Simplified formula for the illuminance in an optical system

Abstract

A formula is derived for the illuminance at any surface in an optical system. By tracing a single ray one can compute the flux density at the image plane or any other position along the ray. The formula involves the ratio of the products of the principal curvatures of the wave front as it approaches each surface to products of the same quantities after the wave front is refracted at each surface. A procedure is presented for determining the required principal curvatures by generalizing the Coddington equations to multiple surfaces for both meridional and skew rays. Results are applicable to both spherical and aspherical surfaces. Since principal radii of curvature specify points on the caustic surfaces, the formula and computation procedure automatically yields the equations for caustic surfaces as a by-product. To illustrate the computation procedure the illuminance and caustic surfaces are derived for an aspherical singlet.