On the resolution enhancement of optical beams with extreme focal depth

A recurring problem in optical design concerns the optimization of the resolution of a beam whose focal depth is several orders of magnitude larger than its wavelength. The cornerstone of such an optimization is the specification of a figure of merit by which the resolution of a typical beam is to be evaluated. In this thesis, the figure of merit takes the form of the mean encircled energy. Aside from providing simple equations for numerical optimization, this merit function allows for a tractible analysis of the behavior of the optimal solution to these equations through asymptotic methods. Such analysis is illustrated here first for the simplest case of symmetric Gaussian beams for both the 2D and 3D cases, where asymptotic analysis is used to find an accurate global approximation for the beam that maximizes mean encircled energy fraction over a given region of interest. For an apertured beam, the equation for the optimal mean encircled energy fraction takes the form of an integral eigenvalue equation, and asymptotic methods are possible for small aperture diameters. On the other hand, when the aperture width is large, it is better to decompose the beam into Hermite- or Laguerre-Gaussian modes. Here, a novel method is derived for computing the matrix whose eigenvector--corresponding to the largest eigenvalue--contains the coefficients of the individual modes in the optimal beam. Much of the analysis presented here is derived for symmetric beams; on the other hand, it is shown that the results for the optimal unapertured beam can be extended to elliptical beams with few modifications. The results in this thesis take on many forms, ranging from approximate expressions for parameters describing the optimal beams to comparisons between the globally optimal beams and corresponding simpler beams. One valuable lesson learned, however, is that although the globally optimal mean encircled energy fraction is rarely more than 10% greater than that for the optimal Gaussian beam, it turns out that in most cases, the beam obtained by an optimization technique specified here has far superior focal depth and resolution properties than those of the optimal Gaussians.