Lattice sums for gratings and arrays

Lattice sums arising in quasiperiodic Green's functions for the Helmholtz e quation, over general two-dimensional arrays are investigated. The array su ms are related to those over a single quasiperiodic line of sources, and th eir difference is be expressed in terms of exponentially convergent series. It is shown that our expressions can be used to generate the sums pertaini ng to the case of photonic gap states, associated with complex quasiperiodi city (Bloch) vectors. The accuracy and computational speed of our expressio ns are illustrated. (C) 2000 American Institute of Physics. [S0022- 2488(00 )00611-3].