Guided and Leaky Modes of Planar Waveguides: Computation via High Order Finite Elements and Iterative Methods

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1 PERS ONLNE, VOL. 6, NO. 7, 669 Guided and Leaky Modes of Planar Waveguides: Computation via High Order Finite Elements and terative Methods D. Stowell and J. Tausch Southern Methodist University, USA Abstract Guided and leaky modes of planar dielectric waveguides are eigensolutions of a singular Sturm-Liouville problem. This paper describes how this problem can be transformed into a quartic eigenvalue problem, which in turn can be converted into a generalized eigenvalue problem. Thus standard iterative methods, such as Arnoldi methods, can be used to compute the spectrum. We show how the shifts in the Arnoldi methods must be selected to obtain convergence to the dominant modes. n addition, by using high-order finite elements, the resulting solutions can be made extremely accurate. Numerical examples demonstrate the speed and accuracy as well as the stability of the method. 1. NTRODUCTON t is easy to characterize the modes of planar dielectric waveguides by an algebraic equation. There are numerous papers that discuss numerical methods that are based on finding the roots of the characteristic function, either by using the argument principle of complex analysis [1 3, 11], or by a continuation method [7]. However, the numerical solution of the equation suffers from numerical instabilities which are caused by the exponential scaling of the characteristic function. t is well known that for this reason the standard methods are often unreliable especially if there are many or thick layers present, if the frequency is large or if there are lossy layers, see, e.g., [5, 9, 1]. n [1], we presented a new variational formulation of the Sturm-Liouville problem that characterizes the guided and leaky modes. After discretization, the variational form reduces to either a quadratic or a quartic matrix eigenvalue problem. Solving the eigenvalue problem is numerically stable, even for waveguides with thick layers or arbitrary index profiles. n [1], we used a low-order discretization scheme and solved the eigenvalue problem by a direct method. This demonstrated the general feasibility of the approach, but the convergence of the higher-order modes is slow and the high cost of the numerical linear algebra limits the applicability to relatively simple structures. n the present paper we address the slow convergence problem by using piecewise high-order polynomial finite elements. Furthermore, we employ an iterative eigensolver that is capable to exploit the sparseness of the finite-element matrices. To ensure that an iterative method converges to a specific eigenvalue one has to shift the problem such that the selected eigenvalue is dominant. We will present a strategy for selecting the shift such that all dominant guided and leaky modes are found. We will conclude with some numerical results that demonstrate the usefulness of our improved method.. VARATONAL FORMULATON n this section, we briefly review the derivation of the variational formulation that characterizes the modes of a dielectric waveguide. For more details we refer to [1]. We consider an infinite medium where the refractive index n(x) = ɛ(x)µ is a function of x and we let z be the direction of propagation. n this case the modes are of the form E(x, z, t) = exp( iωt + iβz)φ(x)ŷ (TE wave) or H(x, z, t) = exp( iωt + iβz)φ(x)ŷ (TM wave). To simplify notations we only discuss the TE waves, as the treatment of TM waves is completely analogous and can be found in [1]. The lateral dependence of the mode is given by the function φ which satisfies φ (x) + ( k n (x) β ) φ(x) =, x R, (1) see, e.g., [8]. Equation (1) is a singular Sturm-Liouville problem, where the propagation constant β is the unknown eigenvalue. The spectrum consists of a discrete part, corresponding to guided modes, and a continuous part, corresponding to radiation modes. There is a third type of eigensolution, known as leaky modes. These are unbounded solutions of (1) that radiate energy away from the

