Array of line sources that produces preselected plane waves

A generalized Poisson summation formula produces an explicit expression for the surface current on a plane that radiates a set of preselected plane waves. Specifically, the surface current is expressed as an integral in the complex plane of a function whose poles determine the plane-wave directions of propagation and whose residues determine the corresponding plane-wave amplitudes. When only a finite number of plane waves are specified, the surface current is a continuous function. If an infinite number of plane waves are specified, the surface current can degenerate into a discrete set of line sources. In such situations, the expression for the surface current determines both the location and strength of the line sources that radiate the infinite set of plane waves. One can specify both propagating and evanescent plane waves. Explicit formulas are derived for line sources that radiate plane-wave sets determined by both periodic and almost-periodic functions. The closed-form expressions can be applied directly to the problem of scattering by gratings. •A source given by an explicit expression radiates a set of preselected plane waves.•One can specify any number of both propagating and evanescent plane waves.•The source is either a continuous current sheet or a discrete set of line sources.•Evanescent waves can be used as free parameters to obtain an optimal source.•The theory can be applied directly to the problem of scattering by gratings.