David L. Shealy

Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis

A set of differential equations is derived which specifies the shape of two aspherical surfaces of a lens system that will convert an incident plane wave with an arbitrary energy profile into collimated radiation with a uniform energy distribution. As an example, a lens system is designed that converts a laser beam with a Gaussian energy profile into an expanded beam with a uniform energy distribution. Off-axis rays are then traced through the lenses in order to analyze the performance of the lens system.

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.

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.

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.

Geometric optics-based design of laser beam shapers

When diffraction effects are not important, geometrical optics (ray tracing, conservation of energy within a bundle of rays, and the constant optical path length condition) can be used to design laser beam shapers by solving beam shaping equations or by optimizing a beam shaping merit function for the configurations, including aspheric elements or spherical-surface gradient-index lenses, which are required to change the input irradiance and phase profile into a more useful form. Geometrical optics-based methods are presented for shaping both rotationally and rectangular symmetric laser beam profiles. Solutions of the beam shaping equations are presented for a two-plano-aspheric lens system for shaping a circular symmetric Gaussian beam into a top-hat output beam profile and a two-mirror system with no central obscuration for shaping an elliptical Gaussian input beam into a Fermi-Dirac output beam profile.

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.

Analytical illuminance and caustic surface calculations in geometrical optics

The analytical illuminance monitoring technique provides an exact expression within the geometrical optics limit for the illuminance over an image surface for light that has passed through a multiinterface optical system. The light source may be collimated rays, a point source, or an extended source. The geometrical energy distributions can be graphically displayed as a line or point spread function over selected image planes. The analytical illuminance technique gives a more accurate and efficient computer technique for evaluating the energy distribution over an image surface than the traditional scanning of the spot diagram mathematically with a narrow slit. The analytical illuminance monitoring technique also provides a closed form expression for the caustic surface of the optical system. It is shown by examining the caustic surface for anumber of lens systems from the literature that the caustic is a valuable merit function for evaluating the aberrations and the size of the focal region.