In this paper, an extension of the effective index method (EIM) to waveguiding structures containing ideal or saturable third-order nonlinear materials is presented. By applying separation of variables to the dominant field component, the complete problem is subdivided into two scalar problems in the lateral and transverse direction, as in the case of the normal EIM. Making use of the strong transverse confinement, as observed in most real waveguide structures, the nonlinear index changes of the various transverse sections can be lumped into nonlinear effective indexes of the equivalent layered planar structures. By using these nonlinear effective indexes in self-consistent field calculations in the transverse direction, a complete approximate solution is obtained. In this way, the amount of computational effort required for the calculation of the effective indexes and field profiles of the waveguides can be reduced significantly.
Krijnen, G. J. M., Hoekstra, H., & Lambeck, P. (1994). A new method for the calculation of propagation constants and field profiles of guided modes of nonlinear channel waveguides based on the effective index method. Journal of lightwave technology, 12(9), 1550-1559. https://doi.org/10.1109/50.320937