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 simulation 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 complete 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 propagation in arbitrary guiding structures. Complete electromagnetic 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 successfully 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

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 effect. 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 matrix 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. 2. Error in the mode Ex field profile for the zero-field boundary condition. The dashed line gives the dimensions of the smallest domain for comparison.

Fig. 3. Error in the mode Ex ﬁeld proﬁle for the decay boundary condition. The dashed line gives the dimensions of the smallest domain for comparison.

Ϫ Ϫ Ϫ ϩ 1 Hy,iϩ1, j Hx,i, jϩ1 Hx,ij 1 Ex,iϩ1, j Ex,iϩ1, jϪ1 Ex,ij Ex,i, jϪ1 ␤ E ϭ ͩ Ϫ ␤ H ϭ ͩ z x, ij ⌬ ⑀z ⌬ ⑀z ⌬ z x, ij ⌬ ⌬ k0 x r,ij x r,ij y k0 x y ϩ H Ϫ H Ϫ ϩ Ϫ H Ϫ Ey,iϩ1, j Ey,iϪ1, j ϩ y,i 1, j ϩ x,i 1, j 1 x,i 1, j ͪ Ϫ ͪ ⑀z ⌬ ⑀z ⌬ ⌬ r,iϪ1, j x r,iϪ1, j y x 1 1 1 2 ϩ Ϫ ϩ ϩ ͫ Ϫ ⑀y ͬ ͫ k ͩ ͪͬH , r,ijk0 Ey,ij , 0 ͑⌬ ͒2 ⑀z ⑀z y,ij k ͑⌬x͒2 k0 x r,ij r,iϪ1, j 0 Ϫ 1 Hy,iϩ1, j Hy,ij Hx,i, jϩ1 1 E ϩ E ␤ E ϭ ͩ Ϫ x,i, jϩ1 x,i, jϪ1 z y, ij z z ␤ ϭ ͩ ⌬ ⑀ ⌬ ⑀ ⌬ zHy, ij k0 y r,ij x r,ij y ⌬ ⌬ k0 y y Ϫ Hy,iϩ1, jϪ1 Hy,i, jϪ1 Hx,i, jϪ1 Ϫ ϩ Ϫ Ϫ Ϫ ͪ Ey,iϪ1, jϩ1 Ey,i, jϩ1 Ey,ij Ey,iϪ1, j ⑀z ⌬ ⑀z ⌬ ϩ ͪ r,i, jϪ1 x r,i, jϪ1 y ⌬x 1 1 1 2 ϩ ͫ ͩ ϩ ͪ Ϫk ͬH , x 0 x,ij ϩ ͫ ⑀ Ϫ ͬ 2 z z r,ijk0 Ex,ij , (3) k ͑⌬y͒ ⑀ ⑀ Ϫ ⌬ 2 0 r,ij r,i, j 1 k0͑ y͒ J. W. Wallace and M. A. Jensen Vol. 19, No. 3/March 2002/J. Opt. Soc. Am. A 613

