Refracting the k-function: Stavroudis's solution to the eikonal equation for multielement optical systems.

The k-function of Stavroudis describes a solution of the eikonal equation in a region of constant refractive index. Given the k-function describing the optical field in one region of space, and given a prescribed refractive or reflective boundary, we construct the k-function for the refracted or reflected field. This procedure, which Stavroudis calls refracting the k-function, can be repeated any number of times, and therefore extends the usefulness of the k-function formalism to multielement optical systems. As examples, we present an analytic solution for the k-function, wavefronts, and caustics generated by a biconvex thick lens illuminated by a plane wave propagating parallel to the symmetry axis, and numerical results for off-axis plane-wave illumination of a two-mirror telescope.

Wavefront and caustics of a plane wave refracted by an arbitrary surface.

A simple expression is given for the k-function associated with the general solution of Stavroudis to the eikonal equation for refraction or reflection of a plane wave from an arbitrary surface. Using this result, we specialize the solution to derive analytic expressions for the wavefront and caustic surfaces after refraction of a plane wave from any rotationally symmetric surface. The method is applied to evaluating and comparing the wavefront and caustic surfaces formed both by a planospherical lens and a planoaspheric lens used for laser beam shaping, which provides understanding of how the irradiance is redistributed over a beam as the wavefront folds back on itself within the focal region.

Laser beam shaping profiles and propagation.

We consider four families of functions--the super-Gaussian, flattened Gaussian, Fermi-Dirac, and super-Lorentzian--that have been used to describe flattened irradiance profiles. We determine the shape and width parameters of the different distributions, when each flattened profile has the same radius and slope of the irradiance at its half-height point, and then we evaluate the implicit functional relationship between the shape and width parameters for matched profiles, which provides a quantitative way to compare profiles described by different families of functions. We conclude from an analysis of each profile with matched parameters using Kirchhoff-Fresnel diffraction theory and M2 analysis that the diffraction patterns as they propagate differ by small amounts, which may not be distinguished experimentally. Thus, beam shaping optics is designed to produce either of these four flattened output irradiance distributions with matched parameters will yield similar irradiance distributions as the beam propagates.