Fourier Expansion of the Beam Propagation Operator in the Eigenvalue Domain

Propagation methods in the frequency domain are powerful tools for the analysis of electromagnetic field propagation within photonic waveguide devices. They can be classified as wide angle beam propagation or eigenvector expansion methods. In comparison to the wide angle beam propagation methods, eigenvector expansion methods inherently include an accurate computation of radiation waves, guided modes, and evanescent waves. Common numerical methods rely on the solution of eigenvalue problems or the solution of matrix equations of high order. Consequently, the computation time needed can be tedious. A method is proposed, which exploits the inherent advantage of eigenvector expansion methods, but reduces the computation time considerably as the resulting propagation algorithm is based on simple matrix vector multiplications, allowing a large number of discretization points. The method takes advantage of the compact eigenvalue spectrum of the system matrix. The operator can be periodically expanded so that a Fourier decomposition can be applied. The expansion is carried out under consideration of the physical interpretation of eigenvalues for radiation waves, guided modes, and evanescent waves.