⑀͕x,y,z͖ where r,ij represents the relative permittivity aver- nent of the computed field profile for three of the cases aged about the Ex,ij , Ey,ij ,orEz,ij component, respec- considered. tively. For small cell sizes, the terms containing (⌬x)2 and (⌬y)2 lead to coefficient matrices that have poor nu- F. Bragg-Resonator Guided-Mode Solution merical conditioning and consequently sparse eigenvalue Figure 5 shows the Bragg-resonator geometry under con- solvers such as ARPACK have difficulty converging. For sideration. This geometry is based on the buried hetero- fine detail, the eigenvalue method may be used on a structure distributed feedback device cross section given ␤ coarse grid to obtain an initial guess for z that subse- in Ref. 12 with parameters specified at a physical tem- quently may be refined with the linear method. This ap- perature of T ϭ 25 °C and a wavelength of ␭ ϭ 1.55 ␮m. proach is presented in Ref. 11 for a cylindrical waveguide. The index of refraction13,14 for the cladding (InP) at this ϭ In this paper, initial guesses for propagation constants temperature is n2 3.15. The core (Ga1ϪxInxAsyP1Ϫy) were computed on a 30 ϫ 60 FD grid using ARPACK, re- is assumed to be matched to the InP lattice with x ϭ 0.2 ϭ ϭ quiring approximately one minute of computation. and y 0.43, giving n1 3.35 (Ref. 15). The refractive index in the grating (n3) is a function of E. Accuracy of the Mode Solution longitudinal position. For the true physical geometry, ϭ ϭ In this section the linear solution method is applied to a the grating would have n3 n1 3.35 and the height rectangular dielectric waveguide whose cross section would be modulated (surface relief ). However, modeling matches the core of the Bragg resonator that will be ana- such fine detail significantly increases the memory re- lyzed in Subsection 2.F. Here, we assess the accuracy of quired for simulation. Instead, we use alternate values the propagation constant and field shape for the propa- for n3 of 3.35 and 3.33 for our grating modulation, which gating mode with respect to boundary-truncation type, corresponds to a 10% modulation in the grating height domain size, and cell size. The basic geometry to be when viewed in terms of an effective permittivity of the simulated is depicted in Fig. 1. Note that error in this grating. Mode solutions are required only in the un- ϭ section is quantified as the fractional deviation from the modulated region (n3 3.35). most accurate numerical solution obtained (largest-size The propagation constants and modal field profiles domain/finest resolution). were computed at six different wavelengths by using the iterative linear method to refine the solution from the 1. Domain Size Dependence eigenvalue/eigenvector method. The wavelengths and Table 1 compares the fractional error in the propagation propagation constants are given in Table 3. FDTD simu- constant for various grid sizes with the zero-field bound- lations required approximately2htosimulate 10 periods ary and decay boundary. The cell size is held constant of modal oscillation. for all computations. Figures 2 and 3 plot the error in the Ex-field component for the zero and the decay bound- 3. THREE-DIMENSIONAL SIMULATIONS ary conditions, respectively. This comparison shows that OF BRAGG-RESONATOR SECTION the error produced by the decay condition is an order of The Bragg resonator can be divided into sections, each magnitude lower than that produced by the zero- having the geometry depicted in Fig. 6. The section truncation condition. The results also indicate that a modest domain size (4500 nm ϫ 4800 nm) gives reason- Ϫ Table 2. Propagation Constant Value and Error ably small error in the propagation constant (10 7) and Ϫ for Various Simulation Cell Sizes field profile (10 3). For all further computations, the decay-truncation approach will be used. ␤ ␭ Nx Ny xy z 0 Fractional Error

Ϫ 2. Cell Size Dependence 30 40 4500 4800 19.95706883 Ϫ2.94 ϫ 10 4 Ϫ Table 2 lists the fractional error in the propagation con- 60 80 4500 4800 19.96163969 Ϫ6.47 ϫ 10 5 Ϫ stant for various cell sizes when the domain size is held 90 120 4500 4800 19.96245221 Ϫ2.40 ϫ 10 5 Ϫ constant. These values show that the propagation con- 120 160 4500 4800 19.96273238 Ϫ9.92 ϫ 10 6 Ϫ stant is more sensitive to the discretization than to the 150 200 4500 4800 19.96286096 Ϫ3.48 ϫ 10 6 180 240 4500 4800 19.96293039 0.00 ϫ 100 domain size. Figure 4 shows the error in the Ex compo-

Fig. 4. Error in the mode Ex ﬁeld proﬁle for the decay boundary condition for various grid resolutions. 614 J. Opt. Soc. Am. A/Vol. 19, No. 3/March 2002 J. W. Wallace and M. A. Jensen

ous derivatives are assumed in z and time. To ensure complete compatibility of the mode shape, we must solve Eq. (1) with finite differences in time and space. Assuming time-harmonic fields, the finite difference (Dt) of any field quantity (F) with respect to time ␻ ⌬ ϭ ␻ ␻⌬ is Dt͓F(x, y, z)exp( j n t)͔ j sinc( t/2) F(x, y, z) ϫ ␻ ⌬ ϭ exp( j n t), where sinc(x) sin(x)/x. Thus k0 in Eq. ␻⌬ (2) must be replaced with k0 sinc( t/2). Similarly, application of finite differences in the propagation direction ␤ ␤ ␤ ⌬ requires replacement of z with z sinc( z z/2). These replacements will be used in Subsection 4.B to assess the accuracy of the complete Bragg solution.

