Nonimaging Optics Winston Episode 1 doc

2.1 The Concepts of Geometrical Optics 72.2 Formulation of the Ray-Tracing Procedure 82.3 Elementary Properties of Image-Forming Optical Systems 112.4 Aberrations in Image-Forming Optica

NONIMAGING OPTICS

NONIMAGING OPTICS

Roland Winston

University of California, Merced, CA

Juan C Miñano and Pablo Benítez

Technical University of Madrid UPM, CEDINT, Madrid, Spain and Light Prescriptions Innovators LLC, Irvine, CA

With contributions by

Narkis Shatz and John C Bortz

Science Applications International Corporation, San Diego, CA

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2.1 The Concepts of Geometrical Optics 72.2 Formulation of the Ray-Tracing Procedure 82.3 Elementary Properties of Image-Forming Optical Systems 112.4 Aberrations in Image-Forming Optical Systems 132.5 The Effect of Aberrations in an Image-Forming System on the

2.6 The Optical Path Length and Fermat’s Principle 162.7 The Generalized Étendue or Lagrange Invariant and the

2.9 Different Versions of the Concentration Ratio 23

3.7 Conclusions on Classical Image-Forming Concentrators 40

4 Nonimaging Optical Systems 43

6.2 Lines of Flow from Lambertian Radiators: 2D Examples 100

6.4 A Simplified Method for Calculating Lines of Flow 1036.5 Properties of the Lines of Flow 1046.6 Application to Concentrator Design 1056.7 The Hyperboloid of Revolution As A Concentrator 1066.8 Elaborations of the Hyperboloid: the Truncated Hyperboloid 1066.9 The Hyperboloid Combined with A Lens 1076.10 The Hyperboloid Combined with Two Lenses 1086.11 Generalized Flow Line Concentrators with Refractive

6.14 Application of the Poisson Bracket Method 1286.15 Multifoliate-Reflector-Based Concentrators 138

6.16 The Poisson Bracket Method in 2D Geometry 1426.17 Elliptic Bundles in Homogeneous Media 144

8.11 Three-Dimensional Ray Tracing of Some RXI Concentrators 2078.12 Comparison of the SMS Concentrators with Other

Nonimaging Concentrators and with Image Forming

11 Global Optimization of High-Performance Concentrators

11.2 Mathematical Properties of Mappings in Nonimaging

11.4 The Effect of Source and Target Inhomogeneities on the

Performance Limits of Nonsymmetric Nonimaging

13.1 Requirements for Solar Concentrators 31713.2 Solar Thermal Versus Photovoltaic Concentrator

13.3 Nonimaging Concentrators for Solar Thermal Applications 32713.4 SMS Concentrators for Photovoltaic Applications 35013.5 Demonstration and Measurement of Ultra-High Solar

13.6 Applications Using Highly Concentrated Sunlight 381

13.8 Solar Thermal Applications of High-Index Secondaries 38713.9 Solar Thermal Propulsion in Space 389

14.4 The Concentrator Error Multiplier 410

APPENDIX A Derivation and Explanation of the Étendue

Invariant, Including the Dynamical Analogy;

A.2 Proof of the generalized étendue theorem 416A.3 The mechanical analogies and liouville’s theorem 418A.4 Conventional photometry and the étendue 419

B.5 Generation of Edge Rays at Slope Discontinuities 429B.6 Offence Against the Edge-Ray Theorem 430

APPENDIX C Conservation of Skew and Linear Momentum 433

C.2 Luneburg Treatment for Skew Rays 434

C.4 Design of Concentrators for Nonmeridian Rays 435

APPENDIX G The Geometry of the Basic Compound Parabolic

APPENDIX I The Truncated Compound Parabolic Concentrator 473 APPENDIX J The Differential Equation for the 2D Concentrator

APPENDIX K Skew Rays in Hyperboloidal Concentrator 481 APPENDIX L Sine Relation for Hyperboloid/Lens Concentrator 483

APPENDIX M The Concentrator Design for Skew Rays 485

M.2 The Ratio of Input to Output Areas for the Concentrator 486M.3 Proof That Extreme Rays Intersect at the Exit Aperture Rim 488M.4 Another Proof of the Sine Relation for Skew Rays 489

1990, he took on the elucidation of nonimaging optics with the same tic vigor and enthusiasm he had applied to imaging optics As a result, nonimag-ing optics developed from a set of heuristics to a complete subject We dedicatethis book to his memory.

characteris-It incorporates much of the pre-1990 material as well as significant advances

in the subject These include elaborations of the flow-line method, designs for scribed irradiance, simultaneous multiple surface method, optimization, and sym-metry breaking A discussion of radiance connects theory with measurement in aphysical way

pre-We will measure our success by the extent to which our readers advance thesubject over the next 10 years

