2106 J. Opt. Soc. Am. A / Vol. 13, No. 10 / October 1996 D. Jenkins and R. Winston Integral design method for nonimaging concentrators
D. Jenkins and R. Winston
Department of Physics and the Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637 Received July 5, 1995; accepted February 27, 1996; revised manuscript received March 20, 1996 We present an integral design method for maximizing concentration onto a given absorber shape. This technique uses a variable-acceptance edge-ray principle to transform nonuniform input and output radiance distributions. It easily recovers familiar designs that transform uniform radiance distributions. The method is simpler to use and more general than previous nonimaging design techniques, such as string methods and flow-line approaches. We show how this technique is adapted to satisfy diverse boundary conditions, such as satisfying total internal reflection or design within a material of graded index. Presented are an analytic solution to the classic 1 ­ 2 concentrator and a novel two-stage, two-dimensional solar collector with a fixed circular primary mirror and nonimaging secondary. This newly developed secondary provides a 25% improvement in concentration over conventional nonimaging secondaries. © 1996 Optical Society of America. 1. INTRODUCTION
Nonimaging optics develops designs for optical devices that approach the theoretical maximum for the concentration of light (ideal concentration). A number of groups have studied both the theory and the applications of nonimaging optics.1 Many types of systems, such as 1 ­ 2 concentrators,2 trumpets,3 and compound parabolic concentrators4 (CPC's), are well known. In certain applications, practical considerations have led to the need for multistage optical systems. Ideal concentrators have very large length-to-aperture ratios when designed to collect over a small angular range. Multistage systems allow system size to be reduced greatly with a slight sacrifice in concentration. In earlier applications the use of classic nonimaging designs was considered for the secondary concentrator that is used in conjunction with an imaging primary. However, the input radiance distribution on the secondary concentrator is nonuniform, which is responsible for some of the shortfall relative to ideal concentration. Aberrations in imaging systems must be accounted for in any general development method. We present a new design method based on simple numerical integration of a geometrically derived differential equation. This method recovers all previously generated designs, advances new solutions, and manages nonuniform input radiance distributions. The general problem addressed by nonimaging systems is the transformation of a source radiance distribution by reflection and refraction to achieve a desired aim, such as maximizing irradiance or concentration on a target. In this paper we consider designs for two-dimensional (2-D) trough concentrators, using a general edge-ray approach. Before exploring the mathematics of our design method, we review the ideal, or maximum, limits of concentration. The thermodynamic limits of concentration are useful checks of any design method. We will describe briefly how the concentration limits for a 2-D system are deter0740-3232/96/1002106-11$10.00 mined. First we show the limits for a classical system and define how phase space concepts and optical analogs to Liouville's theorem can be used to determine the limits for nonuniform distributions.5 A. Conservation Laws Radiance (alternatively called brightness) is the power per unit area per unit projected solid angle falling on a surface element. It is assumed that only one side of the surface element is illuminated. Projected solid angle is calculated by projecting the incident distribution first on the unit sphere surrounding the surface element (thus yielding the solid angle) and then on the plane containing the surface element. The radiance B is conserved along a geometrical light ray's path in a medium of constant index of refraction n. If a light ray traverses different materials, the conserved quantity is B/n in two dimensions and B/n 2 in three dimensions. Specifically, in two dimensions the irradiance I, power per unit area, on a surface is given by
/2 I x
/2 cos d B x, , (1) where is the angle the light ray makes with the surface normal at point x. For classical concentrating systems the source radiance distribution B is B 0 constant for L 1 /2 and zero otherwise. The source 1 and x is immersed in a medium with n 1. This is a uniform radiance pattern. L 1 is the collecting aperture width, and 1 is the extreme angle of the distribution to be concentrated. This input distribution can be transformed (assuming no losses) onto a target with an angular output 2 and size L 2 , immersed in a medium with index n. The irradiance I 1 on the source is given by I1 2B 0 sin
1, (2) and the irradiance I 2 on the target is
© 1996 Optical Society of America D. Jenkins and R. Winston Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A 2107 C2 d n cos sin rim 1 , (5) where rim is the rim angle of the imaging primary and An additional condition is 1 is the acceptance angle. that 1 rim . As rim angles approach 90 deg, concentration decreases precipitously. Aberrations cause the radiance distribution onto the secondary aperture to be highly nonuniform. Classical designs for a uniform box distribution match the actual distribution poorly. Consequent to collecting a larger phase space volume than is required, the concentration of the standard nonimaging secondary is reduced.
