1434 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 10, OCTOBER 1998 Nonlinear PML Boundary Conditions for Nonparaxial BPM in (2) Materials Raimondas Petruskevicius, Gaetano Bellanca, and Paolo Bassi Abstract— Nonlinear perfectly matched layer (NL-PML) boundary conditions for a nonparaxial finite-difference beam propagation method are developed and applied to the simulation of spatial solitary waves in periodically poled second-order nonlinear materials. Their overall better performance is verified comparing their effectiveness with that of linear PML and transparent boundary conditions. Index Terms—Boundary conditions, nonparaxial BPM, PML, quasi-phase matching, spatial solitary waves. I. INTRODUCTION A LL-OPTICAL devices based on cascading effects in second-order nonlinear materials look very promising for next generation optical telecommunication systems [1]. Current success in producing periodically poled lithium nio- bate (PPLN) will overcome phase-matching problems related to birefringence of lithium niobate for these devices using the so-called quasi-phase-matching (QPM) technique [2]. The importance of powerful and flexible tools of analysis to simulate, design and optimize second order nonlinear QPM structures is consequently growing as well. The extension of the beam propagation method (BPM) from the linear (L) to the nonlinear (NL) case is probably the most popular approach used so far [3], [4]. Wide angle formulations must be used to simulate multiple waves travelling at very different angles, wave mixing with large phase mismatching, wide-angle spatial solitary wave interactions and scattering. This can be done, for example, using Pad´ e approximant operators, [3], [5]. A problem still open to discussion concerns the beam interactions with the computational domain boundaries. In fact, spurious reflections from the boundaries can dramatically degrade the quality of results. In linear BPM, the simple transparent boundary conditions (TBC) [6] or the more powerful perfectly matched layers (L-PML) [7], [8] are used. In this letter, NL perfectly matched layers (NL-PML) are proposed to obtain the new NL-PML BC. An example will show the overall better performance of the new BC compared to those of L-PML and TBC for wide angle NL-BPM. Manuscript received April 16, 1998; revised June 29, 1998. This work was supported by the Italian Ministry for University and Scientific and Technological Research and by the National Research Council. The work of R. Petruskevicius at the University of Bologna was supported by the “ICTP Programme for Training and Research in Italian Laboratories.” R. Petruskevicius is with the Institute of Physics, Vilnius LT-2600, Lithua- nia. G. Bellanca and P. Bassi are with the Dipartimento di Elettronica Infor- matica e Sistemistica, University of Bologna, I-40136 Bologna, Italy. Publisher Item Identifier S 1041-1135(98)07132-8. II. FORMULATION NL interactions among fields with the same polarizations and different frequencies (pump), (signal), and (idler) propagating in nonlinear media, such as PPLN, in a QPM configuration ( ), can be described using the nonparaxial scalar Helmholtz equation. For the field envelope of each field it holds (1) where are the free space wavenumbers, and are the index distribution and suitable reference indices at each frequency. The terms account for NL coupling because of , the th-order Fourier coefficient of the nonlinear susceptibility which is modulated by sign reversal with period [9]. The phase- matching condition can be written as , where is the projection of nonlinear grating wavevector along . To develop the NL-PML boundary conditions for nonlinear material, anisotropic conductivities along the coordinate in the PML region are introduced in (1). Performing complex anisotropic mapping of (1) [8], we get the nonlinear propagator operator in the PML region as (2) where , being the vacuum impedance and the constant refractive indexes of PML media for the three waves, usually equal to those of the media adjacent to NL- PML. Note that the L-PML are easily derived simply letting , i.e., neglecting NL coupling terms, in (2). A further degree of freedom concerns the PML outer limit: in both cases it can be either of Transparent (PML-TBC) or Dirichlet (PML- D) type. Introducing Pad´ e; (1, 1) approximants of operators 1041–1135/98$10.00  1998 IEEE