C. Dynamic Range: Scattered-Field Formulation for Waveguides For many waveguide geometries, such as a small section of a Bragg resonator, the field scattered by the obstacle may be significantly weaker than the incident field. In these situations, finite precision arithmetic may produce

Fig. 5. Waveguide cross section for the Bragg resonator. n3 unacceptable error levels. takes on discrete values in the longitudinal direction. To minimize the dynamic range requirements as well as remove the need to absorb the strong incident field at length of 120 nm was chosen since it is approximately ␭/4 the domain boundaries, we utilize the scattered-field for- for the guided mode. mulation in the FDTD implementation.8 This approach, FDTD7,8 was chosen for simulating the 3D Bragg reso- which has been used extensively to model electromagnetic nator because of its computational efficiency and ease of scattering in free space, can also be applied to waveguide implementation. FDTD is generally well suited for free- analysis. In this case, however, the incident field and space propagation and scattering problems. When ap- corresponding geometry are the incident mode and the plied to the Bragg resonator, important considerations in- unperturbed waveguide structure, respectively, where the clude the following: ⑀ waveguide material parameters are represented by i , ␮ ␴ 1. Wideband excitation may not be possible owing to i , and i . If the actual (perturbed) waveguide geom- ⑀ ␮ ␴ wavelength-dependent mode profiles and propagation etry is characterized by material parameters , , and , constants. we may write Maxwell’s equations for the scattered fields ¯ ¯ 2. Discrete stepping in time and finite resolution in (Es and Hs)as the propagation direction will lead to a mode slightly different from that given by Eq. (2), where exact derivatives inc ¯ ץ ¯ ץ Hs H (have been assumed in time and propagation direction. ٌ ϫ E¯ ϭϪ␮ Ϫ ͑␮ Ϫ ␮ ͒ , (4 tץ t iץ The fields scattered by the 3D geometry will be very s .3 weak compared to the incident wave, giving rise to large dynamic range requirements. inc ¯ ץ ¯ ץ 4. The error induced by the absorbing boundary con- Es E ͒ ⑀ ϫ H¯ ϭ ⑀ ϩ ␴E¯ ϩ ͑⑀ Ϫ ٌ tץ t s iץ dition must be acceptably low. This is particularly im- s portant, as the Bragg-resonator frequency response can ϩ ␴ Ϫ ␴ ¯ inc be highly sensitive to errors in the computed fields. ͑ i͒E , (5)

A. Wideband and Sinusoidal Excitation where E¯ inc and H¯ inc represent the incident modal fields. By employing an appropriate time-domain waveform, These equations are identical to Maxwell’s equations for FDTD can obtain the wideband response of many impor- total field with the addition of source terms. For a non- tant structures with a single simulation. Since mode conductive, nonmagnetic medium, we must include a profiles and propagation constants will change as a func- ⑀ Þ ⑀ source term only where i . To illustrate the excitation of excitation wavelength, wideband excitation is not tion of the source in the FDTD code, the standard FDTD well suited for applications that are very sensitive to er- update equation for E is given as ror. Thus in this paper separate simulations were run at x discrete wavelengths to obtain the transmission and re- ⌬ n Ϫ flection of the Bragg section, and interpolation was ap- t Hz,i, jϩ1/2,k Hz,i, jϪ1/2,k Enϩ1/2 ϭ ͩ plied to obtain results at intermediate wavelengths. x,ijk ⑀ ⌬y Subsection 3.C explains how the guided mode at a single n Ϫ wavelength is sourced in an FDTD simulation. Hy,i, j,kϩ1/2 Hy,i, j,kϪ1/2 Ϫ ͪ ϩ EnϪ1/2 (6) ⌬z x,ijk B. Finite-Difference Approximation FDTD approximates continuous derivatives in time and space with finite differences. In Eqs. (2) and (3), continu- ϵ G. (7) J. W. Wallace and M. A. Jensen Vol. 19, No. 3/March 2002/J. Opt. Soc. Am. A 615

Table 3. Propagation Constants and Transmission and Reﬂection Coefﬁcients for the Bragg Section at Various Wavelengths

␭ ␤ ␭ ͉ ͉ Є ͉ Ϫ ͉ Є Ϫ 0 (nm) z 0 R0 R0 (deg) T0,0 1 (T0,0 1) (deg)

1549.0 20.009476 2.7232 ϫ 10Ϫ4 1.27 4.2981 ϫ 10Ϫ4 90.0379 1549.5 20.009300 2.7222 ϫ 10Ϫ4 1.30 4.2952 ϫ 10Ϫ4 90.0375 1550.0 20.009125 2.7212 ϫ 10Ϫ4 1.33 4.2922 ϫ 10Ϫ4 90.0372 1550.5 20.008949 2.7202 ϫ 10Ϫ4 1.36 4.2893 ϫ 10Ϫ4 90.0368 1551.0 20.008774 2.7192 ϫ 10Ϫ4 1.39 4.2864 ϫ 10Ϫ4 90.0364 1551.5 20.008599 2.7182 ϫ 10Ϫ4 1.42 4.2834 ϫ 10Ϫ4 90.0360

problems. To apply the method to waveguide simulation, two different issues must be considered.

