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Analysis of Optical Waveguide Structures by Use of a Combined Finite-Difference

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Analysis of Optical Waveguide Structures by Use of a Combined Finite-Difference 610 J. Opt. Soc. Am. A/Vol. 19, No. 3/March 2002 J. W. Wallace and M. A. Jensen Analysis of optical waveguide structures by use of a combined finite-difference/ finite-difference time-domain method Jon W. Wallace and Michael A. Jensen Department of Electrical and Computer Engineering, 459 Clyde Building, Brigham Young University, Provo, Utah 84602 Received May 4, 2001; revised manuscript received August 7, 2001; accepted August 7, 2001 We present a method for full-wave characterization of optical waveguide structures. The method computes mode-propagation constants and cross-sectional field profiles from a straightforward discretization of Max- well’s equations. These modes are directly excited in a three-dimensional finite-difference time-domain simu- lation to obtain optical field transmission and reflection coefficients for arbitrary waveguide discontinuities. The implementation uses the perfectly-matched-layer technique to absorb both guided modes and radiated fields. A scattered-field formulation is also utilized to allow accurate determination of weak scattered-field strengths. Individual three-dimensional waveguide sections are cascaded by S-parameter analysis. A com- plete 104-section Bragg resonator is successfully simulated with the method. © 2002 Optical Society of America OCIS codes: 000.3860, 000.4430, 060.2310, 230.1480, 230.7370. 1. INTRODUCTION method to provide realistic results for this numerically sensitive problem provides evidence that the method may The ability to design complex optical devices is greatly en- be applied to a wide variety of structures. hanced by simulation tools that characterize wave propa- gation in arbitrary guiding structures. Complete electro- magnetic characterization of such geometries requires two basic tools: (1) a two-dimensional simulation tech- 2. FINITE-DIFFERENCE MODE SOLUTION nique capable of finding propagation constants and mode profiles for arbitrary guiding structures, and (2), a three- Finite-difference methods based on discretizations of the dimensional (3D) electromagnetic analysis method to time-harmonic Helmholtz equation have been success- fully applied for vectorial-mode extraction.5 However, simulate mode propagation in the presence of waveguide such methods often require special treatment of material transitions and discontinuities. For electrically large boundaries, and the mode solutions obtained often devi- structures, limited computer memory and processing ate slightly from modes supported by other discretiza- power often preclude a complete full-wave 3D analysis. tions (FDTD, for example). Full vectorial-mode solutions In many cases, however, simulation efficiency can be have also been demonstrated by applying a Fourier trans- greatly enhanced through appropriate application of net- form to the FD-vector beam propagation method.6 work analysis. Here we employ a straightforward discretization of the Prior work in this area has focused on two-dimensional time-harmonic form of Maxwell’s equations, using Yee’s planar waveguide approximations,1,2 the approximate discretization scheme. An obvious advantage is natural beam-propagation method,3,4 and finite-difference (FD) compatibility with subsequent 3D FDTD simulations. solutions for propagation constants and mode profiles.5,6 The method is free of spurious modes since the FDTD In this paper we employ a simple FD method based on a gridding scheme automatically satisfies Maxwell’s diver- straightforward discretization of Maxwell’s equations to gence relations. Also, if material parameters are speci- determine mode characteristics for waveguides of arbi- fied for each FDTD cell, the gridding arrangement en- trary cross-sectional geometry. The equations may be sures satisfaction of appropriate field continuity solved using sparse eigenvalue/eigenvector methods or it- conditions across material boundaries. eratively by using a sparse linear-equation solver. These modes are used in a full-wave analysis of 3D structures 7,8 with the finite-difference time-domain (FDTD) method A. Discretization of Maxwell’s Equations with Berenger’s perfectly-matched-layer (PML) absorbing Assuming exp( j␻t) time variation and exp(Ϫj␤ z) longitu- 9 z boundary condition. To characterize optically large dinal spatial variation for a propagating mode, we may waveguide devices, S-parameter analysis is employed to write Maxwell’s equations for a nonmagnetic medium as combine the response of individual 3D sections. We demonstrate the three-step method by analyzing