BEAM propagation methods (BPM s) are very powerful

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 3, MARCH Efficient Semivectorial Mode Solvers Frank Wijnands, Thomas Rasmussen, Hugo J. W. M. Hoekstra, Jørn H. Povlsen, and René M. de Ridder, Member, IEEE Abstract A method, based on a semivectorial finite difference scheme, is described to construct modal fields for any twodimensional refractive-index profile which is constant except at abrupt interfaces. The modal fields correspond to eigenvectors of the matrix equation to be solved. In order to find the eigenvectors and their corresponding eigenvalues, the matrix equation is formulated according to the inverse iteration method (IIM). Two versions of the IIM are compared. Further, two matrix equations are compared: one is based on the propagation equation, following from the three-dimensional paraxial wave equation, and the other is the Fresnel equation, leading to the standard eigenvalue equation. A new solution method for the matrix equation is presented. It is a refinement of the alternating direction implicit (ADI) method. This refined ADI method is compared to the standard conjugate gradient (CG) method. Both methods are tested for waveguides having a rectangular core cross section. The refined ADI method is found to be computationally more efficient than the unpreconditioned CG method. Index Terms Alternating direction implicit, complex axis, conjugate gradient, finite difference, imaginary axis, inverse iteration, modesolver, semivectorial. I. INTRODUCTION BEAM propagation methods (BPM s) are very powerful to simulate the propagation of light in structures which cannot be treated analytically. Two frequently used BPM s are the Fourier transform BPM (FTBPM) [1] [3] and the finite difference BPM (FDBPM) [4] [7]. There are various methods to perform a modal analysis for an arbitrary refractive-index structure. Two-dimensional (2-D) transfer matrix methods can be used for 2-D structures [8] [11], whereas three-dimensional (3-D) structures require other methods. In many methods, the field is written as a product of two functions depending on one transverse coordinate only; the problem is solved analytically for these two fields and the solutions are combined. Wellknown methods are the effective index method (EIM) [12], [13] and the Marcatili method [14], but many similar methods have been developed [15], [16]. In many other methods, the 3-D problem is tackled by expressing the solution in terms of Manuscript received December 22, 1995; revised September 16, F. Wijnands was with the MESA Research Institute, University of Twente, 7500 AE Enschede, the Netherlands. He is now with R&D Devices, Hewlett- Packard Fiber Optic Components Operation, Ipswich, Suffolk IP1 5PB, U.K. T. Rasmussen was with the Center for Broadband Telecommunications, Electromagnetics Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark. He is now with NKT Elektronik A/S, NKT Alle 85, DK 2605 Brondby, Denmark. H. J. W. M. Hoekstra and R. M. de Ridder are with the MESA Research Institute, University of Twente, 7500 AE Enschede, the Netherlands. J. H. Povlsen is with the Center for Broadband Telecommunications, Electromagnetics Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark. Publisher Item Identifier S (97) basis functions, which can be Hermite Gauss basis functions or slab waveguide solutions [17], circular harmonics [18], sine functions [19], or a combination of the use of sine functions in one transverse direction and the FDBPM method for the other transverse direction [20]. In a different approach, the problem is solved using a domain integral equation [21]. One class of methods consists of solving an eigenvalue problem with the dimension of the matrix to be diagonalized being equal to the number of grid points of the cross section [22]. Another method, by Yevick and Hermansson [23], consists of propagating a field distribution along an imaginary propagation direction [throughout this paper, the propagation direction is along the (in general complex) axis]. The idea is that the guided mode with the highest effective index gets the maximum amplitude increase during propagation. They use as a propagation scheme the FTBPM. Chen and Jüngling [24] [26] adopt this idea using the FDBPM to calculate modal fields for 2-D cross sections. They use the power method [27] to find the guided modes. Some of us applied the idea of Yevick and Hermansson to find guided modes for onedimensional (1-D) cross sections using the FDBPM as well. The inverse iteration method [28] was used to find the modes. The method works for lossless and absorbing structures [29] as well as for electric field intensity-dependent refractive-index profiles [30], [31]. The FDBPM gives much more accurate results for large index contrasts than the FTBPM (see, e.g., [32] [34]), and, in contrast to the FTBPM, the continuity relations at steps in the refractive index may be incorporated in the FDBPM. The inverse iteration method (IIM) is a very efficient method to find dominant eigenvectors, when good approximations to the eigenvalues and/or the eigenvectors are known. Such approximations may be obtained for optical waveguides by a variety of fast semi-analytical methods, whereof the most general is probably the EIM [12]. It is the aim of this paper to compare the IIM based on the FDBPM and the IIM based on