Fig. 1. Brightness transformation by a classic nonimaging concentrator. I2 2nB 0 sin 2. (3) Therefore the ideal concentration in two dimensions is I 2 /I 1 , or sin sin
2 1 C2­d n . (4) Analogs of the above system with use of a phase space description readily recover this conservation law. Classical phase space of thermodynamical systems uses both spatial and directional coordinates. In 2-D concentrating systems we use an optical analog of Liouville's theorem. Only two parameters are needed to describe a 2-D light ray. These are the position (x) at which the light ray hits a surface element and the unit directional cosine (k x ) tangent to this element. These two parameters define the 2-D phase space for a 2-D system. The area of the phase space hitting both the source and target is conserved. The classical case described previously with n 1 materials is depicted in Fig. 1. An increase in the angular size of the rectangle on the target relative to that on the source causes a corresponding decrease in the spatial extent of the target. The concentration ratio is simply L 1 /L 2 . For a source and target inside different indices of refraction, multiplying k x by the index n yields the limit in Eq. (4). For a nonuniform radiance distribution, the phase space area of the source A phase is found where B 0. This area is less than or equal to the total phase space available on the target. Assuming that the target distribution can be filled completely, the minimum target size of A phase/2 is obtained. Many problems have uniform radiance distributions. But simple ideal concentrators are impractical for small acceptance angles ( 1 1). Their length is proportional to cot 1 . Designs employing two-stage systems, where the first stage is an imaging device, such as a mirror or lens, and a nonimaging secondary have been used to decrease this length greatly. The catch is that aberrations in the system from the imaging primary prevent the ideal concentration limit from being reached. A two-stage system with a focusing primary mirror and a nonimaging concentrator with a fixed acceptance angle is shown in Fig. 2. The theoretical limit is reduced to B. Tailored Designs Recent developments have shown that tailoring a reflector profile is possible with use of variable-acceptance edge rays, and we generalize these techniques to cover most types of concentrator design.6­8 The papers just cited describe the design of illuminators. The same technique can be applied to concentrator design by simply reversing the light ray paths. Only one paper on concentrators that use variable-acceptance-angle tailoring has been published. It employs a two-stage system to increase concentration of a parabolic dish onto a flat absorber.9 We will demonstrate how to tailor for both flat and nonflat absorbers in this paper. Our method readily embraces previously known designs and is general enough to cover variable-acceptance-angle edge rays. We will also show how tailoring can be applied to many more systems. Features of this method comprehend nonflat absorbers, multiple reflection designs, and various constraints, such as satisfying total internal reflection or using gradedindex materials. The method does not use standard string or flow-line techniques. Nor does it require an involute as do many classical designs. In cases where an involute arises, it does so as a consequence of the method. Our method numerically integrates a differential equation in polar coordinates to design concentrators for various absorber shapes and input distributions. It is based on the application of Snell's law of reflection expressed in polar coordinates. This defines a differential equation whose solution specifies the reflector curve. Tailoring the reflector for variable-acceptance edge rays is covered. Whole classes of new concentrator types are generalized so that tailoring can be done for each. The shape of nonflat absorbers is accommodated by changing one variable within the differential equation. There are two types of curvature that are allowed for in Fig. 2. Typical two-stage concentrating system. 2108 J. Opt. Soc. Am. A / Vol. 13, No. 10 / October 1996 D. Jenkins and R. Winston Each shape, regardless of the actual absorber shape, is derived from integrating a differential equation that reflects the edge ray to achieve desired results. The basic underlying equation is simply Snell's law of specular reflection off a curve given in polar coordinates, dR d R tan , (6) Fig. 3. Different types of concentrator: (a) CEC-type, (b) CHCtype. designing nonimaging concentrators depending on whether the reflected edge rays converge (the caustic formed by the reflected rays is real) or diverge (the caustic formed by the reflected rays is virtual).10 An example of the former is a compound elliptical concentrator (CEC), which uses