John C. Bortz

Performance limitations of rotationally symmetric nonimaging devices

An upper limit on concentration for any optical device has previously been derived from the conservation of etendue. In this contribution we derive more stringent upper limits for the efficiency and the concentration of rotationally symmetric optical devices that are a consequence of skewness conservation. If the desired source and target have different skewness distributions, then losses or dilution or both will limit the performance of the optical system. We calculate the limiting curve of efficiency versus concentration and provide a design example that is virtually at this limit. We conjecture that even rotationally symmetric problems may benefit from asymmetric optical systems.

Inverse engineering perspective on nonimaging optical design

We present an inverse engineering topological-axiomatic approach applicable to nonimaging optical design. The reflective and/or refractive surfaces of the optics are sequentially modified within a given parametrization scheme and a constraint set until performance objective global optimality, evaluated upon a system radiometric model is achieved. This formalism permits the study of non-concave, re-entrant, and piecewise continuous reflector and lens configurations that can constructively exploit multiple reflections for maximal energy transfer, beam shaping, or irradiance redistribution. We derive, under a single-reflection approximation, a new variational principle for constructing axially symmetric reflector forms that maximizes energy transfer and can be extended to provide optimal beam shaping. Our derivation accounts for generally shaped sources with arbitrary radiance distributions and for reflection losses. We compare known edge-ray designs with our solutions and find that the 3D CPC concentrator and the 3D involute CPC reflector (operated in reverse as a projector) can be improved upon. We present a projective design, employing a spherical source which makes use of source re-energization through retroreflection. This design achieves a beam radiance greater than that of the (naked) source and requires a re-entrant component, if a requirement for continuity of the reflective surface is imposed.

Generalized functional method of nonimaging optical design

We derive a generalized functional method of nonimaging optical design, applicable in the case of axisymmetric or translationally symmetric geometries utilizing extended sources and multiple reflective and/or refractive optical surfaces. Design methods are described for generation of a specified intensity distribution as a function of angle or a specified irradiance distribution as a function of position along the shape profile of an aspheric target surface. A related method is presented for the design of two successive optical surfaces that transform a given source distribution into either two specified irradiance distributions on successive target surfaces or a specified irradiance distribution and a specified intensity distribution.

Thermodynamic efficiency of solar concentrators

The optical thermodynamic efficiency is a comprehensive metric that takes into account all loss mechanisms associated with transferring flux from the source to the target phase space, which may include losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. We discuss consequences of Fermats principle of geometrical optics and review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We develop an expression for the optical thermodynamic efficiency which combines the first and second laws of thermodynamics. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators. We provide examples illustrating the use of this new metric for concentrating photovoltaic systems for solar power applications, and in particular show how skewness mismatch limits the attainable optical thermodynamic efficiency.The optical thermodynamic efficiency is a comprehensive metric that takes into account all loss mechanisms associated with transferring flux from the source to the target phase space, which may include losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. We discuss consequences of Fermats principle of geometrical optics and review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We develop an expression for the optical thermodynamic efficiency which combines the first and second laws of thermodynamics. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators. We provide examples illustrating the use of this new metric for concentrating photovoltaic systems for solar power applications, and in particular show how skewness mismatch limits the attainable optical thermodynamic efficiency.

Iterative generalized functional method of nonimaging optical design

In a previous paper we derived a generalized functional method of nonimaging optical design, applicable in the case of axisymmetric or translationally symmetric geometries utilizing extended sources and multiple reflective and/or refractive optical surfaces. Variations of this method were described for generation of either a prescribed intensity distribution as a function of angle or a prescribed irradiance distribution as a function of position along the shape profile of an aspheric target surface. A dual-surface functional (DSF) method was described for generation of two successive optical surfaces that transform a given source distribution into either two prescribed irradiance distributions on successive target surfaces or a prescribed irradiance distribution and a prescribed intensity distribution. The RMS difference between the intensity or irradiance distribution produced by a given generalized-functional design and the corresponding prescribed distribution approaches zero in the limit as the optical characteristics of the actual source approach those of the zero-étendue extended source model used in generating the design. In the current contribution, we present an iterative algorithm that can significantly improve the beam-shaping performance of generalized functional designs for sources having non-zero étendue. Three examples are provided, including an iterative DSF design that provides both high irradiance uniformity and étendue-matched flux-transfer efficiency greater than 97%.