1. Propagation into the Perfectly Matched Layer Past work has demonstrated that the PML technique may be applied to a dielectric waveguide by simply extending the waveguide into the PML and ensuring that ␴ ␴ ϭ ⑀ ⑀ ␴ ␴ 1,z(z)/ 2,z(z) 1 / 2 , where 1,z(z) and 2,z(z) are the z-directed conductivities in the PML for the core and clad- ⑀ ⑀ ding and 1 and 2 are the permittivities in core and cladding, respectively.24 This condition ensures that the decay rate of the propagating mode in the core and cladding remains equal. Many other researchers have studied Maxwell’s equations in the PML.18,19,25 Here, we provide a proof suited for our application that if the waveguide naturally extends into the PML, the Fig. 6. Section of the Bragg resonator to be modeled with 3D modes are identical in the PML and normal regions, lead- full-wave analysis. The dashed lines denote the extents of the ing to a (theoretically) reflectionless interface. Note that unit section, lt and ls are physical lengths of the tooth and Bragg section, and ␪ is the angular length of the section in radians at matching of modes does not guarantee absorption. In the ␭ ϭ case of metallic waveguides, for example, we have the dif- 0 1550 nm. ficulty of evanescent modes, and other methods must be employed. The incident/scattered field formulation introduces an ad- Starting with Berenger’s equations for the PML in con- ditional term such that tinuous space, assuming steady-state fields, and substituting equations for electric field into equations for mag- inc ץ ⑀ Ϫ ⑀ ͑ i ͒ Ex,ijk netic field leads to the Helmholtz equation Enϩ1/2 ϭ G ϩ ⌬t ͯ (8) tץ ⑀ x,ijk 2ץ 2ץ 2ץ tϭn⌬t ͩ ϩ ϩ Ϫ ͪ axbx 2 ayby 2 azbz 2 1 Hp zץ yץ xץ ⑀ Ϫ ⑀ ͑ i ͒ ϭ G ϩ ͑ inc, nϩ1/2 Ϫ inc, nϪ1/2͒ Ex,ijk Ex,ijk , (9) ץ ץ ץ ץ ⑀ Hx Hy Hz ϭ b ͩ a ϩ a ϩ a ͪ , (10) zץ y zץ x yץ p xץp where Einc is specified by the known mode shape. The second equality employs a finite difference on the incident where a ϭ 1/( j␻␮ ϩ ␴*), b ϭ 1/( j␻⑀ ϩ ␴ ), and source to ensure compatibility of the mode in the dis- p p p p p ෈ ͕x, y, z͖. A similar relation results for E when the cretized domain. After the FDTD simulation, scattered substitution is performed in reverse. The right-hand fields may be converted to physical total fields by adding side of Eq. (10) is identically 0 when the FDTD gridding the known incident mode shape at each simulation node. scheme is used. In this case, for a PML only in the propagation direction, we obtain D. Absorbing Boundary Condition: Berenger’s Perfectly Matched Layer for Waveguides ␺2ץ ␺2 1ץ ␺2ץ 1 Several absorbing boundary conditions have been applied ͩ ϩ ϩ ͪ ϩ ␺ ϭ 0, (11) 2 ץ ⑀ ␮ 2 ץ 2 ץ ⑀␻2␮ for waveguide termination: methods based on the one- x y zr zr z way wave equation,16,17 PML techniques,9,18,19 modal 20 ␮ ϭ ͓ Ϫ ␴ ␻␮ ͔ ⑀ ϭ ͓ Ϫ ␴ ␻⑀ ͔ eigenfunction expansion approaches, absorption based where zr 1 j z*/( ) , zr 1 j z /( ) , and on discrete time-domain Green’s functions for the ␺ is any field quantity. Assuming the same condition 21,22 23 waveguide, and filter-bank methods. Our problem as Berenger for a reflectionless interface, we let qz ϭ ␮ ϭ ⑀ requires absorption of both guided modes and radiated zr zr . Modal propagation will be of the form ␺ ϭ ␺ Ϫ ␤Ј ␤Ј ϭ ␤ ͱ␮ ⑀ ϭ ␤ fields, suggesting the use of robust PML-based methods. (x, y)exp( j zz) where z z zr zr zqz . Berenger’s PML9 is well suited for free-space scattering Substitution into Eq. (11) gives the Helmholtz equation 616 J. Opt. Soc. Am. A/Vol. 19, No. 3/March 2002 J. W. Wallace and M. A. Jensen

