Design and optimization of an irradiance profile-shaping system with a genetic algorithm method.

We develop a genetic algorithm (GA) optimization method and use it in the design of a refractive-beam profile-shaping system. In this application, we employ the GA to determine the shape of one surface of the primary beam profile-shaping element in our system. The GA is instructed to vary the shape of this surface such that the output intensity profile is flat on a spherical surface some distance away. The GA does this while insuring that only a specified area of the output surface is illuminated. The calculation of the intensity profile is based on geometrical optics and is accomplished exclusively through ray tracing, giving this method broad applicability.

Development and testing of a holographic projection system.

We describe the development and testing of a holographic projection system that is used to produce micro-optical devices. The projector uses a two-dimensional two-phase-level diffraction grating to produce multiple coherent beams and an interferometric optical system behind the grating to recombine the beams to produce interference patterns that have been recorded within a photosensitive substrate. The two different substrates that we used are a diazo imaging material and a bisbenzocyclobutene (BCB) polymeric resin for fabrication of surface relief microstructures. After the exposed photosensitive substrate is developed, the recorded interference pattern forms a micro-optical device. The analysis and testing of these micro-optical devices show promise that this technique can form patterns uniformly over a region of several centimeters in diameter on flat or curved substrates. The experimental testing results of these micro-optical devices demonstrate that this method is a simple and energy efficient system to produce microstructures compared with other methods. These devices may be used as a new generation of directional light filters or monolithic microlenslet arrays that may have applications in communications, display, and components technologies.

Optical analysis of an ultra-high resolution two-mirror soft x-ray microscope.

Promoted by the successful application of multilayer coated optics in soft x-ray imaging experiments in solar physics and projection lithography, several groups have designed, analyzed, fabricated, and are testing Schwarzschild multilayer soft x-ray microscopes. Simulations have indicated that diffraction limited performance of a spherical Schwarzschild microscope operating near 100 Å will be limited to systems with a small numerical aperture of approximately 0.15 and a corresponding resolution, based on the Rayleigh criterion, of 3.3 times the wavelength of the incident radiation. In principle, a two aspherical mirror Head microscope, which satisfies the constant optical path length condition and the Abbé sine condition, should achieve diffraction limited performance for very large numerical apertures. For a practical soft x-ray microscope, surface contour errors, microroughness, reflectance of multilayer coatings, and variation of the angle of incidence over the multilayer substrates become significant factors in degrading system resolution and must be controlled before an ultra-high resolution, two-mirror microscope will be realized. For a 30x reflecting microscope with a numerical aperture ranging from 0.15 to 0.35, the effects on resolution of surface contour errors, tilts, and misalignments of the optics have been studied. Graded spacing of the multilayer coatings on the mirror substrates are required of a fast, two-mirror microscope.

Differential equation design of finite-conjugate reflective systems.

Based on the conditions of aplanatism, a differential equation design method has been applied to the optimization of finite-conjugate, multimirror systems for soft x-ray applications. The shapes of two surfaces in a multimirror system are determined by numerically solving differential equations for the mirror surfaces. The shapes of other mirrors in the system are used to control the remaining aberrations of those systems, such as astigmatism, field curvature, and distortion. Applications include a three-minor projection-lithography system, which has been optimized by this method and compared with results obtained by conventional optical design, and a two-mirror soft x-ray imaging microscopy system, which is predicted to yield a diffraction-limited performance for a large N.A.

Design of gradient-index lens systems for laser beam reshaping.

Refractive systems using two gradient-index lenses have been designed to convert a collimated Gaussian-profile laser beam into a plane wave with a uniform intensity distribution. The axial gradient-index distribution for two lenses is determined by using the energy conservation condition and the constant optical path-length condition. The design consideration and theoretical analysis are presented along with several applications.

Formula for the rms blur circle radius of Wolter telescope based on aberration theory.

A formula for the rms blur circle for Wolter telescopes has been derived using the transverse ray aberration expressions of Saha, Appl. Opt. 24, 1856-1863 (1985), Saha, Proc. Soc. Photo-Opt. Instrum. Eng. 444, 112-117 (1984), and Saha, Proc. Soc. Photo-Opt. Instrum. Eng. 640, 10-19 (1986). The resulting formula for the rms blur circle radius over an image plane and a formula for the surface of best focus are based on third-, fifth-, and seventh-order aberration theory predict results in good agreement with exact ray tracing. It has also been shown that one of the two terms in the empirical formula of VanSpeybroeck and Chase, Appl. Opt. 11, 440-445 (1972), for the rms blur circle radius of a Wolter I telescope can be justified by the aberration theory results. Numerical results are given comparing the rms blur radius and the surface of best focus vs the half field angle computed by skew ray tracing and from analytical formulas for grazing incidence Wolter I-II telescopes and a normal incidence Cassegrain telescope.

Optimization of three- and four-element lens systems by minimizing the caustic surfaces.

