the first stage is an imaging device, such as a mirror or ... Typical two-stage concentrating system. ... back to the source to find the maximum angle that can hit.
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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 u 1 – u 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 u 1 – u 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 I~ x ! 5
E
p /2
2p /2
cos u du B ~ x, u ! ,
(1)
where u 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 5 constant for u u u < u 1 and u x u < L 1 /2 and zero otherwise. The source is immersed in a medium with n 5 1. This is a uniform radiance pattern. L 1 is the collecting aperture width, and u1 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 u 2 and size L 2 , immersed in a medium with index n. The irradiance I 1 on the source is given by I 1 5 2B 0 sin u 1 , and the irradiance I 2 on the target is © 1996 Optical Society of America
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D. Jenkins and R. Winston
Vol. 13, No. 10 / October 1996 / J. Opt. Soc. Am. A
C 22d 5
n cos f rim , sin u 1
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(5)
where f rim is the rim angle of the imaging primary and u 1 is the acceptance angle. An additional condition is that u 1 ! f 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.
I 2 5 2nB 0 sin u 2 .
(3)
Therefore the ideal concentration in two dimensions is I 2 /I 1 , or C2–d 5 n
sin u 2 . sin u 1
(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 5 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 ( u 1 ! 1). Their length is proportional to cot u 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.6–8 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.
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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 5 R tan a , df
Fig. 3. Different types of concentrator: type.
(a) CEC-type, (b) CHC-
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 u 1 – u 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 u 1 – u 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.
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where the coordinates (R, f) represent a point on a curve and a 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. 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 a (R, f) 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. u 1 – u 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 u1 at the entrance aperture and u2 at the exit aperture, there is the classic u 1 – u 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 u 2 . p /2 2 f . In this case the edge ray hits the edge of the target. The only new parameter entering the system is u1(R, f). This is the largest angle a ray makes with respect to the vertical axis that can hit the reflector at point R, f. For standard designs u1(R, f) is a constant for all R, f. 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
h5
S
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D
L1 L2 1 cot u t . 2 2
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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 u 2 a function of R, f. This might be accomplished by designing a concentrator with a target placed behind the exit. Solutions for the standard u 1 – u 2 CPC are presented later in the paper.
Fig. 5. Design of u 1 – u 2 -type concentrator: (b) f < p /2 2 u 2 .
(a) f . p /2 2 u 2 ,
is potentially difficult to perform for complex input radiance distributions. Once u 1 is known, a is found by using simple geometry,
a5
p f 2 u 1 ~ R, f ! 1 4 2
if u 2 .
p 2 f. 2
(7)
This when combined with Eq. (6) is usually not solvable analytically unless u 1 is constant. When the constraint that u 2 , p /2 is added, the edge rays are not always reflected onto the target edge. For u 2 < p /2 2 f , edge rays are not reflected through the origin, and finding a is more difficult. Design in this region is shown in Fig. 5(b). To cause the edge ray to exit at angle u 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 d 5 p /2 2 f 2 u 2 . Adding d to u1 in Eq. (7) yields the desired value for a. By requiring that the complementary edge ray at u 1 1 d hit the edge of the target, the real edge ray at u 1 exits the target with angle u 2 as desired. The equation for a in this case is
a5f1
u 2 2 u 1 ~ R, f ! 2
if u 2
arcsin(1/n) yields a new constraint on a,
a
