title: Fast and accurate propagation of coherent light.
abstract: 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 accuracyϵ>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 anN×Ngrid of output points given anM×Mgrid of input samples. Our algorithm maintains its accuracy throughout the computational domain.
authors.affiliation: Department of Applied Mathematics , University of Colorado at Boulder , Boulder, CO 80309, USA.
authors.forename: R D
authors.initials: RD
authors.lastname: Lewis
authors.forename: G
authors.initials: G
authors.lastname: Beylkin
authors.forename: L
authors.initials: L
authors.lastname: Monzón
keywords: Fraunhofer approximation; Fresnel approximation; Rayleigh–Sommerfeld integral; approximation by Gaussians; quadratures for band-limited functions; unequally spaced fast Fourier transform