Donald G. Burkhard

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.

Flux Density for Ray Propagation in Discrete Index Media Expressed in Terms of the Intrinsic Geometry of the Deflecting Surface

An exact, analytical formula for the flux density (energy per unit area per unit time) over an arbitrary receiver surface for rays which have been reflected from or refracted through an arbitrary curved surface is derived. The change in flux density is associated with the geometrical concentration or spreading of the beam produced by the curvature of the deflecting surface. The formula is expressed in terms of the intrinsic geometry of the deflecting surface (Gaussian curvature, mean curvature, and normal curvature). An equation for the caustic surface is also derived. The general formulas are applied to calculate the flux density on an image plane when light from a point source is reflected from and refracted into a spherical surface. Equations are also given for the associated caustic surfaces. These yield, for paraxial rays, the standard mirror and lens formulas.

Flux density for ray propagation in geometrical optics

A general formula is derived that specifies the illumination (flux density) over an arbitrary receiver surface when light rays are reflected by or refracted through a curved surface. The direction of the deflected ray and its intersection with the receiving surface, used with the equation for the surfaces, lead to a transformation that maps an element of deflecting area onto the receiving area, by means of the jacobian determinant. A formula for the flux density along a ray path follows as a special case. An equation for the caustic surface is obtained from the latter. As an example radiation flux-density contours are calculated for a plane wave reflected from a sphere. Flux density and the caustic surface are calculated for a plane wave reflected onto a plane from a concave spherical lens and also for a plane wave refracted onto a plane through a hemisphere.

Caustic Surfaces and Irradiance for Reflection and Refraction from an Ellipsoid, Elliptic Paraboloid, and Elliptic Cone

General formulas are derived for the caustic surface and irradiance over an arbitrary receiver surface for point source radiation on collimated rays that are reflected or refracted by a curved surface. Specific formulas are obtained for light from a point source that is deflected by an ellipsoid, an elliptic paiaboloid, and an elliptic cone. As a numerical example caustic surfaces are calculated for a concave spherical surface and a concave paraboloid.

Design of reflectors which will distribute sunlight in a specified manner

Abstract A differential equation is obtained for the shape of a reflecting surface which will distribute axially symmetric light intensity into a specified irradiance over a receiver surface which is symmetric about the direction of the incident light. Results are applied to the design of rotationally symmetric solar reflectors and also to a 2-dimensional geometry, that is one in which the reflector is a cylinder with its axis perpendicular to the incident beam. The procedure is used to numerically calculate the shape of reflectors which will uniformly concentrate collimated light and also light from a point source over a flat receiver surface. Results are also applied to determine the shape of a reflector which will distribute collimated light uniformly over the surface of a cylinder and also over a sphere.

Analytical Illuminance Calculation in a Multi-interface Optical System

An analytical expression for the illuminance over a receiver surface and an equation for the caustic surface are derived for rays which have passed through a multi-interface optical system. The results are expressed in terms of the intrinsic geometry of the deflector surface, that is, the Gaussian, mean, and normal curvatures, and the derivative of the incident ray vector over the deflector surface. The original source may be a plane wave, a point source, or an extended source. As a numerical example of this technique, the caustic surface and the line-spread function on different image planes are plotted for an off-axis point source of light whose rays are refracted by a thick double convex lens.

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.

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.

Flat-sided rectilinear trough as a solar concentrator: an analytical study

Formulas are derived for the concentration factor and irradiance distribution at the base of the flat-sided linear trough. Performance is affected by the number of reflections the solar rays undergo before reaching the base, the cone apex angle, and the coefficient of reflection. Results are presented graphically in such a way that one can choose the optimum configuration, which is the minimum material required, to achieve a given concentration factor. Practical concentration factors range from 1.5 to 4 depending on the geometry and coefficient of reflection.

Specular reflection of heat radiation from an arbitrary reflector surface to an arbitrary receiver surface

Abstract General formulas are derived which specify the heat flux over an arbitrary receiving surface for radiation incident upon and specularly reflected from an arbitrary curved surface. The direction of the reflected ray and its intersection with the receiving area provides a transformation which maps an element of reflecting area onto the receiving area through the Jacobian determinant. Results are expressed in terms of the equations of the surfaces. The general formulas are reduced to the special case for which the reflecting area is a surface of revolution.