On the dynamics of two-dimensional array beam scanning via perimeter detuning of coupled oscillator arrays

Arrays of voltage-controlled oscillators coupled to nearest neighbors have been proposed as a means of controlling the aperture phase of one-dimension al (1-D) and two-dimensional (2-D) array antennas. It has been demonstrated , both theoretically and experimentally, that one may achieve linear distri butions of phase across a linear array aperture by tuning the end oscillato rs of the array away from the ensemble frequency of a mutually injection-lo cked array of oscillators. These linear distributions result in steering of the radiated beam. It is demonstrated theoretically that one may achieve s imilar beamsteering in two dimensions by appropriately tuning the perimeter oscillators of a 2-D array. The analysis is based on a continuum represent ation of the phase in which a continuous function satisfying a partial diff erential equation of diffusion type passes through the phase of each oscill ator as its independent variables pass through integer values indexing the oscillators. Solutions of the partial differential equation for the phase f unction exhibit the dynamic behavior of the array during the beamsteering t ransient.