␴ 2ץ 2ץ have a different max according to Eq. (14). This condi- ͩ ϩ Ϫ ␤2 ϩ ␻2␮⑀ͪ ␺ ͑ ͒ ϭ z x, y 0. (12) tion is equivalent to that given in Ref. 24. y2ץ x2ץ Thus the governing equation for propagation constants does not change across the normal/PML or PML/PML in- 2. Transverse Perfectly Matched Layer terfaces. In order to absorb ﬁelds radiated from waveguide discon- To show that the mode shape is also continuous across tinuities, we place the PML on the transverse sides (xˆ and the boundary, we write Faraday’s law in the PML medium yˆ ) of the simulation volume. The Helmholtz equation as will not be the same in the normal and PML regions, lead-

ץ 1 Ϫj␻␮H¯ ͑x, y͒exp͑Ϫj␤Јz͒ ϭ ٌ͓ ϫ E¯ ͑x, y͔͒exp͑Ϫj␤Јz͒ ϩ ͩ ˆz ͪ ϫ ͓E¯ ͑x, y͒exp͑Ϫj␤Јz͔͒ z ץ z T z qz z ␤ q ¯ Ϫ ␻␮ ¯ ϭ ٌ ϫ ¯ Ϫ z z ϫ j H͑x, y͒ ͓ T E͑x, y͔͒ ͓zˆ E͑x, y͔͒, (13) qz

ϫ ٌ where T represents the transverse part of the curl op- ing to aberrations in the mode shape. However, if the ϫ ץ ץ ϩ ץ ץ ϭ ٌ erator ( T ͓ / x͔xˆ ͓ / y͔ yˆ ), and is the cross- field is weak in the PML region, we expect the effect to be product operator. Canceling the qz terms leads to the small. To show this, consider the FDTD simulation vol- same equation as in the non-PML medium. Repeating ume depicted in Fig. 7. The Bragg-resonator waveguide this analysis for Ampere’s law shows that the equations geometry is the same as that shown in Fig. 5. The source for the cross-sectional mode shape are identical in the for the simulations is an electric wall at the left side of the PML and normal regions. volume that is forced to the known mode. A ramp func- When applying the PML, a stepped n-order conduc- tion of Gaussian shape and 99% rise time of 3T (3 peri- ␴ ϭ ␴ ⌬ n tivity gradient is convenient, or (z) max(z/ z) , ods) is applied to avoid initial transients. The steady- ␴ ⌬ where max is the maximum conductivity and z is state fields are extracted by taking the fast Fourier the length of the PML. The decay rate at any point transform (FFT) of a complete period after 10T and using ␬ ϭ ␤ Ϫ ␴ ␻⑀ in the PML is (z) Im͕ z͓1 j (z)/( )͔͖, and the the sample corresponding to the fundamental frequency. accompanying reflection for the complete PML is The results of these simulations have been compared with ϭ Ϫ ␴ ␤ ⌬ ␻⑀ ϩ R exp͕( 2 max z z)/͓ (n 1)͔͖. To determine the the solution given by an ideal one-dimensional (1D) FD conductivity in our FDTD simulation we compute simulation, where discretization applies only in the zˆ direction. The decay boundary condition is used in the FD ␴ ϩ ⑀ ␻ solver on the transverse walls in order to ensure compat- max ͑n 1͒ln R 0 ϭϪ , (14) ibility of the guided mode. ⑀ ␤ ⌬ r 2 z z The results of the computations using the 1D and FDTD solutions were compared in terms of complex-field where R is the specified modal reflection. For each re- envelopes at the center of the simulation (on the line ⑀ gion of different r in the waveguide cross section, we will x ϭ 0, y ϭ 0) and on complex-mode envelopes using Eq. (15). The fractional difference between the ideal 1D solution and the FDTD solution was below 5 ϫ 10Ϫ6 and 1.5 ϫ 10Ϫ5 for the simulations with no transverse PML and simulations with transverse PML, respectively. These results indicate that the PML at the domain edges influences negligible error on the mode behavior.