an ץ 1 Ez electrically large buried heterostructure Bragg resonator H ϭ ͩ j Ϫ ␤ E ͪ , z y ץ x employing a surface relief grating. The ability of the k0 y 0740-3232/2002/030610-10$15.00 © 2002 Optical Society of America J. W. Wallace and M. A. Jensen Vol. 19, No. 3/March 2002/J. Opt. Soc. Am. A 611 Ϫ ץ 1 Hz 1 Hz, iϩ1, j Hz, ij E ϭ ͩ ␤ H Ϫ j ͪ , E ϭ ͩ j Ϫ ␤ H ͪ , y,ij ⑀ ⌬ z x, ij ץ x ⑀ z y rk0 y rk0 x ץ 1 Ez H ϭ ͩ ␤ E Ϫ j ͪ , Ϫ Hy, iϩ1, j Hy, ij 1 ץ y z x k0 x jE ϭ ͩ z,ij ⑀ ⌬ rk0 x ץ 1 Hz ϭ ͩ Ϫ ␤ ͪ H Ϫ H E j H , ϩ z x x, i, j 1 x, ij ץ ⑀ y rk0 x Ϫ ͪ , (2) ⌬y ץ ץ ץ ץ j Ey Ex j Hx Hy H ϭ ͩ Ϫ ͪ , E ϭ ͩ Ϫ ͪ , ⑀ ץ ץ ⑀ z ץ ץ z k0 x y rk0 y x where r now represents the relative permittivity aver- (1) aged over one cell centered about the component on the where E denotes electric-field intensity, H represents left-hand side in each equation. magnetic-field intensity multiplied by the free-space in- ␩ ͕ ͖ B. Boundary Truncation Conditions trinsic impedance 0 , and the subscripts x,y,z denote ϭ ␻ͱ␮ ⑀ The FD method requires a truncation condition for the field polarization. Also, k0 0 0 is the free-space ⑀ field components at the outer domain boundary. Simple wave number and r is the material relative permittivity. To discretize these equations, we use the standard Yee Neumann or Dirichlet conditions are often employed. grid assignment8 collapsed onto a two-dimensional sur- However, such nonphysical boundaries may induce unac- face as shown in Fig. 1. After applying first-order central ceptable error when placed near the guiding structure. differences we obtain To develop an approach that more closely matches the true field behavior, we note that for a cylindrical wave- Ϫ 1 Ez, ij Ez, i, jϪ1 guide the field profile for the HE11 mode decays outside of H ϭ ͩ j Ϫ ␤ E ͪ , (1) ␣␳ ␣ ϭ ␤2 Ϫ ␻2␮ ⑀ 1/2 x,ij k ⌬y z y, ij the core as H1 ( j ), where ( z 1 1) . For 0 ␣␳ Ͼ 2, the function is approximately (1/ͱ␳)exp(Ϫ␣␳), Ϫ 1 Ez, ij Ez, iϪ1, j which provides an approximate functional relationship H ϭ ͩ ␤ E Ϫ j ͪ , between fields that lie on the boundary and those just in- y,ij k z x, ij ⌬x 0 side the boundary. As shown in Subsection 2.E, use of Ϫ this boundary condition significantly reduces the required 1 Ex, ij Ex, i, jϪ1 jH ϭ ͩ size of the computational grid for a given level of accuracy. z,ij ⌬ k0 y Similar localized boundary conditions have been em- Ϫ ployed in the finite-element method for open-boundary Ey, ij Ey, iϪ1, j Ϫ ͪ , waveguides.10 ⌬x Ϫ C. Iterative Linear Method 1 Hz, i, jϩ1 Hz, ij E ϭ ͩ ␤ H Ϫ j ͪ , One approach to solving Eq. (2) is to construct a vector of x,ij ⑀ z y, ij ⌬ ϭ rk0 y field components to obtain the matrix equation Af f0 , ϭ T where f ͓͕Hx,ij͖͕Hy,ij͖͕ jHz,ij͖͕Ex,ij͖͕Ey,ij͖͕ jEz,ij͖͔ ͕ ͖ and • represents a stacking operation. Solving this equation using matrix inversion requires that the forcing ␤ vector f0 be nonzero and z be fixed. We can make the forcing vector nonzero by fixing one of the field compo- ␤ nents at a specific node. We then find the value of z ʈ ␤ ʈ that minimizes the vector norm AЈ( z)f , where AЈ is the coefficient matrix obtained when no forcing is in ef- fect. The minimization is performed by optimization af- ␤ ter specifying an initial guess for z from an approximate analytical solution or using the eigenvalue/eigenvector method outlined in Subsection 2.D. The simulations were performed on a 700-MHz Pentium-based PC with 512 megabytes of memory. The SuperLU package was used to compute sparse linear ma- trix equation solutions. For a 60 ϫ 120 FD grid, each it- eration required 15 s. Approximately 100 iterations were required to obtain 10Ϫ9 accuracy in the propagation constant. D. Eigenvalue/Eigenvector Method Fig. 1. Waveguide geometry used to assess error in the propa- Substituting the expressions for jHz and jEz in Eq. (2) gation constant and field profile found with the FD method. into the remaining four equations and rearranging those Also shown is the assignment of FD grid indices. equations produces the relations 612 J. Opt. Soc. Am. A/Vol. 19, No. 3/March 2002 J. W. Wallace and M. A. Jensen Table 1. Propagation Constant Value and Fractional Error for Various Sizes of the Simulation Domain Cells Size (nm) Zero Bound Decay Bound ␤ ␭ ␤ ␭ Nx Ny xy z 0 Fractional Error z 0 Fractional Error 36 48 2700 2880 19.96636646 2.37 ϫ 10Ϫ4 19.96168858 2.51 ϫ 10Ϫ6 48 64 3600 3840 19.96267660 5.20 ϫ 10Ϫ5 19.96164637 3.96 ϫ 10Ϫ7 60 80 4500 4800 19.96186515 1.13 ϫ 10Ϫ5 19.96163969 6.08 ϫ 10Ϫ8 72 96 5400 5760 19.96168746 2.43 ϫ 10Ϫ6 19.96163867 9.63 ϫ 10Ϫ9 84 112 6300 6720 19.96164903 5.04 ϫ 10Ϫ7 19.96163851 1.55 ϫ 10Ϫ9 96 128 7200 7680 19.96164075 8.94 ϫ 10Ϫ8 19.96163848 2.26 ϫ 10Ϫ10 108 144 8100 8640 19.96163896 0.00 ϫ 100 19.96163847 0.00 ϫ 100 Fig.
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