The Continuum Model for Planar Arrays
As described in Section 2.6, coupled oscillator arrays can be constructed in a planar geometry in which each oscillator is coupled to more than the two nearest neighbors of the linear array case. In that section a Cartesian coupling topology is described in which each oscillator is coupled to four nearest neighbors, and the array boundary is rectangular. In such an arrangement, the phase distributions suitable for beam-steering are obtainable either by detuning the edge oscillators  or by injecting them with external signals with adjustable phase . Both of these approaches are treatable via the continuum model. Further generalizing the planar arrangement, one may use alternative coupling topologies such as the triangular lattice in which each oscillator is coupled to six nearest neighbors and the array boundary is triangular or the hexagonal lattice in which each oscillator is coupled to three nearest neighbors and the array boundary is again triangular [44, 45]. As will be shown in this chapter, these coupling topologies are also treatable using the continuum model.
4.1 Cartesian Coupling in the Continuum Model without External Injection
We begin with Eq. (2.6-3) for a 2M + 1 by 2N + 1 rectangular array with zero coupling phase replacing the discrete indices i and j with the continuous variables x and y, respectively; and we expand the phase function in a two-dimensional Taylor series retaining terms to second order. By ...