E. Mode Extraction Because of mode orthogonality, the complex envelope of the mode with shape Mij can be extracted from the fundamental frequency component of the FDTD simulation Eij,z by using the expression ⌺ ijEij,zMij* A ϭ . (15) z ⌺ ͉ ͉2 ij Mij

4. BRAGG-RESONATOR RESPONSE Fig. 7. FDTD simulation volume used to assess performance of FDTD simulations were run for the wavelengths given in propagation and transverse PML. Table 3 by using simulation volume dimensions nearly J. W. Wallace and M. A. Jensen Vol. 19, No. 3/March 2002/J. Opt. Soc. Am. A 617

Fig. 8. Reflection and transmission of the Bragg resonator, ignoring any sources of error. identical to those of Fig. 7, the only differences being Table 4. Sources of Error in the Bragg-Resonator placement of PML in both the ϩzˆ and Ϫzˆ directions, more Simulation and Approximate Values cells in the zˆ direction to accommodate the geometry shown in Fig. 6, and removal of the electric wall source. Fractional Error in Error Source ␤ Mode Shape R and T A. Reflected and Transmitted Fields z To obtain the modal reflection and transmission coeffi- FD grid truncation 6.1 ϫ 10Ϫ8 1.0 ϫ 10Ϫ6 — cients for the Bragg section at each simulated wave- FD grid resolution Ϫ6.5 ϫ 10Ϫ5 2.0 ϫ 10Ϫ3 — ␪ ϭ ␲ ␭ ϭ Ϫ5 length, consider Fig. 6 with rad at 0 1550 nm. PML reflection ——1.0ϫ 10 To compute modal reflection, scattered fields are stored in (guided mode) the xy plane at z ϭ 0. A FFT is applied to obtain the fun- PML reflection ——1.0ϫ 10Ϫ5 damental frequency component, and the complex enve- (radiated fields) Ϫ ϫ Ϫ4 lope (A0) of the mode is computed with Eq. (15). Since Finite time step 1.7 10 —— the incident mode has unit amplitude and zero phase at Finite resolution in zˆ 6.9 ϫ 1.0Ϫ4 —— ϭ the origin, R0 A0 is the modal reflection coefficient at ϭ ϭ N z 0. The reflection at plane A in Fig. 6 is then given as nator is AT AS , where N is the number of sections. ϭ ␤ Ϫ RA A0 exp͓ j z(lt ls)͔. To obtain the transmission co- After converting AT back to an S-parameter matrix, S11 efficient, scattered fields are stored at z ϭ 120 nm, yield- and S21 give the complex modal reflection and transmis- ing the complex modal envelope A120 . Addition of the sion coefficients for the complete Bragg resonator. The incident mode at z ϭ 120 nm gives the transmission response for N ϭ 104 sections is given in Fig. 8. coefficient from 0 to 120 nm, or T ϭ A 0,120 120 B. Error Quantification ϩ exp(Ϫj␤ l ). The transmission coefficient from z t Small amounts of error in the single section response may plane A to plane B is then given as T A,B cause more appreciable error in the complete Bragg re- ϭ T exp͓ j␤ (l Ϫ l )͔. Since computational investiga- 0,120 z t s sponse. Table 4 lists the primary sources of error and tions have indicated that R , T , and the propagation 0 0,120 their estimated values. The numbers in boldface in the constant follow a linear trend versus excitation wave- table represent error values that are most significant: length, linear interpolation is employed to obtain results increased mode amplitude due to finite FD grid resolution at wavelengths between the simulated data points. and an increase in ␤ due to finite FDTD resolution in the For a fixed wavelength, the Bragg response is com- z propagation direction. The effect of these sources of er- puted with 1D transmission-line analysis.26 The values ror is discussed below. of R0 and T0,120 at the six excitation wavelengths are in- terpolated, RA and TA,B are computed, and an 1. Error Source: Increased Mode Amplitude S-parameter matrix is formed for the symmetric Bragg The finite resolution of the FD method leads to a mode section. Next, the ABCD matrix for the section is com- amplitude that is too high in the region of the waveguide puted (AS) and the ABCD matrix for the complete reso- discontinuity. Since we are applying the scattered-field 618 J. Opt. Soc. Am. A/Vol. 19, No. 3/March 2002 J. W. Wallace and M. A. Jensen

