Focusing Properties of a Three-parameter Class of Oblate, Luneburg-like Inhomogeneous Lenses

Abstract

The focusing attributes of spherical Luneburg lenses, quite apart from their inherent physical fascination, offer the practical possibility of transforming low-gain feed patterns into equiphase aperture distributions of appreciable size. These latter yield, in their turn, considerable gain amplification for the far-field beams which they radiate. Such benefit is coupled moreover with an option, at least in principle, to scan beams without restriction on angular amplitude and without any gain/definition degradation penalty whatsoever. Attractive features of this sort notwithstanding, both their bulk and weight militate against widespread Luneburg lens application, and most certainly so in a satellite, size- and weight-averse setting. The question thus naturally arises whether it might be possible somehow to reduce Luneburg lens size and weight while retaining at least a vestige of its focusing power.We answer this question affirmatively by offering, as a most natural generalization of Luneburg's canonical refractive index profile, a three-parameter family for which the Fermat ray equation can be brought to quadrature in closed form. The three parameters in question are the center and surface dielectric permittivities κ_c and κ_s , respectively, and a geometric aspect ratio β > 1 which gauges the amount of sphere compression into an oblate, ellipsoidal form. With the Fermat solution duly established in this context, we go on to subject the rays to an additional, discontinuous Snell's law refraction at lens surfaces whenever the available choice κ_s > 1 is in fact exercised. The ray pencils encompassed by this framework provide beams with near-perfect collimation (κ_s = 1) and, for κ_s > 1, stigmatic focusing between conjugate foci at finite remove from lens faces. In addition, under the conditions having κ_s > 1 and sufficiently large κ_c , retrograde rays are encountered whose propagation reversal is due either to total reflection at the far lens face, or else to an outright internal bending. While situations of this sort are naturally bereft of any practical utility, and indeed are most unwelcome, nevertheless their presence provides ray pencils with considerable aesthetic allure and a healthy challenge to the mathematical underpinning.Although it is true that ray collimation to infinity is not exact en masse, individual rays can be so aimed through an appropriate choice of, say, κ_c , an inverse problem which we solve numerically ray-by-ray, and then average over the permittivity spread thus encountered, obtaining in this fashion a determinate law of required < κ_c > versus source location. Continuing in this spirit, we jot down a few remarks regarding the difficulty of unraveling the inverse problem of producing that class of refractive index profiles which, for any given oblate spheroid, will assure stigmatic focusing for conjugate foci arbitrarily prescribed along the compression axis. This difficulty is a sad consequence of the symmetry loss in passing from a sphere to its oblate cousins. At the same time the spherical standard, open to resolution in its natural coordinate system, should also succumb to attack in a cartesian setting, even though, clearly, no one would recommend this avenue as a preferred route. The relevance of such an inconvenient alternative derives from our belief that, should it ever be mastered for the sphere, it would at once vanquish also the inverse problem for the associated ellipsoid.We leave aside, by virtue of outright ignorance, all consideration of how it is that one can achieve some given refractive index in the microwave regime, be it through chemical composition or else through artificial dielectric means. Left aside likewise is any mention of Ohmic loss, which, in any practical application, must be acknowledged as a constraint upon one's gain ambitions.