  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.