the eigenvalue wave equation. Two versions of the IIM are discussed. Furthermore, we shall discuss two methods, the conjugate gradient (CG) method and a refined alternating direction implicit (ADI) method, to solve the sparse system of equations arising from the finite difference approximations to the operators in the matrix equation. Both methods are especially well suited for sparse linear matrix equations. The method described in this paper is suitable for any refractive-index profile which, for a given cross section, is constant except at interfaces parallel to the two lateral axes. The reason for this restriction is that the interface has to be either parallel or perpendicular to the dominant field component in order to use the appropriate boundary conditions. If an interface which is not parallel to the two /97$ IEEE

2 368 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 3, MARCH 1997 transverse axes would be approximated by a staircase profile, this would impose a problem on the continuity relations for the dominant field component, since then many grid points will be close to both interfaces parallel and perpendicular to the field component. Notice that due to this restriction, diffused waveguides cannot be treated with our interface conditions. The computational grid in the present paper is uniform. For an optimized nonuniform grid, the number of grid points required for a given accuracy can be reduced by a factor of 3-4 in the case of weakly guiding waveguides [35], which would reduce the required computational memory and computation time considerably. This paper is organized as follows. In Section II, the theory is described, both for nonabsorbing and for absorbing structures. The numerical results for some 2- D rectangular core cross-section waveguides are presented in Section III. A discussion of the results follows in Section IV. II. THEORY The problem is worked out in terms of the electric field. The Maxwell equation for reads We shall solve the problem for the dominant field component for a quasi-te mode and for a quasi-tm mode. Hence, the formulation of the problem is semivectorial. A. Derivation of the Matrix Equation Using FDBPM A scalar field distribution can be written, according to the slowly varying envelope approximation (SVEA), as where is a suitably chosen mean refractive index. In the particular case of quasi-te and quasi-tm, substitution of (2) in (1) leads to (3) and (4), shown at the bottom of the page. Now is assumed to vary slowly as function of [see (2)]. Hence, the term in (3) and (4) can be neglected. Second, the terms containing and are nonzero only near interfaces. These terms can be taken into account by using efficient interface conditions, described in detail in [37]. The field equation to solve is then Now discretise the field, with. Here and are the number of grid points in the and directions, respectively. Introduce the vector (1) (2) (5). Then (5) is integrated for the discretised field, using the Crank Nicolson scheme. The result is [5] with being the identity matrix. Here is a five-band matrix with,, elsewhere. At the lateral boundaries, transparent boundary conditions introduced by Hadley are used [36]. B. Mode Search Using the Inverse Iteration Method (IIM) The problem is to find eigenvalues and corresponding eigenvectors of the matrix defined in (6). This problem can be solved using the IIM [28], [40]. This method involves three steps in each iteration. 1) Solve a matrix equation, giving a new approximation of the eigenvector. 2) Calculate a new approximation of the eigenvalue (it is here that the IIM formulations of [28] and [40] differ). 3) Normalize the new approximation of the eigenvector. We shall compare two formulations of the IIM: 1) based on the propagation equation (6) and, 2) the standard formulation, based on the eigenvalue wave equation. Starting from the th iterated field solution, we calculate the next iteration for formulations 1) and 2), respectively, and it will be shown that, in general, after being properly normalized, they converge to the same modal field. In the IIM based on the propagation equation (6), the matrix equation is where we introduced eigenvalue, which is related to the effective index as. According to the standard formulation of the IIM, (7) should be replaced by the matrix equation according to the eigenvalue wave equation This can be seen as follows. If the waveguide structure is -invariant, we may write the electrical field as a sum of eigenmodes, : where is the free space wavevector, is the effective index of the th mode, and is the amplitude. Thus, (6) (7) (8) (9) quasi-te (3) quasi-tm (4)