Fig. 9. Transmission and reflection amplitude and phase of the Bragg resonator resulting from random phase shifts on A0 and A120 over 32 realizations. The thick gray curves are the mean and the thin black curves are deviation of the individual realizations from the mean. formulation and the error is nearly constant over the 5. CONCLUSION waveguide discontinuity, the scattered fields will increase In this paper we have outlined a method for simulating linearly with the amount of error. To assess the impact complex optical devices by applying three basic modeling of this effect, we reduced the magnitude of the complex- tools: (1) a 2D FD mode solution technique for finding mode envelopes extracted from the FDTD simulations by modes in arbitrary guiding structures, (2) 3D full-wave 0.2% (as suggested in Table 4) and again plotted the re- FDTD analysis of waveguide discontinuities, and (3) net- sponse. The magnitude of fractional change in the reflec- work analysis employing a 1D transmission line model. tion and transmission response was less than 2% and 1%, By using the method, we simulated a large (104 sections) respectively. Bragg resonator. Detailed error analysis indicated that very low error can be obtained for such a problem using this simulation approach. Natural extensions include ␤ 2. Error Source: Increase in z the simulation of distributed feedback lasers by incorpo- ␤ The FD-mode solver estimates values for z that are too rating gain into the 3D full-wave FDTD simulator. ϳ ␤ ␭ ϭ high ( 0.07%). At the value of z for 0 1550 nm, the amount of phase error over the discontinuity length of The authors may be reached at the address on the title 120 nm is 0.06°. The resulting error in the phase of A page or by e-mail, [email protected], and wallacej 0 @ee.byu.edu. and A120 is uncertain owing to the multiple reflections in the discontinuity. However, we may explore the effect of phase error by assuming independent distributions on the phase error for A0 and A120 and produce a number of REFERENCES Monte Carlo realizations. Plotting the response for sev- 1. D. Marcuse, ‘‘Tilt, offset, and end-separation loss of lowest- eral random values will provide an indication of the dis- order slab waveguide mode,’’ J. Lightwave Technol. LT-4, tribution of the error. 1647–1650 (1986). 2. S. I. Hosain, J.-P. Meunier, and Z. H. Wang, ‘‘Coupling effi- For simplicity we assume the phase shift for A0 and A to be uniform on [0°, Ϫ0.06°]. Figure 9 plots the ciency of butt-jointed planar waveguides with simultaneous 120 tilt and transverse offset,’’ J. Lightwave Technol. 14, 901– mean value of transmission and phase of the Bragg reso- 907 (1996). nator along with the response for each of 32 Monte Carlo 3. M. D. Feit and J. A. Fleck, ‘‘Computation of mode properties realizations. The plots show that the most sensitive in optical fiber waveguides by a propagating beam method,’’ quantity is the phase of the reflection coefficient near Appl. Opt. 19, 1154–1164 (1980). 4. W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, ‘‘The resonances, where reflection is nearly zero. This is rea- finite-difference vector beam propagation method: analy- sonable, since we expect numerical difficulty in the com- sis and assessment,’’ J. Lightwave Technol. 10, 295–305 puting phase for a small value. (1992). J. W. Wallace and M. A. Jensen Vol. 19, No. 3/March 2002/J. Opt. Soc. Am. A 619