an edge-ray method that reaches the absorber in one reflection, and the hyperbolic concentrator (CHC) uses virtual foci that may cause edge rays to make multiple reflections before hitting the target absorber. Ideal CEC types converge to a finite length; the CHC reflector may become infinitely long for an ideal concentrator. Compound means that there are generally two sides to a concentrator and that the two sides are in most cases cosymmetric. Combining to two symmetric sides gives a compound reflector. Asymmetric designs are generated by determining each side separately.11 The two types of collector are shown in Fig. 3. Both can be produced using a general integral design method. We show how both are designed with one approach, whereas before, separate techniques were used. The outline of the paper is as follows. In Section 2, the basic differential equation is developed and explained. The various subsections show modifications required for designing a variety of concentrator types. Subsection 2.A illustrates how the equation is applied to 1 ­ 2 concentrator types. Trumpet flow-line concentrators are considered in Subsection 2.B, and Subsection 2.C generalizes to nonflat absorbers. The addition of constraints such as the use of a lens to minimize truncation losses, total internal reflection requirements, and design within a graded-index material are discussed in Subsection 2.D. We show two solved examples in Section 3. An analytic derivation of the classic 1 ­ 2 concentrator is done in Subsection 3.A, and a novel two-stage solar collector that uses a fixed circular mirror demonstrates that a unique secondary can give a concentration increase of 30% over that of a convential CPC. The potential of this new technique is discussed in Section 4, along with comments on designing 3-D concentrators. where the coordinates (R, ) represent a point on a curve and is the angle that a ray from the origin of coordinate system makes with the normal to the curve. The whole design process now depends on finding . A ray reflected from the origin of the system is depicted in Fig. 4. We will show that designing various nonimaging concentrators requires only a convenient origin and finding (R, ) to satisfy the edge-ray principle and various design constraints. Our design goal in this paper is to present a method that achieves the maximum concentration from a given radiance distribution and set of design constraints. A. 1 ­ 2 -Type Design The simplest type of concentrator to design has both a real source and a real target aperture. This is a CECtype concentrator with the edge ray making one reflection to reach the target. For flat phase space distributions with a maximum angle of 1 at the entrance aperture and 2 at the exit aperture, there is the classic 1 ­ 2 CPC. Since the design of this type of concentrator has most edge rays hitting the edges of the target, this is the optimal choice for coordinate origin. Clearly, the choice of origin does not change the overall reflector profile, but judicious choice can facilitate the design procedure. Figure 5(a) shows the concentrator profile and the various design parameters for 2 /2 . In this case the edge ray hits the edge of the target. The only new parameter entering the system is 1(R, ). This is the largest angle a ray makes with respect to the vertical axis that can hit the reflector at point R, . For standard designs 1(R, ) is a constant for all R, . Other types require ray tracing back to the source to find the maximum angle that can hit the reflector at that point. Though easy in principle, this 2. THEORY
This section considers the theory for designing 2-D concentrators that use polar coordinates. 3-D-concentrators can be created by rotating the 2-D design around its axis.4
Fig. 4. Ray from origin reflecting off a curve given in polar coordinates. D. Jenkins and R. Winston Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A 2109 h L1 2 L2 cot 2 t . (9) Since the ideal limit is not generally attainable, the optimum design concentration is found numerically. When the concentration L 1 /L 2 is varied, integration is done starting from either the edge of the source or the target. Standard numerical computation algorithms, such as a bisection technique, can be used until the integrated curve passes through both edges of the source and the target. If h is fixed, then in general there will be losses in concentration. Designing for a nonflat output brightness distribution can be done by making 2 a function of R, . This might be accomplished by designing a concentrator with a target placed behind the exit. Solutions for the standard 1 ­ 2 CPC are presented later in the paper. B. Generalizing to Trumpet The trumpet, or flow-line concentrator, is of the CHC type.12 The edge rays take an infinite number of reflections to reach the target