Generalised formulas for equivalent geodesic and nonuniform refractive lenses

Abstract

In guided wave optics, where wave propagation is confined to a thin dielectric film over a substrate, a geodesic lens is formed by profiling the substrate, most commonly by a circularly symmetric indentation. A standard transformation exists which associates the geodesic ray path over such an identation with the ray in a radially variable refracting medium. Hence the ray trace in a geodesic lens can be obtained from known ray paths in the refractive lens. When this procedure is followed for the focusing lenses known in microwave optics the resultant geodesic surface profiles all fall into the general formulation σ = Aρ + B sin−1 Cρ. Therefore, by a reversal of the process, a generalised formula for the variable refractive index laws can also be obtained. This is found in the relation Cr2/BC − 2r1/BC (ηr)(A/BC −1) + C(ηr)2A/BC = 0, η(r) being the varibale refractive indes). If now aditional forms of ray propagation are considered, such as nonimaging process, which have been ignored in microwave oprical devices, two effects come about. First the range of refractive indices less than unity or even containing poles or zeros arises, and secondly the reslutant law is not necessarily an explicit η, r relation. In the first instance it can be shown that, even with unrealistic refractive indices, relistic geodesic surfaces can stil be possible. In the second, devices are designed with specified ray traces applicable to integrated optical components. The nonexplixit refracticve index laws derived are shown stil to conform with the generalised law given above, and hence the specification of the parameters is direct and consequently the geodesic surface required is derived. The ray configurations give integrated optical components which are the analogues of such microwave components, as the variable power divider, directional coupler, beam mixer, magic T and ring resonator, all in fact that are required for general microwave and optical circuitry.