For an imaging optical system it is, in general, desirable to transform a collection of point sources of light into point images distributed over the focal plane with the appropriate magnification. In practice, this is achieved by varying the lens system parameters such that the spread of a bundle of rays from each object point over the image plane has been minimized. In this study, caustic surfaces are used to construct a merit function that describes the spread of the caustic surfaces from an ideal image point. This caustic merit function has been used to optimize a large collection of three- and four-element lens systems. The performance of the optimized lenses has been evaluated by comparing the rms blur circle radii vs field angle to that of similar lenses designed by conventional techniques. The average rms improvement has been calculated for optimized systems. Results indicate that minimizing the caustic merit function reduces the rms blur radii over the field of view and the total aberrations of the lens systems, particularly for systems with large apertures and wide fields of view.

Caustic merit function for optical design.

The two caustic surfaces, which are in turn the loci of the image points, are formed in the image region by each element of the source of an optical system. Lens optimization is achieved herein by adjusting parameters so that the caustic surfaces formed by rays from selected points of the object coalesce and converge to the corresponding Gaussian image points. A merit function, specifying the sum of the distances between each caustic surface and the corresponding Gaussians image, is defined and minimized subject to the constraints that the system focal length and physical length are constant and that the third-order aberrations of the system are zero. An optimization procedure, based on minimizing the caustic merit function, is used to design a singlet, doublet, and triplet lens system.

Design of a normal incidence multilayer imaging x-ray microscope.

Normal incidence multilayer Cassegrain x-ray telescopes were flown on the Stanford/MSFC Rocket X-Ray Spectroheliograph. These instruments produced high spatial resolution images of the Sun and conclusively demonstrated that doubly reflecting multilayer x-ray optical systems are feasible. The images indicated that aplanatic imaging soft x-ray /EUV microscopes should be achievable using multilayer optics technology. We have designed a doubly reflecting normal incidence multilayer imaging x-ray microscope based on the Schwarzschild configuration. The Schwarzschild microscope utilizes two spherical mirrors with concentric radii of curvature which are chosen such that the third-order spherical aberration and coma are minimized. We discuss the design of the microscope and the results of the optical system ray trace analysis which indicates that diffraction-limited performance with 600 Å spatial resolution should be obtainable over a 1 mm field of view at a wavelength of 100 Å. Fabrication of several imaging soft x-ray microscopes based upon these designs, for use in conjunction with x-ray telescopes and laser fusion research, is now in progress. High resolution aplanatic imaging x-ray microscopes using normal incidence multilayer x-ray mirrors should have many important applications in advanced x-ray astronomical instrumentation, x-ray lithography, biological, biomedical, metallurgical, and laser fusion research.

Theoretical analysis of stable and unstable aspherical laser cavities.

The low- order resonant modes of the stable and unstable aspherical resonators are studied by the Fox- Li method. An integral equation is given with which the effect of varying the aspherical cavity surface parameters on the field distribution, the energy loss and the intensity profile have been studied.

Wave front equation, caustics, and wave aberration function of simple lenses and mirrors.

Using the condition of constant optical path length for rays passing through an optical system, an equation for the wave front is presented in a simplified form. The wave front equation has been explicitly evaluated for a plane wave incident on a spherical reflector or a plano-convex lens. Then, the principal radii of curvature of the reflected or refracted wave front, evaluated directly from the wave front equation, are shown to locate the caustic surfaces of the optical system. From the wave front equation, a closed form expression for the wave aberration function for a plane wave reflected by a spherical mirror or a plano-corvex lens has been evaluated and compared to the results obtained from third-order aberration theory.

Formula for the density of tangent rays over a caustic surface.

The geometrical flux density (irradiance) is singular over caustic surfaces and, therefore, cannot be used effectively as a measure of the concentration of rays at or near the caustic surfaces. A finite substitute measure, the density of rays tangent to the caustic, may be obtained by dividing an element of incident flux by the area of the caustic formed by the associated rays. This gives a measure of the energy density over different regions of the caustic. As an example, the ray density over the caustic is evaluated for collimated light reflected from a spherical mirror. A similar calculation is performed for collimated light refracted by a plano-convex singlet lens. General formulas are presented for computing the ray density over the caustic for reflection of meridional rays by an aspheric surface. Also analytical and numerical algorithms are given for evaluating the ray density over the caustic in a multiinterface optical system.

Imaging characteristics of a conical primary, aspheric secondary x-ray telescope.

The imaging effectiveness of an x-ray telescope consisting of a simple conical primary reflector and an aspheric secondary reflector is compared with a paraboloidal primary, hyperboloidal secondary of the same dimensions (Wolter type II). The rms spot size for the Wolter type II is considerably smaller being 1/28 as large as that of the cone-asphere for incident collimated light 1 min off the optical axis and 1/6 as large for incident light 12 min off the optical axis for a typical configuration. The cone-asphere spot size, on the other hand, is one half as large as that for a hyperboloidal primary, asphere secondary telescope of the same dimensions.

Simplified formula for the illuminance in an optical system.

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.