NONIMAGING OPTICAL

SYSTEMS AND THEIR USES

1

Nonimaging concentrators and illuminators have several actual and some tial applications, but it is best to explain the general concept of a nonimaging con-centrator by highlighting one of its applications; its use of solar energy Theradiation power density received from the sun at the earth’s surface, often denoted

poten-by S, peaks at approximately 1 kWm-2, depending on many factors If we attempt

to collect this power by absorbing it on a perfect blackbody, the equilibrium

tem-perature T of the blackbody will be given by1

(1.1)where s is the Stefan Boltzmann constant, 5.67 ¥ 10-8W m-2°K-4 In this example,the equilibrium temperature would be 364°K, or just below the boiling point ofwater

For many practical applications of solar energy this is sufficient, and it is wellknown that systems for domestic hot water heating based on this principle areavailable commercially for installation in private dwellings However, for larger-scale purposes or for generating electric power, a source of heat at 364°K has alow thermodynamic efficiency, since it is not practicable to get a very large tem-perature difference in whatever working fluid is being used in the heat engine

If we wanted, say, ≥300°C—a useful temperature for the generation of motive

power—we should need to increase the power density S on the absorbing body by a factor C of about 6 to 10 from Eq (1.1).

black-This, briefly, is one use of a concentrator—to increase the power density ofsolar radiation When it is stated plainly like that, the problem sounds trivial Theprinciples of the solution have been known since the days of Archimedes and hisburning glass:2we simply have to focus the image of the sun with an image-forming

system—a lens—and the result will be an increased power density The problems

to be solved are technical and practical, but they also lead to some interesting puregeometrical optics The first question is that of the maximum concentration: How

large a value of C is theoretically possible? The answer to this question is simple

in all cases of interest The next question—can the theoretical maximum tration be achieved in practice?—is not as easy to answer We shall see that thereare limitations involving materials and manufacturing, as we should expect Butthere are also limitations involving the kinds of optical systems that can actually

concen-be designed, as opposed to those that are theoretically possible This is analogous

to the situation in classical lens design The designers sometimes find that acertain specification cannot be fulfilled because it would require an impracticallylarge number of refracting or reflecting surfaces But sometimes they do not knowwhether it is in principle possible to achieve aberration corrections of a certainkind

The natural approach of the classical optical physicist is to regard the problem

as one of designing an image-forming optical system of very large numerical

aper-ture—that is, small aperture ratio or f-number One of the most interesting results

to have emerged in this field is a class of very efficient concentrators that wouldhave very large aberrations if they were used as image-forming systems Never-theless, as concentrators, they are substantially more efficient than image-formingsystems and can be designed to meet or approach the theoretical limit We shall

call them nonimaging concentrating collectors, or nonimaging concentrators for short Nonimaging is sometimes substituted by the word anidolic (from the Greek,

meaning “without image”) in languages such as Spanish and French because it’smore specific These systems are unlike any previously used optical systems Theyhave some of the properties of light pipes and some of the properties of image-forming optical systems but with very large aberrations The development of thedesigns of these concentrators and the study of their properties have led to a range

of new ideas and theorems in geometrical optics In order to facilitate the opment of these ideas, it is necessary to recapitulate some basic principles of geo-metrical optics, which is done in Chapter 2 In Chapter 3, we look at what can bedone with conventional image-forming systems as concentrators, and we show howthey necessarily fall short of ideal performance In Chapter 4, we describe one ofthe basic nonimaging concentrators, the compound parabolic concentrator, and weobtain its optical properties Chapter 5 is devoted to several developments of thebasic compound parabolic concentrator: with plane absorber, mainly aimed atdecreasing the overall length; with nonplane absorber; and with generalized edgeray wavefronts, which is the origin of the tailored designs In Chapter 6, weexamine in detail the Flow Line approach to nonimaging concentrators both for2D and 3D geometries, and we include the description of the Poisson bracketsdesign method At the end of this chapter we introduce elliptic bundles in theLorentz geometry formulation Chapter 7 deals with a basic illumination problem:designing an optical system that produces a prescribed irradiance with a givensource This problem is considered from the simplest case (2D geometry and pointsource) with increasing complexity (3D geometry, extended sources, free-form sur-faces) Chapter 8 is devoted specifically to one method of design called Simulta-neous Multiple Surfaces (SMS) method, which is the newest and is more powerfulfor high concentration/collimation applications Nonimaging is not the opposite ofimaging Chapter 9 shows imaging applications of nonimaging designs Sometimes

the performance of some devices is theoretically limited by the use of rotational orlinear symmetric devices, Chapters 10 and 11 discuss the problem of improvingthis performance by using free-form surfaces departing from symmetric designsthat are deformed in a controlled way The limits to concentration or collimationcan be derived from Chapter 12, which is devoted to the physical optics aspects ofconcentration and in particular to the concept of radiance in the physical optics.Chapters 13 and 14 are devoted to the main applications of nonimaging optics:illumination and concentration (in this case of solar energy) Finally, in Chapter