absorber. As with the 1 ­ 2 concentrator, the target and source distributions both are flat surfaces. The difference is that the target is in the source plane, and the input distribution acts as a virtual source. Figure 7 shows the geometry of the design system. Again, the source brightness distribution is assumed to be known. For a truncated design, the height h is chosen and points A, A are found such that all rays hitting the source pass between them. The edge-ray principle used in this case requires that edge rays hit the target after an infinite number of reflections. Applying Eq. (6) to this case requires some intuition. In classical designs of trumpets, the edge rays reflect toward virtual foci at the edge of the source distributions and keep getting closer and closer but take an infinite number of reflections to actually reach the target. We can actually manipulate the edge ray so it follows this pattern. By choosing the origin of our coordi- Fig. 5. (b) Design of /2 2. 1­ 2 -type concentrator: (a) /2 2, is potentially difficult to perform for complex input radiance distributions. Once 1 is known, is found by using simple geometry,
1 R, 4 2 if 2 2 . (7) This when combined with Eq. (6) is usually not solvable analytically unless 1 is constant. When the constraint that 2 /2 is added, the edge rays are not always reflected onto the target edge. For /2 , edge rays are not reflected through the ori2 gin, and finding is more difficult. Design in this region is shown in Fig. 5(b). To cause the edge ray to exit at angle 2 , we introduce an auxiliary ray (we term it the complementary edge ray) that passes through the edge of the target. This new construct helps place the edge ray correctly. The angle between the complementary edge ray and the true edge ray is /2 2 . Adding to 1 in Eq. (7) yields the desired value for . By requiring that the complementary edge ray at 1 hit the edge of the target, the real edge ray at 1 exits the target with angle 2 as desired. The equation for in this case is
2 1 R, 2 if 2 2 . (8) Note that if 1 is a constant, this equation integrates to a straight line. It is not possible, for most cases, to determine the height h and size of the exit aperture L 2 analytically, assuming that the input distribution is known, i.e., that L 1 fixed. The concentration is generally not known beforehand, as gaps in the phase space distribution at the input may have to be collected. Any unnecessary collection results in less than ideal concentration. Figure 6 shows how to find h in terms of L 2 and L 1 . The truncation angle t is the angle of the edge ray from the end of the source. From trigonometry, Fig. 6. Determining height h from truncation angle t. 2110 J. Opt. Soc. Am. A / Vol. 13, No. 10 / October 1996 D. Jenkins and R. Winston no bisection is required as in the 1 ­ 2 case. Numerical integration of this equation recovers the classic trumpet designs that were derived with a flow-line approach. Unfortunately, this method cannot analytically derive the trumpet shape as the flow-line approach can. But it has the important advantage of being able to be tailored to nonuniform distributions, a feature not hitherto available. Since ideal trumpets have indefinite length, letting R go to infinity results in numerical difficulties. These can be overcome provided that integration of the reflector profile starts from the target edge and R is allowed to go to infinity when tan diverges. In this way, ideal trumpets are covered. The only difficulty is that the optimal concentration must be found by using conservation laws or numerical computation. Fig. 7. tor. Use of a virtual source to design the trumpet concentra- Fig. 8. Multiple-reflection design of the trumpet with use of a single-reflection edge ray: (a) ray coming from the opposite side of the reflector, (b) ray hitting the reflector directly. C. Nonflat Absorbers Generalizing to a nonflat absorber follows along the lines already described. The main design change is that the target or the absorber can block radiation from reaching the reflector. This is accounted for by defining the acceptance angle appropriately (discussed below). In Fig. 9 we show the design method for an arbitrary-shaped convex absorber, using a CEC-type approach. The parameter (R, ) is the angle that the extreme ray from the absorber makes at point R, . If the absorber is circular, then the center of the absorber is the optimal location for the origin and arcsin( /R), where is the absorber radius. The coordinate origin is along the vertical axis of the absorber for simplicity. 