Nonrotationally symmetric nonimaging systems that overcome the flux-transfer performance limit imposed by skewness conservation

Theoretical upper limits on measures of flux-transfer performance due to skewness conservation in rotationally symmetric nonimaging optical systems have recently been discovered and quantified. These limits can have an adverse impact on the performance of projection or coupling optics which collect light from 3D sources. In this contribution we show that these limits can be exceeded by employing nonrotationally symmetric configurations. We consider the problems of maximizing flux transfer from both a homogeneous spherical source and a homogeneous cylindrical source to a homogeneous disk-shaped target of equal etendue. We find that the performance limits due to skewness conservation for these problems can be overcome by numerically optimized reflectors possessing a nonrotationally symmetric star-like cross-section.

Optimal design of a nonimaging TIR doublet lens for an illumination system using an LED source

We consider the problem of designing illumination optics to transfer flux from an LED source into a rectangular target aperture. An important constraint on this design was that it provide a large working distance between the source and the target aperture. An optimized compact nonimaging system consisting of an aspheric totally-internally-reflecting (TIR) lens along with an aspheric singlet was developed to provide high flux-transfer efficiency with the required working distance. A global optimization procedure was used to search a 65-dimensional parameter space for the set of optical-component shape and positioning parameters providing maximum performance, while satisfying the design constraints. A total source-to-target flux-transfer efficiency of 85.0% was achieved by the design, assuming ideal AR coatings on all refractive surfaces.

Consequences of skewness conservation for rotationally symmetric nonimaging devices

An upper limit on concentration for any optical device has previously been derived from the conservation of etendue. In this contribution we derive more stringent upper limits for the efficiency and the concentration of rotationally symmetric optical devices which are a consequence of skewness conservation. If the desired source and target have different skewness distributions, then losses or dilution or both will limit the performance of the optical system. For all skewness values, for which the source contains more radiation than the target, the difference is lost. Conversely, for all skewness values, for which the target contains a large etendue than the source, the difference remains empty and results in dilution. We calculate the limiting curve of optical transfer efficiency versus concentration relative to the maximum concentration possible and provide a design example that is practically at this limit. We also provide another design example that addresses the challenge posed at the last SPIE meeting, namely to transfer the maximum radiation from a Lambertian spherical source to a disk target of equal etendue under a reflector- to-source minimum distance constraint. We conjecture that even rotationally symmetric problems may benefit from asymmetric optical systems.

Relationships between the generalized functional method and other methods of nonimaging optical design

The recently developed generalized functional method provides a means of designing nonimaging concentrators and luminaires for use with extended sources and receivers. We explore the mathematical relationships between optical designs produced using the generalized functional method and edge-ray, aplanatic, and simultaneous multiple surface (SMS) designs. Edge-ray and dual-surface aplanatic designs are shown to be special cases of generalized functional designs. In addition, it is shown that dual-surface SMS designs are closely related to generalized functional designs and that certain computational advantages accrue when the two design methods are combined. A number of examples are provided.

Nonrotationally symmetric reflectors for efficient and uniform illumination of rectangular apertures

A requirement for a uniformly illuminated rectangular aperture is common in optical design, particularly for projection applications. The naive approach to producing uniform illumination over a rectangular aperture is to uniformly fill a circular zone which circumscribes the desired rectangular region. Unfortunately, this technique is wasteful of flux, e.g. a rectangular aperture having as aspect ratio of 1.5:1 collects only 59% of the flux incident on the circumscribing circle. Other approaches such as rectangular light pipes and mosaic lens arrays dilute source etendue, increase the length of the optical train, and can suffer from losses due to multiple reflections. We have previously discovered that the performance limit due to skewness conservation for collecting light from a 3D source and projecting it into a beam can be overcome by a numerically optimized reflector with a nonrotationally symmetric star-like cross-section. In this paper we present preliminary results of our research into designing high- efficiency, single-element nonrotationally symmetric reflectors which provide uniform flux delivery into rectangular apertures.