15 we examine briefly several manufacturing techniques There are several dixes in which the derivations of the more complicated formulas are given

RATIO; THE THEORETICAL MAXIMUM

From the simple argument in Section 1.1 we see that the most important erty of a concentrator is the ratio of area of input beam divided by the area ofoutput beam; this is because the equilibrium temperature of the absorbing body

prop-is proportional to the fourth root of thprop-is ratio We denote thprop-is ratio by C and call

it the concentration ratio Initially we model a concentrator as a box with a plane

entrance aperture of area A and a plane exit aperture of area A¢ that is just large

enough to allow all transmitted rays to emerge (see Figure 1.1) Then the tration ratio is

1.2 Definition of the Concentration Ratio; the Theoretical Maximum 3

Figure 1.1 Schematic diagram of a concentrator The input and output surfaces can face

in any direction; they are drawn in the figure so both can be seen It is assumed that the

aperture A¢ is just large enough to permit all rays passed by the internal optics that have

entered within the specified collecting angle to emerge.

all the operative surfaces, reflecting and refracting, are cylindrical with parallelgenerators (but not in general circular cylindrical) Thus, a typical shape would

be as in Figure 1.2, with the absorbing body (not shown) lying along the trough.Such long trough collectors have the obvious advantage that they do not need to

be guided to follow the daily movement of the sun across the sky The two types

of concentrator are sometimes called three- and two-dimensional, or 3D and 2D,concentrators The names 3D and 2D are also used in this book (from Chapter 6

to the end) to denote that the optical device has been designed in 3D geometry or

in 2D geometry (in the latter case, the real concentrator, which of course exists in

a 3D space, is obtained by rotational or translational symmetry from the 2Ddesign) In these cases we will use the name 2D design or 3D design to differen-tiate from a 2D or a 3D concentrator The 2D concentrators are also called linearconcentrators The concentration ratio of a linear concentrator is usually given asthe ratio of the transverse input and output dimensions, measured perpendicular

to the straight-line generators of the trough

The question immediately arises whether there is any upper limit to the value

of C, and we shall see that there is The result, proved later, is very simple for the

2D case and for the 3D case with an axis of revolution symmetry (rotational centrator) Suppose the input and output media both have a refractive index ofunity, and let the incoming radiation be from a circular source at infinity sub-tending a semiangle qi Then the theoretical maximum concentration in a rota-tional concentrator is

con-(1.3)Under this condition the rays emerge at all angles up to p /2 from the normal

to the exit face, as shown in Figure 1.3 For a linear concentrator the ing value will be 1/sin qi

correspond-The next question that arises is, can actual concentrators be designed withthe theoretically best performance? In asking this question we make certain ide-

Cmax = 1sin2qi

Figure 1.2 A trough concentrator; the absorbing element is not shown.

alizing assumptions—for example, that all reflecting surfaces have 100% tivity, that all refracting surfaces can be perfectly antireflection coated, that allshapes can be made exactly right, and so forth We shall then see that the following answers are obtained: (1) 2D concentrators can be designed with the theoretical maximum concentration; (2) 3D concentrators can also have the theo-retical maximum concentration if they use variable refractive index material or apile of infinitely thin surface waveguides properly shaped; and (3) some rotationalsymmetric concentrators can have the theoretical maximum concentration In case(3) it appears for other types of design that it is possible to approach indefinitelyclose to the theoretical maximum concentration either by sufficiently increasingthe complexity of the design or by incorporating materials that are in principlepossible but in practice not available For example, we might specify a material ofvery high refractive index—say, 5—although this is not actually available withoutlarge absorption in the visible part of the spectrum.

The application to solar energy utilization just mentioned has, of course, lated the greatest developments in the design and fabrication of concentrators Butthis is by no means the only application The particular kind of nonimaging con-centrator that has given rise to the greatest developments was originally conceived

stimu-as a device for collecting stimu-as much light stimu-as possible from a luminous volume (thegas or fluid of a Cˇ erenkov counter) over a certain range of solid angle and sending

it onto the cathode of a photomultiplier Since photomultipliers are limited in sizeand the volume in question was of order 1 m3, this is clearly a concentrator problem(Hinterberger and Winston, 1966a,b)

Subsequently the concept was applied to infrared detection (Harper et al.,1976), where it is well known that the noise in the system for a given type of detec-tor increases with the surface area of the detector (other things being equal)

Figure 1.3 Incident and emergent ray paths for an ideal 3D concentrator with symmetry about an axis of revolution The exit aperture diameter is sin qitimes the exit aperture diam- eter; the rays emerge from all points in the exit aperture over a solid angle 2p.