Abstract This work investigates an alternative numerical scheme for the solution of an exact formulation based on Geometrical Optics (GO) principles to synthesize offset dual reflector antennas. The technique is suited to solve a second-order nonlinear partial differential equation of the Monge-Ampère type as a boundary value problem. An iterative algorithm based on Newton's method was developed, using axis-displaced confocal quadrics to locally represent the subreflector surface, thus enabling an analytical description of the partial derivatives within the formulation. Such approach reduces discretization errors, as exact expressions for the mapping function and its derivatives are analytically determined. To check the robustness of the methodology, an offset dual-reflector Gregorian antenna was shaped to provide a Gaussian aperture field distribution with uniform phase within a superelliptical contour. The shaped surfaces were further interpolated by quintic pseudo-splines and analyzed by Physical Optics (PO) with equivalent edge currents to validate the synthesis procedure at 11,725 GHz.

We derive an amplitude equation for the stationary instability for the isothermal double diffusive system with cross diffusion. A quintic term is computed in terms of rescaled cross-diffusion constants. This quintic term is stabilizing in the vicinity of the tricritical point. The existence of a tricritical point indicates the presence of a hysteresis loop whose parameters are also presented.

Derivamos una ecuación de amplitud para la inestabilidad estacionaria del sistema isotérmico doble difusivo con difusión cruzada. Se calcula un término a quinto orden en términos de las constantes re-escaladas de difusión cruzada. Este término es estabilizador en la vecindad del punto tricrítico. La existencia de un punto tricrítico indica la presencia de un ciclo de histeresis, cuyos parámetros se presentan en esta carta.