Beam steering dynamics in coupled oscillator based phased arrays with coupling delay

A continuum solution for the phase evolution of a coupled oscillator array with coupling delay, when one oscillator is step detuned in time, is derived from the discrete model solution previously published. From this Green's function a differential equation is derived for the Laplace transform of the phase dynamics. The solution of this differential equation is expressed as a sum of the eigenfunctions of the differential operator for a finite length array with one oscillator detuned. The dynamic behavior of the aperture phase for beam steering is obtained by superposing two of the Green's function solutions corresponding to antisymmetric excitation at each end of the array. The far zone beam steering behavior is obtained by summing the elemental radiated signals. Generalization to planar arrays appears to be straightforward. An alternative more flexible formulation suitable for early times is also presented.