spectral parameter µ = a k andhaving known particular non-vanishing solution u = c √ xI m ( aβx ). However the direct applicationof the SPPS method for the obtained equation leads to certain difficulties. First, for the practicallyused fibers the value aβ is quite large (in the examples below it is within the range 20 − u . Second, since we are looking for the values of theparameter k satisfying (2.6), we cannot take advantage of the property of the SPPS representationto be the most accurate near the origin. To overcome these difficulties we apply the spectral shifttechnique.Let us assume that the refractive index n is real valued. By n we denote the maximum of n in the core. Note that n > n . The main equation (2.3) can be written in the following form − u ′′ + (cid:18) m − / x + a (cid:0) β − k n ( ax ) (cid:1)(cid:19) u = a (cid:0) k − k (cid:1) n ( ax ) u (5.1)where k = β/n . Then q ( x ) := a (cid:0) β − k n ( ax ) (cid:1) ≥ x ∈ (0 , µ := a (cid:0) k − k (cid:1) and r ( x ) := n ( ax ) we find that equation (5.1) is an equation of the form (4.1)with q satisfying the nonnegativity condition from the preceding section. Hence the solution u constructed as explained above does not have other zeros on [0 ,
1] except at x = 0 and the boundedsolution of (5.1) has the form (4.6).Consequently, the characteristic equation (2.5) can be written as follows2 K m (cid:16) a q β − k n (cid:17) u ′ (1) ∞ X k =0 µ k e X (2 k ) (1) − u (1) ∞ X k =1 µ k e X (2 k − (1) ! − (cid:18) (1 + 2 m ) K m (cid:16) a q β − k n (cid:17) − a q β − k n K m +1 (cid:16) a q β − k n (cid:17)(cid:19) × u (1) ∞ X k =0 µ k e X (2 k ) (1) = 0 . (5.2) Remark . Solution of equation (5.2) gives values of the parameter k for a fixed β . For someapplications it can be more convenient to find the values of the parameter β satisfying characteristicequation (2.5) for a fixed k . For this, equation (2.3) can be written in the form − u ′′ + (cid:18) m − / x + a (cid:0) β − k n ( ax ) (cid:1)(cid:19) u = a (cid:0) β − β (cid:1) u, where β = k n . Then q ( x ) := a ( β − k n ( ax )) ≥ x ∈ (0 ,
1] and the procedure describedabove can be applied. 6
Numerical implementation and examples
Based on the results of the previous sections we can formulate a numerical method for computingthe guided modes of the fiber. We suppose that some range for the parameter β is given, say β ∈ [ β , β ]. If instead a certain segment of wavelengths [ λ , λ ] is given, one can find the range of β ’s from (2.6): β ∈ (cid:20) πn λ , πn λ (cid:21) . (6.1)1. For each β belonging to a mesh over [ β , β ] perform steps 2–62. Start with m = 0 and perform steps 3–6 until no new propagation mode be found.3. Compute a particular solution u of equation (4.2) according to (4.3) as well as its derivativeaccording to (4.5).4. Use partial sums of the series (4.6) and (4.8) to obtain an approximation of the functionappearing on the left-hand side of the characteristic equation (5.2).5. Find its zeros satisfying propagation condition β n < k < β n . (6.2)6. If the interval for the spectral parameter µ in (5.1) defined by (6.2) is large (e.g., max µ > k found for different values of β to obtain dependencies k ( β ) for different propagation modes.Before considering numerical examples let us explain how the numerical implementation of theSPPS method was realized in this work. All calculations were performed with the help of Matlab2013b in the double precision machine arithmetics in a PC with Intel i7-3770 processor. The formalpowers e X ( j ) were calculated using the Newton-Cottes 6 point integration formula of 7 th order, see,e.g., [10], modified to perform indefinite integration. We choose M equally spaced points coveringthe segment of interest and apply the integration formula to overlapping groups of six points. Itis worth mentioning that for large values of the parameter m a special care should be taken nearthe point 0, because even small errors in the values of e X (2 j − after the division by u ∼ x m +1 lead to large errors in the computation of e X (2 j ) on the whole interval [0 , e X (2 j − in several points near zero to their asymptotic values.This strategy leads to a good accuracy. The computation of the first 100 to 200 formal powersproved to be a completely feasible task, and even for M being as large as several millions thecomputation time of the whole set of formal powers is within seconds. In the presented numericalresults we specify, among others, two parameters: N is the number of computed formal powers e X ( j ) ,and M is the number of points taken on the considered segment for the calculation of integrals.Using the calculated formal powers we evaluated the approximation of the characteristic equation(5.2), constructed a spline passing throw the obtained values and found its zeros using the Matlabcommand fnzeros . 