Modal analysis of periodic planar phased arrays of apertures

A general method is established for the evaluation of the driving point admittance of a radiating aperture, fed by a waveguide of the same cross section as the aperture, in an infinite periodic planar phased array. The array may have an arbitrary element lattice and it may be covered by a dielectric layer. The coefficients of the waveguide modal expansion and of the Floquet series representing the electromagnetic field in the waveguide and in the radiation half-space, respectively, are determined by approximately enforcing the boundary conditions in the array plane through an application of Galerkin's method. By eliminating from the set of equations thus obtained the complex amplitudes of the waveguide modes and of the Floquet harmonics, the driving point admittance can be expressed as the ratio of two determinants of order N and N-1 (N being the number of the waveguide modes utilized), whose elements contain truncated bidimensional series, structurally similar to the well-known grating-lobe series. The expression allows relatively simple numerical computations if the Fourier transforms of the waveguide vector mode functions are known in closed form (as they are for rectangular or circular elements). The variation of the power reflection loss with scan angle has been numerically calculated for various array configurations. The results are in some cases substantially different from those predicted through the conventional grating-lobe series technique, which is based on the assumption of one-mode elements.