FAST BPM METHOD TO ANALYZE NONLINEAR OPTICAL DEVICES

Transcription

1 FAST BPM METHOD TO ANALYZE NONLINEAR OPTICAL DEVICES J. de Oliva Rubio 1 and I. Molina Fernandez 1 1 Departamento de Ingenierıa de Comunicaciones Escuela Tecnica Superior de Ingenieros de Telecomunicacion Universidad de Malaga 9071 Malaga, Spain Recei ed February 1998 ABSTRACT: A new finite-difference BPM method based on the application of the Runge Kutta ( RK) technique altogether with transparent boundary conditions ( TBCs) is proposed. Numerical experiments are carried out to erify the compatibility of TBCs with this method, and to compare its performance with the standard Crank Nicholson scheme when simulating nonlinear optical de ices John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: , Key words: nonlinear beam-propagation method; nonlinear optical wa eguide; transparent boundary conditions; Runge Kutta method I. INTRODUCTION Simulation of integrated optical devices is commonly performed by means of scalar beam-propagation methods Ž BPM. based upon the resolution of the Fresnel wave equation 1. Although, historically, BPM techniques based on the FFT were first introduced and exploited, today, finite-difference schemes Ž FDBPM. are the most widely used due to their superior ability to analyze strongly guiding and nonlinear structures. To date, the most popular FDBPM method is based upon the discretization of the scalar Fresnel equation by means of the implicit Crank Nicholson Ž CN. finite-difference scheme 1 together with the transparent boundary conditions Ž TBCs.. This method, which will be referred to as CN FDBPM in this paper, leads to a sparse nonlinear system of equations which must be solved in each propagation step. Explicit finite-difference BPM methods have also been developed 3, 4 which avoid the time-consuming algorithms for the solution of the nonlinear system of equations. These methods are not unconditionally stable and, to our knowledge, they have not been used in conjunction with TBCs, so absorbent boundary conditions Ž ABCs. were used, which considerably increase the numerical magnitude of the problem and reduce the method efficiency. In this paper, a fourth-order Runge Kutta Ž RK. explicit method is developed for the solution of the Fresnel equation which allows TBCs to be successfully applied, thus drastically reducing the computational effort. This method relies on the discretization of the transversal operator with a second-order centered finite-difference scheme Žthe same as in the CN algorithm., leading to a system of nonlinear ordinary differential equations in the longitudinal variable, which is then solved by means of the RK technique. As this method employs a finite-difference discretization for the transversal operator and an RK strategy for the longitudinal one, it will be called in this paper the RK FDBPM technique. Although, for the sake of simplicity, only the scalar problem has been analyzed in this paper, since the proposed method is transversally identical to the standard CN FDBPM technique, its extension to already existing semivectorial and vectorial formulations can easily be performed. The RK method has been previously used for the resolution of the nonlinear Fresnel equation 5, but, to our knowledge, this is the first time that this method has been applied in conjunction with TBCs, and that a detailed comparison between RK and CN FDBPM has been performed. Assessment of the proposed technique is carried out in two steps. First, to study the behavior of TBCs within the RK FDBPM algorithm, two previously proposed problems for testing the boundary conditions are analyzed: diffraction and oblique incidence on a boundary of a Gaussian beam propagating through a linear homogeneous medium. Second, the ability of the RK FDBPM technique to analyze nonlinear structures is established and compared with the standard CN FDBPM. This is carried out by simulating the propagation of a three-dimensional Gaussian beam in an optical slab with a nonlinear Kerr-type saturable substrate 6, and studying the functioning modes of a nonlinear directional coupler 7. A rigorous comparison of the CN FDBPM and RK FDBPM methods is obtained by choosing mesh sizes which guarantee the same numerical dispersion in both cases 8. II. THE NONLINEAR CN FDBPM METHOD The application of the finite differences implicit in CN scheme plus the TBCs to the resolution of the scalar Fresnel equation produces a nonlinear system of equations that must be solved at every propagation step. The fastest way to solve it is to linearize the system using the known value