The reflector antenna design problem requires to solve a second boundary value problem for a complicated Monge-Ampere equation, for which the traditional discretization methods fail. In this paper we reduce the problem to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes an linear optimization problem and algorithms in linear programming apply.

On the design of a reflector antenna 2

A numerical double-freeform-optical-surface design method is proposed for beam shaping applications. In this method, both the irradiance distribution and the wavefront of the output beam are taken into account. After numerically obtaining the input-output ray mapping based on Energy conservation using the variable separation method, the two freeform optical surfaces can be constructed simultaneously and point by point corresponding to the ray mapping based on Snell’s law and the constancy of the optical path length. The method is only applicable for separable irradiance distributions. However, such a restriction is fulfilled by many practical laser beam shaping examples. Moreover, the restriction can simplify the computation considerably. Therefore, the method may be quite useful in practice, although it is not applicable to more general cases. As an example, the method was applied to design a two-plano-freeform-lens system for transforming a collimated 20 mm Gaussian laser beam (beam waist: 5mm) into a uniform 10 × 40 mm2 rectangular one without changing the wavefront. Simulation results show that we can obtain a dual lens beam shaping system with the relative root mean square deviation of the irradiance ranging from 0.0652 to 0.326 and the power ratio concentrated on the desired region ranging from 97.5% to 88.3% as the output beam transfers from 0mm to 1000mm.

Beam shaping system design using double freeform optical surfaces

We design a mirror to form a directivity diagram defined as a vector function of one argument. An analytical solution for the problem of generating a line-shaped directivity diagram from a point source is derived. The mirror calculation reduces to solving an ordinary differential equation. A mirror to generate the line-shaped directivity diagram of angular size 120° and uniform intensity is designed.

Designing a mirror to form a line-shaped directivity diagram

A novel method of designing a mirror surface to generate a directivity diagram represented as a vector function of one argument is presented. A general relationship for the mirror surface for an arbitrary illuminating beam wavefront is derived as an envelope of a parametric family of surfaces. Each surface in the family transforms the input beam into a beam with plane wavefront of desired direction. For the spherical illuminating beam the mirror surface is given as the envelope of the family of rotational paraboloids. The envelope is represented as a family of curves given by the intersections of paraboloids with circular cones of rays from the point source.

Designing reflectors to generate a line-shaped directivity diagram

Surface reflectance of real-world materials is now widely represented by the bidirectional reflectance distribution function (BRDF) and also by spatially varying representations such as SVBRDF and the bidirectional texture function (BTF). The raw surface reflectance measurements are typically compressed or fitted by analytical models, that always introduce a certain loss of accuracy. For its evaluation we need a distance function between a reference surface reflectance and its approximate version. Although some of the past techniques tried to reflect the perceptual sensitivity of human vision, they have neither optimized illumination and viewing conditions nor surface shape. In this paper, we suggest a new image-based methodology for comparing different anisotropic BRDFs. We use optimization techniques to generate a novel surface which has extensive coverage of incoming and outgoing light directions, while preserving its features and frequencies that are important for material appearance judgments. A single rendered image of such a surface along with simultaneously optimized lighting and viewing directions leads to the computation of a meaningful BRDF difference, by means of standard image difference predictors. A psychophysical experiments revealed that our surface provides richer information on material properties than the standard surfaces often used in computer graphics, e.g., sphere or blob.

Perceptually motivated BRDF comparison using single image

(1980). The Optical Design of Reflectors. Optica Acta: International Journal of Optics: Vol. 27, No. 9, pp. 1305-1305.

W.T. Welford

Papers

The Optical Design of Reflectors