5. P. Lu¨ sse, P. Stuwe, J. Schu¨ le, and H. Unger, ‘‘Analysis of magnetic-field equations,’’ IEEE Trans. Electromagn. Com- vectorial mode fields in optical waveguides by a new finite pat. EMC-23, 377–382 (1981). difference method,’’ J. Lightwave Technol. 12, 487–493 17. K. K. Mei and J. Fang, ‘‘Superabsorption—a method to im- (1994). prove absorbing boundary conditions,’’ IEEE Trans. Anten- 6. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, nas Propag. 40, 1001–1010 (1992). ‘‘Full-vectorial mode calculations by finite difference 18. S. D. Gedney, ‘‘Anisotropic perfectly matched layer absorb- method,’’ IEE Proc.: Optoelectron. 141, 281–285 (1994). ing medium for the truncation of FDTD lattices,’’ IEEE 7. K. S. Yee, ‘‘Numerical solution of initial boundary Trans. Antennas Propag. 44, 1630–1639 (1996). value problems involving Maxwell’s equations in isotropic 19. C. M. Rappaport, ‘‘Perfectly matched absorbing boundary media,’’ IEEE Trans. Antennas Propag. AP-14,302–307 conditions based on anisotropic lossy mapping of space,’’ (1966). IEEE Microwave Guid. Wave Lett. 5,90–92 (1995). 8. A. Taflove and S. C. Hagness, Computational Electrody- 20. M. Okoniewski, M. A. Stuchly, M. Mrozowski, and J. DeMo- namics: The Finite-Difference Time-Domain Method erloose, ‘‘Modal PML,’’ IEEE Microwave Guid. Wave Lett. 7, (Artech House, Boston, 2000). 33–35 (1997). 9. J.-P. Berenger, ‘‘A perfectly matched layer for the absorp- 21. S. L. Dvorak, ‘‘Exact, closed-form expressions for transient tion of electromagnetic waves,’’ J. Comput. Phys. 114, 185– fields in homogeneously filled waveguides,’’ IEEE Trans. 200 (1994). Microwave Theory Tech. 42, 2164–2170 (1994). 10. H. E. Hernańdez-Figueroa, F. A. Fernańdez, and J. B. 22. M.-J. Park and S. Nam, ‘‘FDTD modal field generation and Davies, ‘‘Finite element approach for the modal analysis of absorption in homogeneous waveguides with arbitrary open-boundary waveguides,’’ Electron. Lett. 30, 2031–2032 cross section,’’ in Proceedings of the 1996 IEEE Antennas (1994). and Propagation Society International Symposium (Insti- 11. J. W. Wallace and M. A. Jensen, ‘‘An FD/FDTD method for tute of Electrical and Electronics Engineers, New York, optical waveguide modeling,’’ in Proceedings of the 2000 1996), Vol. 2, pp. 1308–1311. IEEE Antennas and Propagation Society International 23. C.-N. Kuo, T. Itoh, and B. Houshmand, ‘‘Synthesis of ab- Symposium (Institute of Electrical and Electronics Engi- sorbing boundary condition with digital filter bank,’’ in Pro- neers, New York, 2000), Vol. 3, pp. 1380–1383. ceedings of the 1996 IEEE Microwave Theory and Tech- 12. F. Girardin, G.-H. Duan, and A. Talneau, ‘‘Modeling and niques Society International Symposium (Institute of measurement of spatial-hole-burning applied to amplitude Electrical and Electronics Engineers, New York, 1996), Vol. modulated coupling distributed feedback lasers,’’ IEEE J. 2, pp. 1043–1046. Quantum Electron. 31, 834–841 (1995). 24. M. Chen, B. Houshmand, and T. Itoh, ‘‘FDTD analysis of a 13. S. Adachi, ‘‘Model dielectric constants of GaP, GaAs, GaSb, metal-strip-loaded dielectric leaky-wave antenna,’’ IEEE InP, InAs, and InSb,’’ Phys. Rev. B 35, 7454–7463 (1987). Trans. Antennas Propag. 45, 1294–1301 (1997). 14. D. E. Aspnes and A. A. Studna, ‘‘Dielectric functions and 25. M. A. Gribbons, W. P. Pinello, and A. C. Cangellaris, optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, ‘‘Stretched coordinate technique for numerical absorption of and InSb from 1.5 to 6.0 eV,’’ Phys. Rev. B 27, 985–1009 evanescent and propagating waves in planar waveguiding (1983). structures,’’ in Proceedings of the 1995 IEEE Microwave 15. B. Jensen and A. Torabi, ‘‘Refractive index of quaternary Theory and Techniques Society International Symposium In1ϪxGaxAsyP1Ϫy lattice matched to InP,’’ J. Appl. Phys. 54, (Institute of Electrical and Electronics Engineers, New 3623–3625 (1983). York, 1995), Vol. 1, pp. 31–34. 16. G. Mur, ‘‘Absorbing boundary conditions for the finite- 26. G. Gonzalez, Microwave Transistor Amplifiers (Prentice- difference approximation of the time-domain electro- Hall, Upper Saddle River, N.J., 1997).