of the field at every point of the previous propagation step to estimate the nonlinear refraction index in the actual propagation step. If better accuracy is desired, it is possible to iterate, using the new field obtained, to make a better estimation of the nonlinear refraction index and recalculate the field. In the described technique, a system of linear equations must be solved at every iteration of the nonlinear method. The system solution may be achieved by LU decomposition, but this is a very expensive method in terms of memory and CPU time, so it is preferable to exploit the sparse characteristic of the system matrix using, for example, the Gauss Seidel Ž GS. method 9. This method is the one that has been used in this paper to perform the CN FDBPM algorithm. The use of the Crank Nicholson scheme together with the Gauss Seidel method, for the usual mesh sizes in nonlinear optical problems, reduces the memory requirements at least two orders of magnitude and the consumed CPU time one order of magnitude, depending upon the accuracy imposed on the GS method. In most cases of interest, one single iteration of the non-linear method is enough to get sufficiently accurate results. The reason is that the nonlinear coefficient of the usually employed materials is very small, as happens with the propagation step, so that the error made in the estimation of the nonlinear coefficient by means of the field in the previous propagation step is very small. Regardless of the improvement introduced by the GS method, the need to solve at least one linear system at every propagation step considerably increases the computation time, especially when very fine meshes are used to reduce numerical dispersion. III. THE NONLINEAR RK FDBPM METHOD WITH TBCS The computational effort of the previously described technique can be reduced if an explicit scheme is employed; however, the stable nonlinear explicit methods proposed to 418 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August

2 date have the disadvantage of being incompatible with the TBCs 4, 8. Absorbent boundary conditions considerably increase the transversal magnitude of the numerical problem, and thus the CPU time and memory needed. To have an explicit stable nonlinear method compatible with the TBCs, the application of the fourth-order Runge Kutta method to the solution of the scalar Fresnel equation has been developed. Under the slowly varying approximation, the propagation of optical beams in a nonlinear Kerr saturable-type waveguide is governed by the Fresnel equation A A A Ž. jk N k n x, y, z, A N 0 0 A z x y Ž 1. Ž. where n x, y, z, E is the nonlinear refractive index given by A L NL n Žx, y, z, E. n Ž x, y, z. C Ž x, y, z. Ž. 1 A and A ' NL E is the envelope of the transversal electric field normalized to the medium nonlinear coefficient. In Ž., nl is the linear part of the refractive index, CNL accounts for the spatial distribution of the nonlinear Kerr saturable effect, and is 1 for saturable media and 0 for nonsaturable ones. The first step in the developed technique is to discretize the transversal partial derivatives using a second-order centered finite-difference scheme. In doing so, the Fresnel differential equation is transformed into the following system of nonlinear ordinary differential equations: da 1 A A A dz jk0 N x p, q p 1, q p, q p 1, q A A A p, q 1 p, q p, q 1 y Ž. k n N A Ž 3. 0 p, q p, q which is well suited to the application of the fourth-order Runge Kutta method. To do it, the transversal mesh points are consecutively numbered from left to right and from down to up; the resulting transversal operator is labeled FŽz, AŽ z.., A A 1, 1, A 1,,..., A 1, Nx,..., A, 1,..., A, Nx, A Ny,1, A Ny,,..., A Ny, Nx being the vector whose components A k are the values Ap, q arranged as described, i.e., with k p qn x, Nx being the width and Ny the height of the transverse mesh. In doing so, Eq. Ž. 3 becomes Ž. da z dz where the transversal operator is Ž Ž.. Ž. F z,a z 4 1 FŽ z,a. MŽz, A. A, Ž 5. jk N 0 M being the pentadiagonal matrix obtained from the transversal discretization. For the mesh points interior to the contour, its elements are 1 1 Mk, k 1, Mk, k Nx, x y Ž. Mk, k k0 np, qž z, Ap, q z. N, x y Ž 6. while for the mesh points at the contour, the field is calculated with the TBCs that relate the value of the field at any point in the contour with the value of the field at the adjacent point interior to the contour. For example, at any point on the right boundary, the TBC condition is Ž. Ž. A A exp jk x 7 Nx 1, q Nx q x where ANx 1, q is the value of the field at a point