1 is the extreme angle from the source distribution that hits the reflector at the design point. It is obtained by ray tracing backward as described above, but with the difference that it is not blocked by the absorber. We desire that the edge ray at 1 reflect so that it hits just tangent to the top of the absorber. We use a complimentary edge ray as we did in nate system at the edge of the source distribution and causing edge rays to be reflected toward the virtual foci, we can design for an infinite-reflection edge-ray collector, using a one-reflection design method. All that is required is that we define the acceptance angle 1(R, ) appropriately. There are two parts along the integration curve that need to be considered. These two regions are depicted in Fig. 8. Starting from the origin, a ray is drawn from the opposite focus through the reflector curve. If this ray does not pass through A, A , then the angle for 1(R, ) is given by the angle of this ray. This is because the reflector on the other side is sending all of its edge rays toward this focus. If the ray from the opposite focus passes through A, A , then the maximum angle onto the reflector is found by ray tracing backward toward the source. This angle 1 is used for the integration of Eq. (6). The angle is determined by Eq. (7). The technique for this integration is to start from the outside edge of the reflector and integrate inward, because these points on the reflector curve are known. The final target aperture is arrived at by the integration, and Fig. 9. Concentrator design with use of an auxiliary edge ray for a nonflat absorber. D. Jenkins and R. Winston Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A 2111 able gap may allow significant increases in concentration with minimal loss. D. Adding Constraints The approach outlined above presents the general technique for design of the three major types of concentrator. We now show how to modify the design equation to account for additional constraints. This is similar to adding the requirement that 2 /2 in the 1 ­ 2 system design. The constraints that we explain further are the addition of a lens to the design, the requirement that total internal reflection (TIR) be satisfied, and the requirement that the design be in the context of a graded-index material. Adding a lens allows a concentrator to be much more compact. Also, it is relatively easy to incorporate another lens into the design method. To account for the lens, 1 and height h are changed. The acceptance angle along the reflector 1 is obtained by a backward ray trace through the lens to yield the maximum angle that is inside the input distribution, and numerical integration proceeds as before. The new height h is found by determining how the truncation angle t is deviated by the lens. This new angle is required to hit the edge of the absorber to produce the optimal height. As before, truncating height results in a loss of concentration. The requirement that all rays satisfy TIR is used to advantage in dielectric TIR concentrators to minimize reflection losses and heating. For a 1 ­ 2 concentrator, the edge ray makes an angle , with the normal of the reflector greater than the critical angle. In Fig. 11 we show how this ray is made to satisfy TIR. Requiring that arcsin(1/n) yields a new constraint on ,
1 Fig. 10. Reverse ray tracing to find the maximum angle onto the reflector with 1 b. Subsection 2.A to simplify the geometry. This auxiliary ray goes through the origin as before, and simple trigonometry gives the equation for ,
1 R, 2 R, . (10) However, a new difficulty arises in addition to finding (R, ). The absorber can now intercept rays from the source and prevent them from reaching the reflector. Designing the reflector to collect these rays results in an unnecessary loss in concentration. Thus the problem requires finding the extreme angle ray hitting the reflector that is not blocked. Figure 10 shows how to backward ray trace to find the maximum angle. Knowing the angle b , the maximum angle that cannot hit the absorber, one searches between /2 and b to find 1 . Note that if no light hits directly onto the reflector from the source, then it becomes necessary to ray trace backward, using multiple reflections, until a ray that intersects the source is discovered. This requires that the integration start from the outside of the source and proceed inward, because otherwise the multireflection ray trace cannot be done, as the reflector profile on top is as yet undetermined. An important point is that an involute is not assumed for the bottom of the concentrator, because the integral equation is defined for all and gives the maximum concentration. For flat phase space distributions the integral method recovers the involute automatically, and the results are the same as those obtained with geometrical techniques.4 As with the trumpet, analytic solutions for the reflector shape are replaced by a constructive technique that addresses a wider class of problems. The gap between absorber and reflector is not fixed and is determined by the value of R at 0. A ray trace is required for us to learn whether this gap results in a significant loss. Later in the paper, we describe a new secondary concentrator designed for a circular primary mirror that has a very large gap with very low losses. This concentrator does not use an involute. Allowing a vari- 4 arcsin 1/n , (11) where n is the index of refraction of the medium. Designing a concentrator with use of a circular or elliptical profile lens yields the same shapes as previously achieved with different techniques.13 Fig. 11. Requiring TIR, i.e., arcsin(1/n). 2112 J. Opt. Soc. Am. A / Vol. 13, No. 10 / October 1996 D. Jenkins and R. Winston R L2 cos cos 2 2 1 /2
