Fast and accurate propagation of coherent light

We describe a fast algorithm to propagate, for any user-specified accuracy, a time-harmonic electromagnetic field between two parallel planes separated by a linear, isotropic and homogeneous medium. The analytical formulation of this problem (ca 1897) requires the evaluation of the so-called Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is very large, it can be accurately approximated by asymptotic methods. In the large intermediate region of practical interest, where the oscillatory Rayleigh-Sommerfeld kernel must be applied directly, current numerical methods can be highly inaccurate without indicating this fact to the user. In our approach, for any user-specified accuracy ϵ>0, we approximate the kernel by a short sum of Gaussians with complex-valued exponents, and then efficiently apply the result to the input data using the unequally spaced fast Fourier transform. The resulting algorithm has computational complexity [Formula: see text], where we evaluate the solution on an N×N grid of output points given an M×M grid of input samples. Our algorithm maintains its accuracy throughout the computational domain.