Normalized differential cross section for measurement of refractive ray distribution and focusing quality of a spherical lens: A probability model

Abstract

From the viewpoint of statistical physics, the characteristics of macroscopic matter are dictated by the probability distribution of numerous random micro-states. Drawing a parallel in geometric optics, we consider the focusing effect of a spherical lens as a statistical outcome of a vast number of randomly incident rays. We propose the concept of the normalized differential cross section (NDCS), which we interpret as the probability density for the deflection angle of a random ray, to describe the angular distribution of refracted rays. When an optical screen is positioned behind the lens, the NDCS can provide an analytical solution for the brightness distribution on the screen, allowing for the assessment of the lens's focusing quality. To validate this probability model, we have derived distribution laws and numerical characteristics for the deflection angles of rays passing through a Luneburg lens and a half Maxwell fish-eye lens. The analytical results align perfectly with simulations. Our theory circumvents the cumbersome computation of the diffraction field, particularly for gradient-index (GRIN) lenses, and sets a precedent for employing probability to analyze the distribution of refracted rays and focusing effects.