2 PERS ONLNE, VOL. 6, NO. 7, 67 stack. For more information on leaky modes in planar waveguides we refer to the recently published survey articles [6] and [13]. The refractive index n(x) is a piecewise constant function with discontinuities at x,..., x J and function value n(x) = n j in the j-th layer. t follows that a mode has the general form φ(x) = { exp(iα x)φ(x ), x x cos(α j (x x j ))φ(x j ) + sin(α j (x x j ))/α j φ (x j ), x j x x j+1, exp( iα J (x w))φ(x J ), x x J, where α j is the lateral propagation constant given by α j = k n j β. By considering the exponential form of φ in the semi-infinite layers, it follows that the modes are solutions of the non-standard Sturm-Liouville problem φ (x) + ( k n (x) β ) φ(x) =, x (, w), (3) φ (x ) iα φ(x ) =, (4) φ (x J ) + iα J φ(x J ) =. (5) To derive the variational formulation, we first multiply (3) by a test function, use integration by parts and apply the boundary conditions (4), (5) to obtain ψ φ ( + k n β ) ψφ iαj ψ(xj )φ(w) iα ψ(x )φ(x ) = (6) Conceivably, one could discretize (6). However, the result would be a nonlinear eigenvalue problem. We circumvent this problem by making use of the change of variable suggested by []. ζ = 1 i ( ) k n β + k n J β. (7) () A bit of algebra reveals that ( ) δ α = i 4ζ + ζ and ( ) δ α J = i 4ζ ζ (8) where δ = k n J k n. From the definitions of α, α J we have β = k (n + n J ) α + α J. (9) f we let q(x) = k [ n (x) 1 (n + n J )], we can introduce the bilinear forms a(ψ, φ) = ψ φ q ψφ, () a ± (ψ, φ) = ψ(x J )φ(x J ) ± ψ(x )φ(x ). (11) with these notations the variational formulation of (3) is: Find ζ and φ such that for all test functions ψ δ 4 δ (ψ, φ) + ζ 16 4 a (ψ, φ) + ζ a(ψ, φ) + ζ 3 a + (ψ, φ) + ζ 4 (ψ, φ) = (1) holds. This is a quartic eigenvalue problem in the variable ζ. Once the eigenvalues ζ have been found, β and be recovered using (8) and (9). f the refractive indices in the semi-infinite layers are equal, then δ = and (1) reduces to the quadratic eigenvalue problem. a (ψ, φ) + ζa + (ψ, φ) + ζ (ψ, φ) =. (13)

3 PERS ONLNE, VOL. 6, NO. 7, DSCRETZATON AND MPLEMENTATON A discretization of (1) or (13) can be obtained by letting the trial and test function be restricted to a finite dimensional subspace span [ϕ 1,... ϕ n ]. This yields the quartic matrix eigenvalue problem: find ζ such that A + ζa 1 + ζ A + ζ 3 A 3 + ζ 4 A 4 (14) is singular. We note these matrices are sparse, since from (11) and from () A 1 = δ 4 diag( 1,,...,, 1), A 3 = diag(1,,...,, 1), (15) A (i, j) = δ4 16 (ϕ i, ϕ j ), A (i, j) = (ϕ i, ϕ j) (qϕ i, ϕ j ), A 4 (i, j) = (ϕ i, ϕ j ). (16) By using the companion matrix, the quartic eigenvalue problem is converted into an equivalent generalized eigenvalue problem Ax = ζbx, where A = and B =. (17) A A 1 A A 3 Since the matrices involved are sparse and large, the problem is ideally suited for solution by an iterative method. terative methods converge to eigenvalues near the extremes of the spectrum. However, by introducing the shift σ, (17) can be transformed to the equivalent shifted problem Bx = 1 (A σb)x. (18) λ σ An iterative method applied to (18) will produce approximations that converge to the eigenvalues near σ. By determining where the eigenvalues of interest are found in the complex ζ-plane we can find a good choice of σ. t is known that the eigenvalues of (1) accumulate at ζ = and ζ = ±i, see [13]. The ζ s near the accumulation points are not of interest because they lead to a large value of β. The condition for a guided mode is that α and α J are purely imaginary and negative for exponential decay of the mode in the semi-infinite layers. From (8) it follows that the corresponding ζ-values are real and satisfy ζ > δ/. A 4 R 1 σ 1 R R 3 σ Figure 1: Eigenvalue distribution for a typical quartic eigenvalue problem. Also shown are the search regions in the complex plane as well as a choice of shifts for the eigenvalue problem.