on the boundary, ANx q is the field at the adjacent left point, and the value of k x is calculated from the previous step field at the points Ž N 1, q. and Ž N, q. from the expression x x Ž. Ž. A A exp jk x. 8 Nx, q Nx 1, q x To get the field AŽ z z. from AŽ z. 0 0, the RK algorithm is applied; notice that the evaluation of the right-hand side of the nonlinear system of ordinary differential equations is easily performed by multiplying a sparse matrix Ž M. by a vector, Ž A. which is a fast operation, involving fewer than 5NxNy multiplications. IV. NUMERICAL RESULTS Numerical simulations have been carried out to study the behavior of the TBCs within the RK FDBPM algorithm and the performance of the method when analyzing nonlinear structures. The TBCs behavior has been tested by comparing simulation results with analytical solutions for the diffraction, and oblique incidence on the computational boundary, of a Gaussian beam propagating through a linear homogeneous medium. The mesh size in both transverse coordinates and the propagation step for these experiments are shown in the figure captions. In Figure 1, analytical and numerical results for the diffraction of a Gaussian beam propagating in the z-direction are plotted together, and it can be seen that they are almost indistinguishable. Figure shows a comparison of the normalized power in the computational window obtained by the numerical and analytical methods. These results, which are comparable to those obtained in, show that the behavior of the RK FDBPM dealing with the outflow of the beam power by the four boundaries at the same time is excellent; furthermore, the behavior is fairly good even when the fraction of power remaining in the computational window is very small. In Figure 3, the incidence on a boundary of a Gaussian beam propagating with a tilt angle of.5 with respect to the z-axis is shown and compared with analytical results for different propagation distances. It can be seen that the beam, which propagates almost diffractionless, passes through the boundary almost completely without distortion. Figure 4 shows a comparison between the numerically and analytically calculated power of the beam in the computational window MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August

3 Figure 1 Gaussian beam diffraction in a linear homogeneous medium, 1550 nm, x y z 1 m, N 1, spot size 0.5 m. numerical RK FDBPM with TBCs, analytical Figure Variation with distance of the normalized power in the computational window for Gaussian beam diffraction with the same data as Figure 1. analytical, numerical as a function of the propagation distance. It can be seen that very good agreement between both results is obtained up to the distance where 75% of the initial power has been radiated. After that distance, the numerically calculated power in the computational window is greater than it should be due to reflection from the TBCs. However, it must be noted that the numerically calculated power tends asymptotically to zero with the propagation distance, resulting in a reflection coefficient of , calculated as the remaining normalized power in the computational window when the beam has completely passed through the boundary Ž z 4000 m.. Obviously, the reflection coefficient obtained with this technique Ž TBCs and RK FDBPM. is not as good as the one obtained with the TBCs and the CN FDBPM for which they were originally developed, but, as will be shown in the next paragraphs, it can be enough for many practical situations. From previous results, the correct behavior of the TBCs when applied to the proposed RK FDBPM technique has been stated; however, where the proposed method shows its superior performance is when dealing with nonlinear structures where an accurate description of dispersion and nonlinearity must be achieved. To compare the behavior of the CN FDBPM and the RK FDBPM techniques when analyzing nonlinear structures, two previously reported problems have been simulated: the propagation of a three-dimensional Gaussian beam in an optic slab with nonlinear saturable Kerr-type substrate 6, and the study of the nonlinear directional coupler working modes 7. The excitation and slab s refractive indexes are the same as those of the references, and are reproduced in the figure captions for convenience. Figure 5 shows the result of propagation of a Gaussian beam in a slab with a nonlinear saturable substrate 6 for z 70. Figure 5Ž. a has been obtained by means of the CN FDBPM technique, while Figure 5Ž b. has been obtained with the RK FDBPM technique. In both cases, the mesh sizes were