3 WIJNANDS et al.: EFFICIENT SEMIVECTORIAL MODE SOLVERS 369 the wave equation reduces to (10) Using the same notation for the discretized field and the matrix as before, this equation may be written as (11) Equations (7) and (8) converge to the same solution as can be seen as follows. Suppose in (7) that is an approximate eigenvector of the matrix with approximate eigenvalue, then the right-hand side of (7) may be rewritten as. Thus the right hand side of (7) is nearly proportional to. This implies that the solutions to (8) and (7) are nearly proportional. Therefore, the sequence of eigenvector approximations of the two matrix equations (7) and (8) will in general converge to the same eigenvector solution, provided the sequence converges. Hence, we expect these two matrix equations to give quite similar results (which has been confirmed by tests). We choose to work with the propagation equation (7) for the following reasons. One reason is its convenience. When using the propagation equation, the same algorithm can be used both in the mode-finding procedure and in the usual real-axis propagation part. The other reason is connected to the accuracy of the splitting mechanism in the ADI method (see Section II-C). For the Laplace equation, it can be proved that the error in the ADI method is of order if the transverse grid spacing [42]. For the Fresnel equation, where an extra refractive-index term is present, it can be proved that the error term of order is present as well, whereas for the eigenvalue wave equation the corresponding error term is of order. For these reasons, the propagation equation (7) is chosen as the matrix equation to be used in our mode solver. This is the first result of our paper. The next problem is to decide on the expression to be used for calculating the subsequent approximations of the eigenvalues. Two versions of the IIM are discussed. In the IIM approach of [40], a new approximation of the eigenvalue is calculated by the Rayleigh quotient (12) with inner product. Finally, is renormalized such that. The IIM as described in [28] works slightly different. For convenience, (7) is written in the form (13) Starting with a normalized vector and trial eigenvalue, for each next iteration step is replaced by (14) and is renormalized such that. In the IIM described by (7), assuming that the structure is (nearly) lossless, the propagation is along a (nearly) imaginary propagation axis. This can easily be seen by expanding in modal fields,.if and are real, then is (almost) real for a (nearly) lossless structure, since. Consequently, must be (nearly) real [this is true in both formulations of the IIM, (12) and (14)]. The observation that in (7) concludes the argument. For this reason, in the IIM, an adapted form of Hadley s transparent boundary conditions, has to be used at grid points located at a lateral boundary. On solving the matrix equation (6), the values of at the boundaries are given as (15) for. The question which of the two IIM versions, (12) or (14), is to be preferred is strongly related to the ADI-matrix equation solver, to be discussed in Section II-C. There we will compare the two IIM versions. C. Two Methods for Solving the Matrix Problem The first method for solving matrix equation (7) is based upon the alternating direction implicit (ADI) method. The ADI method makes use of the special form of the matrix equation (7) [28], [42], [43]. Solving the Fresnel equation (5) discretely, introducing an intermediate field, and performing a Crank Nicholson integration yields the following ADI equations for (7) [42]: with (16) (17) elsewhere (18) elsewhere (19) Here. For grid points near an interface, the coefficients are adapted according to the EIC s described in [37]. For grid points at a lateral boundary, adapted transparent boundary conditions are used, as described in (15). The computational advantage of the ADI method is the fact that the two matrix equations (16) and (17) are both tridiagonal matrix problems. Hence, the number of operations per propagation step is strictly of order, the number of transverse grid points. If the propagation

4 370 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 3, MARCH 1997 equation (7) is solved using the ADI method, the stepsize, as prescribed by the IIM, can be very large. But then, due to the error term of order, our ADI method will be inaccurate for large stepsizes. In order to have an equation solver which is both efficient and accurate, the following refinement procedure has been applied. Once we have chosen a value for, as prescribed by the IIM, we have to solve the matrix problem. By regarding this matrix equation as a propagation equation (along a complex axis), it will be solved much more accurately, if one propagates times with a constant stepsize of, instead of one propagation step of. One could argue that, due to this procedure, the IIM becomes meaningless since, after determining a value for according to the IIM, we subsequently subdivide this value. Notice, however, that the value for, determined according to the IIM, determines the propagation equation to be solved. The only purpose of the refinement strategy is to solve this given equation much more accurately. This will be referred to as the refined ADI method and is the second result of this paper. The ADI method described above is a noniterative method. Hadley applied an iterative ADI method, both in a scalar [45] and in a fully vectorial [46] formulation. A general description of the iterative ADI method can be found, e.g., in [44]. For the Laplace equation, it has been proved rigorously that the noniterative ADI method is unconditionally stable [42]. For the propagation equation, in which a refractive-index term is present as well, we found numerically that the noniterative ADI method is stable. The reason for choosing the noniterative ADI instead of the iterative ADI method is that in our tests the stability of the iterative ADI was much more delicate and depends heavily on, for example, the choice of optimalization parameters. Now that the ADI method has been described, we are in a