  The integral equation method is widely used in numerical simulations of 2D/3D
acoustic and electromagnetic scattering problems, which needs a large number of
values of the Green's functions. A significant topic is the scattering problems
in periodic domains, where the corresponding Green's functions are
quasi-periodic. The quasi-periodic Green's functions are defined by series that
converge too slowly to be used for calculations. Many mathematicians have
developed several efficient numerical methods to calculate quasi-periodic
Green's functions. In this paper, we will propose a new FFT-based fast
algorithm to compute the 2D/3D quasi-periodic Green's functions for both the
Helmholtz equations and Maxwell's equations. The convergence results and error
estimates are also investigated in this paper. Further, the numerical examples
are given to show that, when a large number of values are needed, the new
algorithm is very competitive.
