The role of aberrations in the relative illumination of a lens system

Abstract

Several factors impact the light irradiance and relative illumination produced by a lens system at its image plane. In addition to the cosine-fourth-power radiometric law, image and pupil aberrations, and light vignetting also count. In this paper, we use an irradiance transport equation to derive a closed form solution that provides insight into how individual aberration terms affect the light irradiance and relative illumination. The theoretical results are in agreement with real ray tracing.

Image irradiance in the absence of aplanatism and isoplanatism

“…By way of reversibility, if all of the rays from these SSFs were to travel towards the lens, they would fill the exit pupil, so the image irradiance for those rays would be determined by integrating over the exit pupil. This integration is precisely what is done to obtain relative illumination in computer-aided lens design programs [12][13][14][15][16]. However, calculations for relative illumination in lens design programs generally either assume that the object is sufficiently extended, or otherwise, if ray aberrations are present, then these aberrations would first be corrected prior to determining the relative illumination [12].…”

“…where x and y are coordinates at the image plane, E is the image irradiance given by radiometry, V is the volume of the PSF, P(x − x , y − y ) is the PSF, and G(x , y ) is the image distribution (with unit irradiance) predicted by geometrical optics. Note that E in equation ( 3) is of course generally regarded as a function of x and y (such as that given by the relative illumination of a lens system, which would be computed through any of the methods discussed in references [12][13][14][15][16]), but the explicit writing of E(x, y) in equation ( 3) has been omitted in order to minimize clutter in the expression. Equation ( 3) is not just an expression of the relative distribution of the image given by convolution that is discussed in the subject of Fourier optics.…”