For wave propagating in periodic structures made of semiconductor material, the build-in grating or corrugation induces dielectric or refractive index perturbation, and such perturbation results in an interaction between the forward and backward propagating waves. Historically, various
approaches like coupled wave theory and Bloch wave analysis have been adopted. Although these methods were proven to be equivalent , researchers have been keen on coupled wave theory because it is easier to understand, and because numerical algorithms can be implemented to solve the equations .
In a homogeneous, source-free and lossless medium, any time harmonic electric field must satisfy the vector wave equation
where the time dependence of the electric field is assumed to be , n is the refractive index and is the free space propagation constant.
In a semiconductor laser which has a transversely and laterally confined structure, the electric field must satisfy the one-dimensional homogeneous wave equation such that
Consider a multi-dielectric stack in which periodic corrugations are formed along one boundary as illustrated in Fig 8.1.