Dynamical diffraction in sinusoidal potentials: uniform approximations for Mathieu functions

Eigenvalues and Fourier space eigenfunctions of Mathieu's equation are found in the short wavelength limit using a uniform approximation (method of comparison with a `known' equation having the same classical turning-point structure) applied to the Raman-Nath (RN) equation. The uniform approximation used here relies upon the fact that by passing into Fourier space the Mathieu equation can be mapped onto the simpler problem of a double-well …