Abstracts

A recurring problem in optical design concerns theoptimization of the resolution of a beam whose focal depth isseveral orders of magnitude larger than its wavelength. Thecornerstone of such an optimization is the specification of afigure of merit by which the resolution of a typical beam is to beevaluated. In this thesis, the figure of merit takes the form ofthe mean encircled energy. Aside from providing simple equationsfor numerical optimization, this merit function allows for atractible analysis of the behavior of the optimal solution to theseequations through asymptotic methods. Such analysis is illustratedhere first for the simplest case of symmetric Gaussian beams forboth the 2D and 3D cases, where asymptotic analysis is used to findan accurate global approximation for the beam that maximizes meanencircled energy fraction over a given region of interest. For anapertured beam, the equation for the optimal mean encircled energyfraction takes the form of an integral eigenvalue equation, andasymptotic methods are possible for small aperture diameters. Onthe other hand, when the aperture width is large, it is better todecompose the beam into Hermite- or Laguerre-Gaussian modes. Here,a novel method is derived for computing the matrix whoseeigenvector - corresponding to the largest eigenvalue - containsthe coefficients of the individual modes in the optimal beam. Muchof the analysis presented here is derived for symmetric beams; onthe other hand, it is shown that the results for the optimalunapertured beam can be extended to elliptical beams with fewmodifications. The results in this thesis take on many forms,ranging from approximate expressions for parameters describing theoptimal beams to comparisons between the globally optimal beams andcorresponding simpler beams. One valuable lesson learned, however,is that although the globally optimal mean encircled energyfraction is rarely more than 10% greater than that for the optimalGaussian beam, it turns out that in most cases, the beam obtainedby an optimization technique specified here has far superior focaldepth and resolution properties than those of the optimalGaussians.