Accelerated algorithm for calculating spectra of aperiodic gratings

To compute the spectrum of aperiodic Bragg gratings, we address a direct scattering problem for the Helmholtz equation. Numerically, this problem necessitates the calculation of transfer matrix products, which involves multiple multiplications of matrix elements – polynomials that depend on the spectral parameter of the problem. We propose an accelerated algorithm to solve the scattering problem with second-order accuracy. This algorithm leverages the integral approach for discretization, a duplication strategy, the convolution theorem, and the fast Fourier transform. The computational complexity of this approach is asymptotically O(N log2N) arithmetic operations (multiplications) for a discrete grid of size N. Numerical simulations corroborate the algorithm’s second-order accuracy and its high computational speed, in accordance with the estimates obtained.