1 /2 , (13) which is a straight line whose direction corresponds to /2 ( 2 When /2 1 )/2. 2 , Eqs. (6) and (7) yield ln R /2 d
2 /2
2 R tan 1 4 2 . (14) Evaluating the integral and simplifying gives R R /2 cos2
2 /2 /4
1 2 1 2 1 1 /2 /2 (15) cos2 cos sin R
Fig. 12. Design in graded-index material. ray is used. An auxiliary edge /2 2 1 , 1 One can use a graded-index material provided that one knows the index profile beforehand. The graded-index material normally acts similarly to a lens, and one finds 1 from a backward ray trace through the medium. The difference is that a ray trace to hit the edge of the absorber is also needed. In Fig. 12, we find 2(R, ) and use Eq. (8) to find for all values of 2 . The use of graded-index materials may be impractical at present, but examples may be found in nature. We are currently studying the eye cones of Limulus, commonly known as the horseshoe crab. Experimental data show that the original comparison with a standard CPC does not account for the imaging properties of the graded index of refraction within each cone.14,15 which is a parabola with direction of axis corresponding to /2 /2 1 . Using 1 as the final point on the curve determines the height for the ideal concentrator. The curve given in rectangular coordinates with its origin shifted to the center of the target is given by R( )cos L2/2, R( )sin . The entrance width L 1 equals L 2 sin 2/sin 1 , which matches the theoretical limit. B. Novel Secondary for Circular Primary In the past, 2-D solar collector systems with a cylindrical circular mirror had limited utility because of the significantly lower concentration of approximately 2­3 compared with the value of 25­30 from commercial parabolic troughs.16,17 The symmetry properties of the circular mirror allow the primary mirror to be fixed, and only the secondary concentrator or absorber need pivot to track the Sun. This advantage, however, is offset by the large 3. SOLVED EXAMPLES
We have some example solutions to show that the technique can derive previous designs and can be applied to the design for complex systems that may involve many stages. The classic 1 ­ 2 profile is derived analytically, and numerical results are given for the design of a novel two-stage solar concentrator with a fixed circular primary mirror and a unique tailored secondary. We have also solved for CPC's with circular absorbers, dielectric TIR concentrators, and trumpets and have shown that the solutions agree with previous results. These solutions cannot be performed analytically and are omitted here in the interest of conciseness. A. 1 ­ 2 Concentrator The analytic solution for a 1 ­ 2 concentrator where 1 2 is here presented. Combining Eqs. (6) and (8) and then integrating yields the equation for R( ) for /2 2, ln R L2 d
0 tan 2 1 2 . (12)
Fig. 13. Rays reflected off a circular arc. which the reflected rays are tangent. Note the envelope to For 1 and 2 constant, Eq. (12) gives the analytic solution for R( ), D. Jenkins and R. Winston Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A 2113 13. We prefer to call it an envelope because the aberrations off the primary are very large (we use an approximately 30-deg arc). The shape of the envelope turns out to be important for positioning the aperture of the secondary. We find the equation for the curve by intersecting two parallel rays separated by an infinitesimal amount after they reflect off the primary. The parameterized equations for the envelope in Fig. 13 in rectangular coordinates are X Y R sin3 , R cos3 , (16) (17) 3 R cos 2 Fig. 14. Effect of tracking tolerances on proper secondary positioning. L/R is the ratio of lever arm length to arc radius; W/R is the ratio of secondary width to arc radius. where R is the primary arc radius of curvature and defines the angular position that the ray reflects from the primary arc and parameterizes the position on the envelope to which the ray is tangent. Note that when 0, each ray passes through the paraxial focus. For the secondary aperture to be positioned for a set of incident parallel rays, each edge must intersect the caustic. For a given width of the secondary W 2X, the