7 .168 · · · · · - - - m Figure 1: The graphs of the absolute value of the characteristic function for the problem fromSubsection 6.1 for different values of m . As a first case of study, we present the computation of the propagation constants β for a gradedindex multimode fiber with a refractive index given by n ( x ) = ( n (1 − n x α ) / , ≤ x ≤ n x > n and n is defined as∆ n = n − n n . For α = 2 the exact solution of the problem is known allowing us to verify the accuracy of theproposed method. The specific values that were used in order to be able to compare our resultswith the ones in the literature and with an exact solution were n = 1 . n = 1 . a = 25 µ m,and λ = 0 . µ m. The parameters associated to the numerical implementation of the SPPS methodincluded the approximation of the functions with M = 100001 points and using N = 100 formalpowers e X ( j ) for the truncated series. For this example it was natural to apply Remark 5.1. Sincethe range of the new spectral parameter µ = a ( β − β ) defined by (2.6) is approximately [0 , µ n = (200 + 4 i ) n . On each step a 1000 points mesh was usedto approximate the characteristic equation.Our program has found 121 propagation constants ending with m = 20, the overall computationtime was about 3 minutes. The number of modes is in a good agreement with that obtained fromthe WKB approximation [23, 3.7.2]. Indeed, taking into account that there are 2 conventionalmodes corresponding to m = 0 and 4 corresponding to m ≥ m Exact SPPS0 1.177122819807467 1.1771228198074670 1.175978672140716 1.1759786721407160 1.174833410211082 1.1748334102110830 1.173687030756757 1.1736870307567550 1.172539530501824 1.1725395305018180 1.171390906243938 1.1713909062439310 1.170241157428142 1.1702411574246990 1.169090334155869 1.1690903343453290 1.167939131914112 1.1679391339031060 1.166794286172046 1.1667942940973070 1.165728772787623 1.16572880826658510 1.171390906148080 1.17139090614808010 1.170241154528824 1.17024115452882410 1.169090277987715 1.16909027798771410 1.167938398976696 1.16793839897669510 1.166787247703049 1.16678724770305010 1.165655359303853 1.165655359303857(b) (c)Table 1: Mode-propagation constants for the fiber from Subsection 6.1 computed using the SPPSmethod, exact values and the results from [20] obtained using WKB and FEM methods. Table (a):first 20 modes; table (b): modes from 36 to 50; table (c): modes corresponding to m = 0 (“themost inaccurate”) and to m = 10 (last m for which Wolfram Mathematica could find the exactcharacteristic equation). All values must be multiplied by 10 m − .9odes. In Table 1 we show our results together with the exact values and the values from [20]obtained with the use of the WKB and the Finite Element (FEM) methods. The exact values werecomputed with the help of the commands DSolve and
FindRoot from Wolfram Mathematica 8.0.It is worth mentioning that Mathematica was only able to compute the propagation constants formodes up to m = 10. That explains the missing exact values for modes 42 and 49 in Table 1. Theoutstanding accuracy achieved by the SPPS method can be appreciated. In Table 1 (c) we mentionthat the propagation constants obtained for m = 0 were the most inaccurate. Such phenomenonhas already been observed in [8, Example 7.5] and the accuracy can be improved by taking morepoints to represent the formal powers e X ( j ) . The accuracy of the obtained propagation constants forlarger values of m was excellent even taking less points representing the formal powers. It is alsoremarkable that the propagation constants conform clusters so that the modes in the same groupwill propagate with very similar constants. We illustrate this additionally on Figure 1 where thegraphs of the absolute value of the characteristic function (2.5) are presented for different values of m . The SPPS method allows us not only to find the propagation constants for a particular wavelengthbut also to analyze the behavior of the propagation constants for a range of wavelengths.For a numerical illustration we considered the profile (6.3) for α = 1 (the so-called triangularprofile) with n = 1 . n = 1 . a = 12 . µ m. On Figure 2 we present the graphs of thenormalized propagation constant b = ( β /k ) − n n − n versus normalized frequency V = ak p n − n for different propagation modes. The parametersof the SPPS implementation were M = 10001, N = 100 and we used 140 different values of V to calculate the propagation constants. We refer the reader to [12] where the normalized cut-offfrequency V ≈ .
