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.

广义泊松求和公式为辐射一组预选平面波的平面上的表面电流生成明确的表达式。具体来说，表面电流被表示为一个函数的复平面​​中的积分，该函数的极点决定了平面波的传播方向，其余数决定了相应的平面波振幅。当只指定有限数量的平面波时，表面电流是一个连续函数。如果指定了无限数量的平面波，则表面电流可以退化为一组离散的线源。在这种情况下，表面电流的表达式决定了辐射无限平面波的线源的位置和强度。可以指定传播和渐逝平面波。对于辐射由周期函数和近似周期函数确定的平面波集的线源，推导出显式公式。闭式表达式可以直接应用于光栅散射问题。