length of the optimal pivoting lever arm is L Y. Maximum collection occurs for this lever arm length because rays do not cross the envelope to which they are tangent. On each side of the symmetry axis there is an envelope function, so rays tangent to one side pass through the envelope on the opposite side. For smaller lever arms more of Fig. 16. Standard 55-deg acceptance CPC with C 1.15. Fig. 15. Brightness distribution hitting the aperture of the secondary mirror. aberration of the cylindrical mirror. We want a nonimaging secondary to restore some of the concentration lost to aberrations of the circular primary. Designing the secondary requires that the location and the aperture of the secondary be specified. This involves an elementary study of how rays reflect off a circular arc. One notices a caustic, or envelope, pattern, to which the reflected rays are tangent. This caustic is what is used to describe the profile in imaging optics and is shown in Fig. Fig. 17. Tailored concentrator with C 1.45. 2114 J. Opt. Soc. Am. A / Vol. 13, No. 10 / October 1996 D. Jenkins and R. Winston Fig. 18. Brightness distribution from the primary mirror collected by a tailored concentrator. the rays passing through the opposing envelope are lost, and no more rays are collected to compensate for them. As seen from Fig. 13, the density of rays tangent to the envelope is much higher than the density of those traversing it. Thus increasing the lever arm so that the width no longer intersects the envelope results in a net loss of intercepted rays. The angular spread of the Sun and tracking tolerances are now taken into account. A tracking angle track is introduced as the maximum angle of deviation of a ray off the primary mirror for any cause. These may include the angular spread of the Sun, tracking tolerances, and nonspecular reflection off the primary mirror. The effect of the tracking angle is modeled by rotating the envelope functions by track and requiring the intersection as before. Ray tracing with this model allows selection of the optimum lever arm. As track increases, L decreases, resulting in lost collection. Figure 14 shows L as a function of W, R, and track . We find that W/R in the range of 0.04/0.05 works best and allows the primary mirror to be seen well. A value of track of 1 deg (18 mrad) gives wide tolerances and little loss in performance. This procedure differs from that used in parabolic troughs where the tracking tolerances are input parameters rather than design requirements. An overall scale for the system is chosen by housing the secondary inside a 150-mm-outer-diameter quartz tube that is evacuated to minimize thermal losses. Placing the aperture at the center of the tube gives W 143 mm (5.6 in). Setting R 3.23 m (127 in.), we find that L 1.74 m. All the light rays incident on a 55-deg-arc of the primary are collected on the secondary aperture. This gives a concentration of the primary of 21. Next the radiance distribution onto the secondary aperture is ray traced, and, as illustrated in Fig. 15, a very nonuniform pattern results. A Gaussian distribution of angular deviations is used, with a standard deviation of 0.5 deg. This model assumes that the tracking has no offset and that other sources of angular spreading are uncorrelated. By calculating the phase space area that actually needs to be collected, we find the theoretical concentration limit of the secondary to be 2.5. However, a standard CPC with a 55-deg-acceptance, no reflector­absorber gap, and an entrance aperture located at the center of the evacuated tube, concentrates only to 1.15. Note that truncation losses prevent ideal concentration. This concentrator is shown in Fig. 16. The main reason for the shortfall is that there are gaps in the brightness distribution, and the maximum angle as a function of position on the aperture also varies. With use of the design technique for circular absorbers in Subsection 2.C, the acceptance angle at each point along the reflector is found by reverse ray tracing. The concentrator that results gives a concentration of 1.45 and is shown in Fig. 17. The large gap is checked by a ray trace of the design and results in a loss of only 0.1%. In Fig. 18 we show the radiance distribution at the secondary that is collected. We believe that this is the optimum secondary concentrator for this type of system. Prototypes of this collector system are being built and should be tested within the year. To demonstrate how tailoring affects the phase space that is collected, Fig. 19 shows which parts of a Lambertian distribution are collected. Also, each radiance bin has a gray-scale level associated with the number of reflections it undergoes before hitting the absorber. Since the direct radiation onto the absorber is not reflected, it Fig. 19. Brightness distribution