Asymptotic series solution of the paraxial equation in layered media

The paraxial wave equation for the electromagnetic field in a medium with layered index of refraction variation is solved by successive integrations to obtain an asymptotic series in
k^{-1}
. This solution is valid for complex
k
(lossy media). For a sinusoidal variation of index a compact form is obtained which always converges; consequently, using numerical methods and applying superposition, we may solve in arbitrary index variations with limited spatial spectral content. For other types of variation, e.g., Gaussian, the series is seen to converge only for values of the Fresnel parameter 1.