position to compare the two versions of the IIM, (12) and (14). Suppose one uses the IIM based on (14) in order to find a converging sequence of eigenvalues with the ADI method as matrix equation solver. The sequence converges only if. For matrix equation (7), using (16) and (17) results in (20), shown at the bottom of the page. Since we assumed that converges, may be considered as an approximate modal field. Therefore, (20) may be rewritten as In other words: (21) (22) Hence, if the term cannot be neglected, will not converge to an eigenvalue of the matrix. If one works with (12), this problem does not arise, since then is determined by direct multiplication with the original, unsplitted, matrix (hence, the approximations in the ADI method affect the choice of only in the sense that has been determined using the ADI method). The same argument holds for what was called the refined ADI method where the propagation is performed in subsequent steps of. In terms of the eigenvalues, for each of these steps, is replaced by. In order to get a converging sequence, it should hold that (23) where the term contains all terms of order If this term cannot be neglected, will not converge to an eigenvalue of the matrix. Again, the IIM based upon (12) does not have this problem, since then is determined by direct multiplication with. The consequence of this is that, for the (refined) ADI method using the propagation equation (7), the only sensible IIM is the one based upon (12). This is our third result. In the next section for the matrix equation (7), using the IIM based upon (12), results for the refined ADI method will be compared to results obtained with the CG method [28], [41], [42], which is a very efficient iterative method for solving a sparse matrix equation. In the CG method, the number of operations per iteration is of order, the number of transverse grid points. The number of iterations per propagation step is of order as well, in order to solve the matrix problem (numerically) exactly. In practice, a quite accurate solution is already achieved after a smaller number of iterations. Hence, the number of operations per propagation step is of order,. This number can be decreased considerably by making the CG method preconditioned [28], [41], [42]. For example, the ORTHOMIN method [41] is a preconditioned CG method of order, hence. In fact, the biconjugate gradient method [28], [41], [42] has to be used since the matrix might not be positive definite and the matrix is not symmetric if the structure inside the computational window is inhomogeneous. For our calculations, an unpreconditioned biconjugate CG method was implemented. III. RESULTS In this section, the two matrix equation solvers are tested for some rectangular-core nonabsorbing 2-D refractive-index profiles. Our results are compared to results of [16], [18], [20], [21]. In all examples, the launched field is the lowest order mode constructed with the EIM. Although in this paper the emphasis is on the lowest order mode, in principle all guided modes can be found by launching an approximate modal field constructed by means of the EIM. Another approach would be (20)

5 WIJNANDS et al.: EFFICIENT SEMIVECTORIAL MODE SOLVERS 371 (a) Fig. 1. Geometry of example described in text. A rectangular film, having width a; b in x; y directions, respectively, with refractive index n f, is surrounded by a substrate with refractive index n s. to add up all modal fields determined with the EIM. The field resulting from that procedure is the starting field. In this way, all modal fields would be contained in the starting field to a considerable extent. Once a mode has been found, the launched field is orthogonalized with respect to the mode just found. Tests have shown that the latter method is to be preferred, since one is not sure, when launching a certain mode calculated with the EIM, that this field will converge to the corresponding modal field using our method. Indeed, we found cases in which a higher order mode calculated with the EIM converged to the fundamental mode. The first example represents the strongly guiding case. The structure consists of a core with a rectangular cross section having width 2 m in the directions, respectively, with refractive index 1.5, surrounded by air ( 1.0). Fig. 1 shows the geometry of the transverse cross section. First, we define the normalized effective index and the normalized frequency as (24) Fig. 2 shows as function of the square of the lateral grid spacing for two different values of. Here, is varied by changing the wavelength for fixed geometry. The (b) Fig. 2. Normalized effective index B of the fundamental (a) quasi-te and (b) quasi-tm mode as function of the squared grid spacing (1x) 2 =(1y) 2, both for the CG method and the ADI method, for two values of the normalized frequency V. In all calculations, the interfaces are in the middle between grid points. results are given both for the fundamental quasi-te [Fig. 2(a)] and quasi-tm [Fig. 2(b)] mode, for both the CG method and the ADI method. The computation time as function of the number of transversal grid points is plotted in Fig. 3 both for the quasi-te [Fig. 3(a)] and the quasi-tm [Fig. 3(b)] mode, for both the CG method and the ADI method. Figs. 2 and 3 are based on the same data. In all calculations, the interfaces have been put in the middle between grid points.

