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.

Numerical implementation of generalized Coddington equations for ophthalmic lens design

“…The main contributions in this point have been related to the inclusion of differential geometry aspects in Coddington approach, which has allowed an alternative derivation of the classical equations [11][12][13]. A matrix version of the GCE has also been proposed, and its equivalence with Equations (1) and (2) established [14].…”

“…The aim of this work is to describe in detail the numerical implementation of the equations proposed for ophthalmic lens design using the GRT approach [13,14,21]. The work has been divided in five sections plus this introduction.…”

“…The principal curvatures are these two extreme values of geodesic curvature. GRT explains what happens to the principal directions and principal curvatures of the wavefront after transfer and refraction [13,14,21]. For transfer, the principal directions of the local wavefront are unchanged, and their centers of curvature remain fixed.…”

“…These equations are obtained from the directional derivative of the vector form of Snell's law and from the Frenet equations for space curves [13]. GRT equations provide the refracted wavefront curvatures on the p direction (ρ′ u ) and on the q direction (ρ′ ν ) and σ′, the torsion of the refracted wavefront, using…”

Using refraction caustics to monitor evaporation of liquid drop lenses

“…The first approach employs the eikonal approximation for the deflected wavefield and expresses the caustic in terms of the metric properties of the deflecting surface. 3 - 7 This approach has been widely used in the analysis of the aberrations of imaging systems. The second approach classifies and describes certain types of caustics using the mathematics of catastrophe theory.…”