General solution of the problem of perfect geodesic lenses for integrated optics

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Abstract
The problem of designing waveguide geodesic lenses for integrated optical processing is considered. Geodesic lenses are, at present, the best solution whenever crystals of high refractive index must be used as substrates. A good geodesic lens has to be constituted by a surface of rotation, coupled without discontinuities with the planar optical circuit. The present paper derives, by an analytical procedure, the general expression for the profile of aspherical geodesic lenses able to form perfect geometrical images of the points of two given concentric circles on each other. In particular, the equations that characterize a family of lenses having two conjugate foci external to the lens depression are given. The case of one external and one internal focus is only outlined because it is less important in practice. The advantage of this analytical method is represented by the clear theoretical approach that allows a large flexibility in the lens design without long expensive computations.

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