381 of the single-mode operation was computed for the triangular profile fiber.Our results agree with this value.
The SPPS method was used to evaluate the group velocity (3.1) and the dispersion coefficient (3.3)for the triangular profile (with the refractive index defined by (6.3) with α = 1). The parametersof the fiber were taken from [3] and were the following: n = 1 . , n = 1 . , a = 3 . µ m.For the computation we used 1 . ≤ λ ≤ . µ m, 1000 values for k , 100 values for β , M = 10001and N = 100. The elapsed time was 0.56 minutes. The group velocity is shown on Figure 3, andthe dispersion parameter is shown on Figure 4. In both graphs the refractive index profile of thefiber was included in a subplot. The refractive index depends on the wavelength. Such dependence is usually described by theSellmeier equation, see, e.g., [1, (7.126)] or by its generalizations allowing graded-index fiberswhere dopant concentrations depend on the radius, see [11], [14], [25]. The SPPS method appliedas described in Remark 5.1 can be used to study the combined material and waveguide dispersion(chromatic dispersion).As an example we considered a dispersion-flattened triple clad fiber (fiber 2 from [4]). Thedopant concentrations of the fiber are the following: core – 9.1 m/o P O , cladding 1 – 13.510 b −1 −0.5 0 0.5 11.441.46 Refractive index profilex n ( x ) LP mℓ mode
01 11 02 2112 3103
Figure 2: Normalized propagation constant for different propagation modes for the triangular profilefrom Subsection 6.2. −6 Group velocity λ v g LP LP −1 0 11.511.521.53 Refractive index profilex n ( x )
11 01
Figure 3: Group velocity for a fiber having a triangular refractive index profile from Subsection6.3. The refractive index profile is displayed in the embedded figure.11 .9 1 1.1 1.2 1.3 1.4 1.5 1.6x 10 −6 −300−250−200−150−100−500 Dispersion Coefficient λ [m] D [ p s / ( k m n m ) ] LP LP −1 0 11.511.521.53 Refractive index profilex n ( x ) Figure 4: Dispersion coefficient for a fiber having a triangular refractive index profile from Subsec-tion 6.3. The refractive index profile is displayed in the embedded figure.m/o B O , cladding 2 – quenched silica, cladding 3 – 4.1 m/o GeO , the radiuses are r = 2 . µ m, r = 3 . µ m and r = 4 . µ m correspondingly. The Sellmeier coefficients for all mentioned materialswere taken from [1, Table 7.3]. For the computation we used 1 . ≤ λ ≤ . µ m, 100 values of k , 200 values for the search of β ’s, M = 10001 and N = 100. The elapsed time was 0.2 minutes.Dispersion coefficient is shown on Figure 5 , and the group velocity is shown on Figure 6. In bothgraphs the refractive index profile of the fiber was included in a subplot. As can be seen from thegraph, the dispersion is within ± · nm) for 1 . ≤ λ ≤ . µ m. The SPPS method is developed and applied to problems of wave propagation in graded-index opticalfibers. The numerical examples presented show that the method provides a remarkable accuracyachievable within seconds. It shares with the widely used WKB approach the possibility to workwith an analytic representation for the solution and for the characteristic function of the problem,offering at the same time the accuracy superior to other available purely numerical methods suchas finite element method. We believe that the SPPS approach will become a standard tool foranalysis and design of optical fibers and other inhomogeneous cylindrical waveguides.
Acknowledgements
R. Castillo would like to thank the support of the SIBE and EDI programs of the IPN as well asthat of the project SIP 20140733. Research of V. Kravchenko and S. Torba was partially supportedby CONACYT, Mexico via the projects 166141 and 222478.
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