1.1 NONIMAGING COLLECTORS

Nonimaging concentrators and illuminators have several actual and some potential applications, but it is best to explain the general concept of a nonimaging concentrator by highlighting one of its applications; its use of solar energy. The radiation power density received from the sun at the earth’s surface, often denoted by S, peaks at approximately 1kWm−2, depending on many factors. If we attempt to collect this power by absorbing it on a perfect blackbody, the equilibrium temperature T of the blackbody will be given by1

     (1.1)

where σ is the Stefan Boltzmann constant, 5.67 × 10−8Wm−2°K−4. In this example, the equilibrium temperature would be 364°K, or just below the boiling point of water.

For many practical applications of solar energy this is sufficient, and it is well known that systems for domestic hot water heating based on this principle are available commercially for installation in private dwellings. However, for larger-scale purposes or for generating electric power, a source of heat at 364°K has a low thermodynamic efficiency, since it is not practicable to get a very large temperature 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 blackbody by a factor C of about 6 to 10 from Eq. (1.1).

This, briefly, is one use of a concentrator—to increase the power density of solar radiation. When it is stated plainly like that, the problem sounds trivial. The principles of the solution have been known since the days of Archimedes and his burning 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 pure geometrical 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 concentration be achieved in practice?—is not as easy to answer. We shall see that there are limitations involving materials and manufacturing, as we should expect. But there are also limitations involving the kinds of optical systems that can actually 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 a certain specification cannot be fulfilled because it would require an impractically large number of refracting or reflecting surfaces. But sometimes they do not know whether it is in principle possible to achieve aberration corrections of a certain kind.

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 aperture—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 would have very large aberrations if they were used as image-forming systems. Nevertheless, as concentrators, they are substantially more efficient than image-forming systems 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’s more specific. These systems are unlike any previously used optical systems. They have 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 the designs 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 development of these ideas, it is necessary to recapitulate some basic principles of geometrical optics, which is done in Chapter 2. In Chapter 3, we look at what can be done with conventional image-forming systems as concentrators, and we show how they necessarily fall short of ideal performance. In Chapter 4, we describe one of the basic nonimaging concentrators, the compound parabolic concentrator, and we obtain its optical properties. Chapter 5 is devoted to several developments of the basic compound parabolic concentrator: with plane absorber, mainly aimed at decreasing the overall length; with nonplane absorber; and with generalized edge ray wavefronts, which is the origin of the tailored designs. In Chapter 6, we examine in detail the Flow Line approach to nonimaging concentrators both for 2D and 3D geometries, and we include the description of the Poisson brackets design method. At the end of this chapter we introduce elliptic bundles in the Lorentz geometry formulation. Chapter 7 deals with a basic illumination problem: designing an optical system that produces a prescribed irradiance with a given source. This problem is considered from the simplest case (2D geometry and point source) with increasing complexity (3D geometry, extended sources, free-form surfaces). Chapter 8 is devoted specifically to one method of design called Simultaneous Multiple Surfaces (SMS) method, which is the newest and is more powerful for high concentration/collimation applications. Nonimaging is not the opposite of imaging. Chapter 9 shows imaging applications of nonimaging designs. Sometimes the performance of some devices is theoretically limited by the use of rotational or linear symmetric devices, Chapters 10 and 11 discuss the problem of improving this performance by using free-form surfaces departing from symmetric designs that are deformed in a controlled way. The limits to concentration or collimation can be derived from Chapter 12, which is devoted to the physical optics aspects of concentration 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 appendixes in which the derivations of the more complicated formulas are given.

1.2 DEFINITION OF THE CONCENTRATION RATIO; THE THEORETICAL MAXIMUM

From the simple argument in Section 1.1 we see that the most important property of a concentrator is the ratio of area of input beam divided by the area of output beam; this is because the equilibrium temperature of the absorbing body is proportional to the fourth root of this ratio. We denote this 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 concentration ratio is

     (1.2)

In the preceding definition, it was tacitly assumed that compression of the input beam occurred in both the dimensions transverse to the beam direction, as in ordinary lens systems. In solar energy technology there is a large class of systems in which the beam is compressed in only one dimension. In such systems all the operative surfaces, reflecting and refracting, are cylindrical with parallel generators (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 2D design). In these cases we will use the name 2D design or 3D design to differentiate from a 2D or a 3D concentrator. The 2D concentrators are also called linear concentrators. The concentration ratio of a linear concentrator is usually given as the 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...