Three-Dimensional Wide-Angle Parabolic Equations with Propagator Separation Based on Finite Fourier Series

Abstract

A possibility of constructing wide-angle diffraction models using Fourier series decomposition of the propagation operator of one-way wave equations is investigated. The propagation operator is considered as a function of the propagation step, reference wavenumber, and transversal Laplacian operator, which appears under the square-root of the pseudodifferential operator in the theory of one-way equations. It is shown that in this operator formalism, Fourier series decomposition approximates the one-way propagator by a weighted sum of exponential propagators, whose structure is similar to the propagator of the standard or small-angle parabolic equation. The exact propagator is modified using Hermite interpolation polynomials in order to achieve two crucial properties that guarantee fast convergence of the Fourier series: propagator periodicity and continuity of its derivatives. It is demonstrated that for three-dimensional diffraction problems, contrary to the standard split-step Padé approach, the proposed wide-angle propagation model allows for using efficient numerical methods and operator splitting procedures available for the standard parabolic equation. As a result, it is possible to organize computations separately along each of the two coordinate axes that are perpendicular to the predominant direction of wave propagation.