x 0.075, y 0.3, and z 0.01, which have been chosen to keep the numerical dispersion under % for at least 95% of the bandwidth of the Gaussian excitation 8. In this experiment, the computational effort of the CN FDBPM technique is twice the effort of the RK FDBPM Žthe CPU times on an HP workstation were 4h, 38 min for the CN FDBPM and, 44 min for the RK FDBPM.. Comparing both figures, it is clear that the CN solution presents a higher penetration of the beam in the substrate, while in the RK solution, a greater amount of energy remains trapped by the film. To find which of the previous results is better, the same simulation has been repeated with both techniques with reduced mesh sizes Žnumerical dispersion below 0.06%, x 0.05, y 0.05, and z 0.01., both methods yielding similar solutions. The RK FDBPM technique presented in Figure 6 Ž CPU time 1 h, 50 min., showing good agreement with previously published data 6. As compared with 40 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August

4 Figure 3 Gaussian beam propagation through a TBC, 1550 nm, x y z 1 m, N 1, spot size 10 m. numerical Ž RK FDBPM with TBCs., analytical Figure 4 Variation with distance of the normalized power in the computational window for Gaussian beam propagation through a TBC with the same data as in Figure 3. analytical, numerical the plots of Figure 5Ž. a and Ž. b, it can be stated that, when dealing with nonlinear structures, the RK FDBPM technique gives more accurate results and is more computationally efficient than the standard CN FDBPM technique with equal mesh sizes. This result is due to the fact that the nonlinearity is solved with fourth-order precision in the RK algorithm, while it is only second order in the CN method. To further validate the performance of the proposed technique when dealing with nonlinear structures, a directional coupler with a nonlinear substrate 7 has been analyzed. In Figures 7Ž. a and Ž. b, propagation in this device, when excited with a Gaussian field profile closely resembling the mode trapped in a single arm of the coupler, is presented. Figure 7Ž. a corresponds to simulation results obtained by means of the CN FDBPM technique Ž CPU time 17 h, 1 min., while Figure 7Ž b. has been obtained with the RK FDBPM technique Ž CPU time 3 h, 6 min.. Transversal mesh sizes have been fixed to k0 x 0.1, k0 y 0.1, k0 z in both cases, which assures that the numerical dispersion is Figure 5 Results of the propagation of a Gaussian beam in a nonlinear slab at z 70. Ž. a CN FDBPM with TBCs. Ž. b RK FDBPM with TBCs. x 0.075, y 0.3, z 0.01, A , x m, w0 197 m, a 1, N 1.644, d Excitation: Ax,y,z 0 A Ž. exp Ž x x. Ž y a. w Refractive index: n c.647 x 0 Ž. n x d n.647 A Ž1 A. s x d n x, y, A f MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August

5 Figure 6 Gaussian beam propagation in a slab with nonlinear substrate. RK FDBPM with reduced mesh size: x 0.05, y 0.05, z Excitation and refraction index are the same as those in Figure 5. under % for 95% of the excitation bandwidth. When compared with previously published results Ž 7 reprinted in Fig. 8 with permission., it can easily be seen that the RK FDBPM exhibits much better agreement than CN FDBPM. As the numerical dispersion has been kept very low and the same for both simulations, the better performance of the RK technique can only be explained as a consequence of the nonlinear fourth-order precision of the RK technique. V. CONCLUSIONS An RK FDBPM method to analyze nonlinear 3-D optical devices under the scalar slowly varying envelope approximation has been presented, and its performance has been compared with the standard Crank Nicholson technique. The method is based on the finite-difference discretization in the transversal coordinates and on the application of the fourthorder Runge Kutta technique in the propagation coordinate, and this allows transparent boundary conditions Ž TBCs. to be included. Two different sets of numerical experiments have been carried out: 1. to determine the compatibility of the RK FDBPM method with the TBCs,. to establish the ability of the proposed method to deal with nonlinear devices. From the first set of experiments, it is shown that, although the reflection coefficient of the TBCs obtained with the proposed technique is not as good as the one attainable within the CN FDBPM, the TBCs in conjunction with the RK FDBPM lead to a continuous outflow of energy from the computational window, giving a reflection coefficient that can be sufficiently small for many practical situations. On the other hand, the reduction of the size of the computational window allowed by the use of TBCs greatly increases the numerical efficiency of the proposed RK FDBPM algorithm. The second set of experiments clearly evidences the greater accuracy and computational efficiency of the RK FDBPM Figure 7 Gaussian beam propagation through the left arm of a nonlinear directional coupler. Ž. a CN FDBPM with TBCs. Ž. b RK FDBPM with TBCs. k0 x 0.1, k0 y 0.1, k0 z 0.003, N 0.5, k0 1, A , k0x0 1.5, k0 0.7, k , k d 1.5, k a 1. Excitation: Ax,y,z 0A Ž. exp Ž x x. 4 y 4. Refractive index: n Žx, y, A. ½.5 A d a x d a 0.5 A d a x d a 4 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August

