We investigate the scattered light properties of a Luneburg lens approximated as a series of concentric shells with discrete refractive indices. The stepped Luneburg lens has been previously modeled at microwave wavelengths with full solutions for the electromagnetic field equations when the lens is of comparable size to the wavelength. We investigate the properties of a Luneburg lens at optical wavelengths using a geometric ray tracing technique. We develop a stack-based ray tracing algorithm with the PYTHON programming language that tracks all reflected and refracted rays generated at each optical interface. The code shows that a Luneburg lens with 40 steps and a refractive index power-law exponent of 0.55 will produce images of nearly all naked eye (<6) magnitude stars with an enclosed energy of 50% at a spatial resolution of 3.2 degrees. We find 72 cases of blended stars where a star with magnitude <6 falls within 3 degrees of angular separation from a star with magnitude < 1. The optical stepped Luneburg lens has promising applications for low-cost, continuous all-sky monitoring to obtain transit light curves of bright, nearby stars.
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