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An improved wide-angle finite-difference beam propagation method (WA-FD-BPM) formulated in terms of complex Padé approximants is proposed to investigate nonlinear optical waveguides. The formalism utilizes the Crank-Nicholson scheme. An adaptive update of the reference index and an iterative algorithm in every propagation step are utilized to accelerate convergence. Stability problems relative to high-order approximants are also addressed in this work.Index Terms-BPM, Crank-Nicholson, finite difference, nonlinear waveguide, wide-angle.

Resumo-Uma nova formulação numérica para simulação computacional da propagação de onda eletromagnética em guias ópticos com materiais magnetoópticos é descrita neste artigo. A formulação é baseada no método de propagação de feixe de ângulo largo (WA-BPM) usando diferenças finitas (FD) e o esquema Crank-Nicholson (CN) para a discretização da equação de onda, expandida por meio dos aproximantes de Padé. Um acoplador direcional não-recíproco para aplicações em sistemas de comunicações ópticas é investigado neste trabalho.Palavras-chave-comunicações ópticas, dispositivos nãorecíprocos, acoplador direcional, método da propagação do feixe, diferenças finitas, propagação não-paraxial, aproximantes de Padé.I. INTRODUÇÃO ARA a implementação de redes ópticas de comunicação que permitam taxas de transmissão de sinais cada vez mais altas, um diversificado conjunto de dispositivos ativos e passivos tem sido objeto de pesquisas em todo o mundo. São lasers, fotodetectores, amplificadores ópticos, chaves, moduladores, filtros, e tantos outros. Dentre estes, os que empregam materiais magnetoópticos formam uma importante categoria de dispositivos, na qual os mais utilizados são os isoladores [1]-[5] e os circuladores [5], [6]. Os isoladores são utilizados para impedir que a luz refletida em algum trecho do sistema retorne para a fonte óptica. Eles estão presentes nos sistemas que empregam amplificação óptica. Os circuladores fazem parte dos esquemas de extração e inserção de sinais em sistemas que utilizam multiplexação por divisão em comprimento de onda (WDM-wavelength division multiplexing). A operação dos dispositivos não-recíprocos é baseada na diferença entre as constantes de propagação do modo magnético transversal (TM) para as duas direções de Este trabalho foi parcialmente financiado pela CAPES (bolsa de estudos de mestrado), CNPq (proc. 300834/97-7), FAPESP (proc. 97/12996-9) e PRONEX (proc. 41/96.0921/00).

A wide-angle finite-difference beam propagation method for the solution of coupled nonlinear wave equations is introduced in this paper. The formalism is expanded in terms of Key words: beam propagation method (BPM); finite difference (FD); wide-angle; Crank-Nicolson (CN); Padé approximants INTRODUCTIONThe demand for bandwidth in telecommunication systems has grown at considerably high rates in the last few years due to the widespread use of multimedia and Internet resources around the world. Fiber-optic networks and WDM techniques are considered the most convenient solutions to accompany this evolution. On the other hand, the switching time limitations of electronic and electrooptic devices have brought increased attention to purely optical switching by means of nonlinear materials. In such materials, light propagation is governed by the Kerr effect, by which the refractive index changes with the applied optical power density according to a quadratic function [1, 2]. A large variety of devices utilizing nonlinear materials have been proposed in the literature, such as logical gates, optical switches, power dividers, couplers, Y junctions, and Mach-Zehnder interferometers [3][4][5][6]. Guided-wave structures based on nonlinear materials have been investigated by numerical techniques that are based mostly on finite elements and finite differences [7][8][9] (and references therein). The combination of these techniques with the beam propagation method allows one to determine not only the parameters of the electromagnetic wave, but also to visualize the wave propagation through the structure. On the other hand, the analysis of such structures has been restricted to the solution of the nonlinear Helmholtz equation under the paraxial approximation, which implies low aperture angles as well as low refractive index contrast.Several techniques were proposed in the literature to circumvent this limitation. The most celebrated technique is based on Padé approximants as suggested by Hadley [10]. Wide-angle analysis of nonlinear waveguide has been recently reported in the literature by Flamino et al. [2] and Yasui et al. [11], both based on the Crank-Nicolson scheme. Unfortunately, these formalisms do not handle nonlinear-coupled wave equations.In this paper, an improved finite-difference beam propagation method (FD-BPM) formulation based on Padé (1,1) approximants and the Crank-Nicolson scheme is utilized to solve the coupled wave nonlinear equation. The proposed wide-angle formalism is validated with a nonlinear X-coupler. An iterative algorithm is utilized in every propagation step in order to optimize convergence and reduce computational time [8,9]. Additionally, a transparent boundary condition (TBC) is utilized on the edges of the computational window to avoid reflections back to the region of interest [12,13]. This paper is organized as follows: Section 2 describes the extension of the wide-angle beam propagation method to the analysis of waveguide structures with two simultaneous propagating waves, section ...