A continuum model of the dynamics of coupled oscillator arrays for phase-shifterless beam scanning

The behavior of arrays of coupled oscillators has been previously studied b y computational solution of a set of nonlinear differential equations descr ibing the time dependence of each oscillator in the presence of signals cou pled from neighboring oscillators. The equations are sufficiently complicat ed in that intuitive understanding of the phenomena which arise is exceedin gly difficult. We propose a simplified theory of such arrays in which the r elative phases of the oscillator signals are represented by a continuous fu nction defined over the array. This function satisfies a linear partial dif ferential equation of diffusion type, which may be solved via the Laplace t ransform. This theory is used to study the dynamic behavior of a linear arr ay of oscillators, which results when the end oscillators are detuned to ac hieve the phase distribution required for steering a beam radiated by such an array.