The wave equations for the inhomogeneous planar optical waveguides can be derived from the Maxwell's equations. If we take z as the propagation direction and let to represent the frequency of laser radiation, we have the following wave equations for one-dimensional inhomogeneous planar waveguides (Tamir, 1990)
for TE modes and
for TM modes. The planar waveguide we are considering here has a refractive index that varies continuously in the x direction. For the planar optical waveguide shown in Figure 8.1, our problem is to find the refractive index profile function in the core for a set of specified propagation constants.
We assume that this planar waveguide has a refractive index profile guiding N modes. The propagation constants are , in which is the value of n(x) as and n1 = sup n(x). Designing an optical waveguide is analogous to the inverse problem encountered in quantum mechanics. We are trying to get the potential function from the given bound states and scattering data. The wave equation for the TE modes can be easily transformed to an equivalent Schrodinger equation
We can see in our case the potential function V(x) is continuous and . The TE mode cases have been solved by Yukon and Bendow (1980) and Jordan and Lakshmanasamy (1989).
We now need to transfer the wave equation for the TM modes to Schrodinger-type equation to apply the inverse scattering method. In (8.2), the first derivative of Ex can be eliminated if we let The wave equation then becomes