Spectral representations for rapid evaluation of periodic green’s functions

Spectral representations are developed for the Green functions of 3D problems with a 1D periodicity. The periodic Greenpsilas functions are represented in terms of continuous transverse spectrum rather than conventional discrete (Floquet) longitudinal spectrum. In the framework of these expansions, the periodic Greenpsilas functions are obtained in terms of a small number of direct source contributions, a small number of pole (Floquet mode) contributions, and a rapidly convergent integral that can be evaluated using a small number of quadrature nodes. The presented scheme works seamlessly for any practical periodicities and observer location within the periodic unit cell.