Non-Uniform Gratings That Radiate Preselected Plane Waves

A generalized Poisson summation formula produces exact expressions for the locations and strengths of line currents that radiate preselected plane waves. The line currents can be in free space or placed above a perfectly conducting ground plane. Closed-form expressions are obtained for the line current locations and amplitudes when the plane-wave parameters have periodic amplitudes and propagation vectors. A corresponding wire-impedance solution is derived that employs thin-wire approximations and takes into account all multiple interactions between wires. Remarkably, this solution holds even when the required impedance values are nonperiodic. The plane-wave parameters within one period determine the period of the impedance values and, thus, determine how many different types of loaded wires are required. For general nonperiodic plane-wave parameters, the line-source locations and amplitudes can be determined from an FFT approach involving a complex-contour integral, and the corresponding wire impedances can be computed numerically. With these impedance solutions, the number of degrees of freedom for the planar grating has been increased significantly to allow for the creation of a wide range of plane-wave fields.