Flux Density for Ray Propagation in Discrete Index Media Expressed in Terms of the Intrinsic Geometry of the Deflecting Surface

Abstract

An exact, analytical formula for the flux density (energy per unit area per unit time) over an arbitrary receiver surface for rays which have been reflected from or refracted through an arbitrary curved surface is derived. The change in flux density is associated with the geometrical concentration or spreading of the beam produced by the curvature of the deflecting surface. The formula is expressed in terms of the intrinsic geometry of the deflecting surface (Gaussian curvature, mean curvature, and normal curvature). An equation for the caustic surface is also derived. The general formulas are applied to calculate the flux density on an image plane when light from a point source is reflected from and refracted into a spherical surface. Equations are also given for the associated caustic surfaces. These yield, for paraxial rays, the standard mirror and lens formulas.