6 Propagation Method, IEEE J. Quantum Electron., Vol. 8, Jan. 199, pp Y. Chung and N. Dagli, An Explicit Finite Difference Beam Propagation Method: Application to Semiconductor Rib Waveguide Y-Junction Analysis, Electron. Lett., Vol. 6, No. 11, 1990, pp Y. Chung and N. Dagli, Analysis of z-invariant and z-variant Semiconductor Rib Waveguides by Explicit Finite Difference Beam Propagation Method with Nonuniform Mesh Configuration, IEEE J. Quantum Electron., Vol. 7, Oct. 1991, pp H. Liu and W. Wang, Beam Propagation Analysis of the Non- Linear Tapered Optical Waveguide, IEEE Microwa e Guided Wa e Lett., Vol. 5, Feb. 1995, pp D. Mihalache, D. M. Baboiu, D. Mazilu, L. Torner, and J. P. Torres, Gaussian-Beam Excitation and Stability of Three-Dimensional Non-Linear Guided Waves, J. Opt. Soc. Amer. B, Vol. 11, No. 7, 1994, pp A. B. Aceves, A. D. Capobianco, B. Constantini, C. De Angelis, and G. F. Nalesso, Beam Dynamics in Non-Linear Coupled Slab Waveguides: Three-Dimensional Variational Analysis, J. Opt. Soc. Amer. B, Vol. 11, July 1994, pp J. de Oliva Rubio and I. Molina Fernandez, Tecnicas de Diferencias Finitas para el Analisis de Estructuras Opticas Tridimensionales No-Lineales, Actas del XI Simp. Nacional de la Union Cientıfica Internacional del Radio Ž URSI., Madrid, Spain, Sept. 1996, Vol., pp W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, London John Wiley & Sons, Inc. CCC NUMERICAL DISPERSION IN THE FINITE-ELEMENT METHOD USING THREE-DIMENSIONAL EDGE ELEMENTS Figure 8 Results obtained in 7 for the Gaussian beam propagation through the left arm of a nonlinear directional coupler. Reprinted with permission of the OSA and the authors. when analyzing nonlinear devices, which can lead to an 80% reduction of the computational time with respect to the standard CN FDBPM technique. From the presented results, it can be stated that the proposed technique can be a very fast and inexpensive simulation tool for the analysis and design of slightly powerradiating 3-D nonlinear optical devices with small requirements on the reflection coefficient from the numerical boundaries. ACKNOWLEDGMENT The authors wish to thank the Optical Society of America and the authors of 7 for their permission to reprint Figure 8, and especially Prof. Capobianco for providing them with high-quality PostScript copies of the figure. This work was supported by the Spanish CICYT, Project TIC C REFERENCES 1. H. P. Nolting and R. Marz, Results of Benchmark Tests for Different Numerical BMP Algorithms, J. Lightwa e Technol., Vol. 13, Feb. 199, pp G. R. Hadley, Transparent Boundary Condition for the Beam Gregory S. Warren 1 and Waymond R. Scott, Jr. 1 1 School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia Recei ed 8 January 1998 ABSTRACT: The discretization inherent in the ector finite-element method results in the numerical dispersion of a propagating wa e. The numerical dispersion of a time-harmonic plane wa e propagating through an infinite, three-dimensional, finite-element mesh composed of hexahedral and tetrahedral edge elements is in estigated in this work. The effects on the numerical dispersion of the propagation direction of the wa e and the electrical size of the elements are in estigated. The numerical dispersion of the tetrahedral edge elements is found to be dependent upon the polarization of the plane wa e propagating through the mesh. In addition, the dispersion of the tetrahedral elements is significantly smaller than the dispersion of the hexahedral edge elements. Both elements are found to ha e a phase error that con erges at the rate of O[( h ) ] John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: 43 49, Key words: finite-element method; numerical dispersion; wa e propagation I. INTRODUCTION The finite-element method is a popular technique in computational electromagnetics for solving three-dimensional vector field problems. For this type of application, edge ele- MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 6, August