6 372 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 3, MARCH 1997 (a) (a) (b) Fig. 3. Total computation time in CPU-seconds as function of the number of transverse grid points for the same data as in Fig. 2 for the fundamental (a) quasi-te and (b) quasi-tm mode. If the interface would have a general position with respect to the grid points as is decreased, then would show some fluctuations while converging [37]. From Fig. 2, it becomes clear that for both the ADI method and the CG method converges to the same value, in general with the same rate of convergence, as the lateral grid spacing is decreased. Fig. 3, however, shows that the refined ADI method is most efficient. It should be noted that the CG method was implemented in an unpreconditioned fashion. We allowed (b) Fig. 4. Normalized effective index B of the fundamental (a) quasi-te and (b) quasi-tm mode as a function of normalized frequency V for our refined ADI method and for four methods from literature, for a rectangular core geometry described in the text. The parameter V is varied by varying the wavelength. maximally 200 iterations per propagation step. We expect that for the unpreconditioned CG method the computer time grows faster than linearly as function of. This is reflected in Fig. 3. From Fig. 3, one can see that for the ADI method the computer time is a smooth function of, whereas for the CG method there are quite some fluctuations. The reason is probably that the field did not converge sufficiently at each propagation step. This was also reflected in the shape of the

7 WIJNANDS et al.: EFFICIENT SEMIVECTORIAL MODE SOLVERS 373 modal fields, which showed some irregularities for the CG method, in contrast to the field shapes for the ADI method. Fig. 3 also shows that for the CG method the curves for quasi- TE [Fig. 3(a)] are smoother than for quasi-tm [Fig. 3(b)]. We think the reason is that the aspect ratio of the waveguide is. Hence, the interface region at which the field is discontinuous is twice as long for quasi-tm than for quasi-te. The asymmetry in the matrix is caused by off-diagonal terms connected to grid points near interfaces at which the field is discontinuous. So, for quasi-tm, the asymmetry in occurs for more coefficients than for quasi-te. We expect that in case the asymmetry is larger, the convergence will be slower. This is probably the reason for the larger fluctuations for the quasi-tm case. Another observation is that for the CG method, the curves in Fig. 3 are much smoother for than for. In the latter case, the field is much more confined. Probably this causes more instability, especially at the interface where a discontinuity appears. Fig. 4 shows the normalized effective index as a function of the normalized frequency both for the fundamental quasi-te [Fig. 4(a)] and quasi-tm [Fig. 4(b)] mode, calculated with the ADI method. In the calculations, the transversal grid spacing m and the number of grid points is 256. Also plotted in Fig. 4 are the results of [16], [18], [20], and [21], denoted as Sharma Goell, DSM, and DIM, respectively. From Fig. 4, it can be concluded that there is good agreement between our results and literature, in particular with the DIM method [21], which is known to be very accurate. In the second example, the weakly guiding case is considered. The structure consists of a core with a square cross section having width in the directions, respectively, with refractive index 1.45, surrounded by a substrate with The wavelength 1.55 m. Fig. 5 shows the normalized effective index for the fundamental quasi-te mode as function of the normalized frequency. For this calculation, is varied by varying ( ) for fixed wavelength. Since the structure is invariant when interchanging the role of and, the quasi- TE and quasi-tm mode are degenerate in this case. In Fig. 5, the results of [18] are plotted as a reference. Again, good agreement is found between our results and literature. IV. CONCLUSION A method, based on a semivectorial finite difference scheme, has been described to construct modal fields for any 2-D refractive index profile which is constant except at abrupt interfaces, the interfaces being parallel to the transverse axes. Two versions of the IIM, differing in the choice of the next approximate eigenvalue, have been discussed. A computational stability argument has been given in favor of one of these two methods. Two matrix equations have been discussed: a propagation matrix equation and an eigenvalue wave equation. The two matrix equations have been shown to be quite similar as far as finding modes is concerned. Due to the smaller error term as function of the propagation step and for convenience we have focussed on the propagation equation. Two methods for solving the matrix equation have been tested: a new Fig. 5. Normalized effective index B of the fundamental quasi-te mode as function of normalized frequency V for the ADI method, for a square core geometry described in the text, with wavelength = 1.55 m. The parameter V is varied by varying the width of the core. Notice that due to the geometry, the quasi-te and quasi-tm modes are degenerate. refined ADI method (based on the standard ADI method in combination with dividing the propagation step into a number of smaller propagation steps) and the unpreconditioned biconjugate CG method. 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