A method to construct modal fields for an arbitrary one- or two-dimensional intensity dependent refractive index structure is described. An arbitrary starting field is propagated along an imaginary axis using the Finite Difference Beam Propagation Method (FDBPM) based upon the Slowly Varying Envelope Approximation (SVEA). First the modes are found for the linear part of the refractive index structure. By suitably choosing the complex value of the propagation step, one mode is maximally increased in amplitude. After the nonlinearity has been put on, two methods are applied to find the modes for the nonlinear structure. One method is the same as the method used for the linear part, in the other method the propagation step is left unchanged. The applicability of the method is discussed and illustrated by a calculation on a waveguide with one-dimensional cross section having Kerr-type nonlinearity.
Wijnands, F. H. G. M., Wijnands, F., Hoekstra, H., Krijnen, G. J. M., & de Ridder, R. M. (1995). Numerical solution method of nonlinear guided modes with a finite difference complex axis beam propagation method. IEEE journal of quantum electronics, 31(5), 782-790. [10.1109/3.375923]. https://doi.org/10.1109/3.375923