from a Lambertian source collected by a tailored concentrator. Black bins hit the absorber directly. Gray indicates light collected after one or more reflections. White shows rays not collected. D. Jenkins and R. Winston Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A 2115 cannot be tailored. This effect results in a 20% loss in concentration. Each multiple-reflection region is separated by a gap-loss region. There is little radiance associated with the gap loss, allowing large absorber­reflector gaps with minimal loss. In a design case like this, leaving the gap as a variable parameter and not using an involute to start the reflector profile is apparently the way to approach maximal concentration. Starting with no absorber­reflector gap would limit concentration to 1.25. This new secondary gives the system an overall concentration of 30.5. The concentration is comparable to that obtained using commercial parabolic troughs, but this system has higher tracking tolerances. More significantly, only the secondary needs to pivot to track the Sun, while the primary mirror remains fixed. Design of secondary concentrators in conjunction with a parabolic primary, a Fresnel lens, or a Fresnel mirror has potential to increase concentration and tracking tolerance. One may also consider tailoring the primary mirror to give radiance distributions that will maximize secondary concentration and hence overall performance. Some interesting concentrators have been recently developed21,22 that fold the light path by using first refraction, then reflection, and finally TIR. These concentrators are not reproducible by a straightforward application of our technique, as they require tailoring multiple surfaces simultaneously. Therefore generalization of our methods to multiple surfaces would be of interest. The tailoring of 3-D concentrators has not been addressed by this paper. Rotationally symmetric concentrator design may work with backward ray tracing to produce an effective projected acceptance angle. The failure of the 3-D CPC to reach the ideal limit suggests that more work needs to be done in this area.23 The trumpet, a CHC-type concentrator, works ideally in three dimensions. It would be interesting to see if tailored CHC concentrators can reduce or eliminate 3-D losses. ACKNOWLEDGMENTS
We thank Josh Bliss for his help in the design method that uses circular absorbers. Thanks to Gabor Horvath for discussions involving the eye cones of Limulus. We are grateful to our colleagues Jeff Gordon, Ari Rabl, and Harald Ries for sharing their work with us before its publication. Todd Greene is thanked for his help in editing this paper. This work is supported by U.S. Department of Energy grant DE-FG02-87-ER-13726. A new paper by H. Ries and W. Sprikl24 came to our attention too late for us to comment about it in this paper. 4. DISCUSSION
The two examples and the ease of application of the new techniques should encourage research into better concentrators for many systems. We have derived many of the older classical designs of nonimaging optics and have shown the potential for unique designs for complex input radiance distributions. The application to the fixed primary mirror two-stage system dramatically demonstrates the advantage of the integral design method over conventional nonimaging designs. This collector has concentration comparable to a parabolic trough, but only the secondary receiver has to track. Other important considerations allow for greater tracking tolerances and make possible the specification of tracking tolerance as a design parameter. As seen in the secondary design, the shape of the absorber plays an important role in setting concentration limits. Use of a complementary edge ray that passes through the origin of the coordinate system readily accommodates variations in absorber shapes. By tailoring the absorber shape to better match the input distribution, one may be able to increase concentration significantly. More work in lens design to minimize concentrator height while maintaining near-ideal concentration would be useful.18 Also, the reflector­absorber gap loss reduction solutions can use an integral design method for very small gaps. More work can be done in this area. Currently, conventional methods work well by adding a gap-loss reduction portion to a reflector.19 The basis for this paper is the edge-ray principle, which has been used to design nonimaging concentrators for many years. This paper spells out in general how to use an edge ray for most types of nonimaging concentrator. Recent work has shown that the method is optimal for uniform radiance distributions.5,20 Expanding on this may allow one to prove that this method is optimal for nonuniform distributions as well. REFERENCES AND NOTES
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