4 PERS ONLNE, VOL. 6, NO. 7, 67 The leaky modes have complex β and hence complex ζ. However, not every complex solution of (1) will be a leaky modes. Whether a complex mode is leaky or non-physical depends on the following conditions. For substrate leaky modes kn J < Re(β) < kn. For full leaky modes β satisfies Re(β) < kn J. See, [4, 8]. These conditions restrict the eigenvalue search to certain portions of the complex ζ plane. Such regions are depicted in Figure 1. The region labeled R 1 is the search region for the guided modes. R is the region for the substrate leaky modes, and R 3 is where eigenvalues associated with full leaky modes are to be found. The circle is of radius δ, where δ is defined in the previous section. The outer boundary of R 3 is given as a parameter, and allows the search to find all modes up to a given order. The inner boundary of R is also given as a parameter. The appropriate shifts can be chosen using an understanding of these regions. There are two important considerations in the choice of σ. First, we want to ensure that we find all dominant modes. Second, we want to avoid the accumulation point(s). One way to accomplish these objectives is to use two shifts, σ 1 and σ, and execute two searches. The first search will begin in R 1 and will include a portion of R. The second will begin in R 3 and will also include a portion of R. The search is complete when the union of the searches covers all three regions. To choose σ 1, use that fact that guided modes satisfy kn < β < kn max. We have found a good choice of σ 1 to be the value of ζ that corresponds with β = kn max. By (8) and (9), this gives σ 1 = B + B δ4 4 where B = k (n +n ) J k n max. Then, let σ = iσ 1. See Figure 1. We conclude with two comments. Generally speaking, the eigenvalues of interest in R are found near the outer radius. The inner radius can then be enlarged, if needed. f there is a question as to whether some propagation constants were missed in the search, the inner radius can be adjusted accordingly and the search run again. Finally, for the quadratic eigenvalue problem, since δ =, the outer radius of R shrinks to zero, there are no accumulation points, and therefore, the search is simplified. 4. NUMERCAL RESULTS We demonstrate our method on the waveguide structure with the following parameters (w j denotes the width of the jth layer): n = 1.5, n 1 = 1.66, n = 1.6, n 3 = 1.53, n 4 = 1.6, n 5 = 1. w 1 = w = w 3 = w 4 =.5, k = This structure has been studied in the literature [7]. n this case, since the characteristic function is stable, the propagation constants can be found using Newton s method. These values will be used for testing the method. We have implemented our method using polynomial bases of degrees 1, and 4. For the interpolation nodes on each element, the Lobatto nodes are chosen. The computed eigenvalues are then compared with the known values TE TE 1 TE TE 3 TE 4 TE 5-1 First Order Second Order Fourth Order Refinement (a) Refinement (b) Figure : Convergence results. The FE computed eigenvalues are compared with known, exact values. (a) Convergence results using piecewise polynomial basis functions of degrees one, two and four. (b) Convergence results for the first six modes. n this experiment, piecewise-quadratic polynomials were used for the basis functions.

5 PERS ONLNE, VOL. 6, NO. 7, 673 n the first experiment, we compare the convergence for different values of p. Figure (a) shows the convergence of T E 4 and T E 5 for p = 1,, 4. The experiment is designed so that at each step the matrices are of the same size for each order. For example, to begin, we have p = 1, n e = 4, p =, n e =, and p = 4, n e =. The number of elements is doubled at each refinement. Figure (b) shows results from he second experiment. Here we demonstrate the convergence of the first six eigenvalues of the same structure for p =. 5. CONCLUSON We have derived a method which is numerically stable and is able to compute all modes up to a given order. Since the discretized version is a polynomial eigenvalue problem, there is no need for a priori knowledge of the location of the modes in the complex plane. However, the physical properties of the modes can be used, in conjunction with an iterative method, to find the eigenvalues quickly and accurately. ACKNOWLEDGMENT This work was in part supported by the National Science Foundation under Grant DMS-915. REFERENCES 1. Anemogiannis, E. and E. N. Glytsis, Mutlilayer waveguides: Efficient numerical analysis of general structures, J. of Lightwave Tech., Vol., , Anemogiannis, E., E. N. Glytsis, and T. K. Gaylord, Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method and wavevector density method, J. of Lightwave Tech., Vol. 17, , Chen, C., P. Berini, D. Feng, S. Tanev, and V. Tzolov, Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media, Optics Express, Vol. 7, No. 8, 6 7,. 4. Chilwell, J. and. Hodgkinson, Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides, J. Opt. Soc. Amer. A, Vol. 1, No. 7, , Hsueh, W. and J. Lin, Stable and accurate method for modal analysis of multilayer waveguieds using a graph approach, J. Opt. Soc. Amer. A, Vol. 4, 85 83, Hu, J. and C. R. Menyuk, Understanding leaky modes: Slab waveguide revisited, Advances in Optics, Vol. 1, No. 1, 58 6, Petracek, J. and K. Singh, Determination of leaky modes in planar multilayer waveguides, EEE Photonics Tech. Letters, Vol. 14, No. 6, 8 81,. 8. Marcuse, D., Theory of Dielectric Optical Waveguides, Academic Press, New York and London, Mehrany, K. and R. Rashidian, Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures, J. Opt. Soc. Amer. B, Vol., , 3.. Smith, R. E., G. W. Forbes, and S. N. Houde-Walter, Unfolding the multivalued planar waveguide dispersion relation, EEE J. Quantum Elect., Vol. 9, No. 4, 31 34, Smith, R. E., S. N. Houde-Walter, and G. W. Forbes, Mode determination for planar waveguides using the four-sheeted dispersion relation, EEE J. Quantum Elect., Vol. 8, No. 6, , Stowell, D. and J. Tausch, Variational formulation for guided and leaky modes in multilayer dielectric waveguides, Technical Report, Southern Methodist University, Tausch, J., Mathematical and numerical techniques for open periodic waveguides, Wave Propagation in Periodic Media Analysis, Numerical Techniques and Practical Applications, n M. Ehrhardt, Editor, 51 74, Bentham, 9.