Classical calculation of differential cross section for a beam deflected by a concentric refractive index field

Ray tracing is a fundamental geometric-optics issue which gives a single ray path but seldom presents the collective behavior of light. The optical field distribution usually involves the calculation of an electromagnetic field and is rarely discussed from the perspective of geometric optics. However, in this paper, we show for a concentric medium with spherically symmetric refractive index, how the relative angular distribution of refractive beams can be obtained from the pure classical geometric optics method. As a measurement of the distribution, we introduce the concept of the differential cross section (DCS), which can be calculated from the relation between aiming distance and deflecting the angle of the ray. We present a general method to solve this relation from both Snell's law in a constant medium and the optical Binet equation (OBE) in a gradient-index (GRIN) medium. Even without observing the collective traces, the DCS can independently give a quantitative description for the deflected light density of concentric media at different directions. It may act as a reference index for the design of beam deflector.