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].