Mathematical modeling of the solid angle function, part II: transmission through refractive media

Abstract

Previously, we developed a classical solid angle function that is valid only when the light is traveling within a homogeneous medium. As soon as the light path contains a refractive interface, the direct solid angle formula is invalid. A different approach must be used if one is to include refraction effects in the solid angle formulation. The variables of integration are given more of a physical interpretation than a geometrical one: by using the emitting point instead of the detection aperture as the basis for the coordinates system, we are able to use the symmetry of the light distribution to simplify the bounds of integration. With carefully chosen coordinate changes, we are thus able to obtain an expression for the solid angle subtended by a circular aperture from a point source situated in a different optical medium. The final refracted solid angle formula also includes the expression of Fresnel's transmission coefficient.