Nonimaging Optics Winston Episode 8 doc

The points S and S¢ of the input bundle of the concentrator 5 (shown in Figure 8.20) are at infinity. This concentrator is not a typical case of one designed for a source at infinity because the points A and A¢ are not coincident with D and D¢. Table 8.3 shows some geometrical characteristics and the 3D ray-tracing results of the RX concentrators #1 to #4. The second row in this table shows the values T t when the shadow of a surface placed at the receiver is taken into account. Observe that this surface intercepts some rays in their trajectories from the refrac- tive surface to the reflective surface. The third row shows the values of T t when the reflection losses at the mirror (reflection coefficient 0.91) and the Fresnel losses at the dielectric interface (refractive index n = 1.483) are also considered. Figure 8.21 shows the transmission function of the RX #3 (Figure 8.19). This is a function of r S /SS¢ and r A /AA¢ (r S is the distance from a point of the circle SS¢ to the symmetry axis, and r A is the same for a point of the circle AA¢). The trans- mission function gives the percent power reaching the receiver relative to the power emitted from the ring r S , r S + dr S (of the circle SS¢) toward the ring r A , r A + dr A (of the circle AA¢). The radiance of all the rays is assumed to be identical for this calculation. If the 3D concentrator were ideal, then the transmission func- tion would be 1 if r S /SS¢£1 and r A /AA¢£1, and it would be zero elsewhere. The 8.8 Three-Dimensional Ray Tracing of Some RX Concentrators 199 #2 #3 #4 -40 -20 0 20 40 120 S ¢ S A A ¢ receiver reflector 100806040200 Figure 8.19 RX concentrator #2, #3, and #4. The input and output bundles are the same as in Figure 8.18. Other data of these concentrators are shown in Table 8.3. 50 100 -50 -100 50 100 0 -50-100 2ø 2ø A A ¢ S S' D D ¢ M M ¢ R ¢ R #5 0 Figure 8.20 RX concentrator #5. The points S and S¢ of the input bundle of this concen- trator are at infinity. total transmission T t appearing in Table 8.3 can also be defined as the percent power reaching the receiver relative to the power emitted by the circle SS¢ toward the circle AA¢. Calculation of the transmission function (and T t ) by averaging the number of rays reaching the receiver must be carefully made. In this average each ray r x should be weighted by the étendue of the pencil of rays represented by r x . For instance, assume that the circles SS¢ and AA¢ of the input bundle are divided into small regions of area dA S and dA A , respectively, and that the rays traced are those linking the centers of the small regions of AA¢ with the center of the small regions of AA¢. Then, each ray represents a pencil of rays whose étendue is (8.5) where D is the distance between the centers of both regions and q is the angle formed by the ray and the z-axis. As the XRs, the rotational symmetric RX performs well for maximal concen- tration when the average angular spread of the input bundle is small. The geo- dE D dA dA SA = cos 2 2 q 200 Chapter 8 Simultaneous Multiple Surface Design Method Table 8.3. Geometrical Characteristics and 3D Ray-Tracing Results of Selected RX Concentrators. Concentrator Number Concentrator Characteristics 1 2 3 4 Total transmission T t (%) 99.4 98.2 98.3 95.7 T t considering shadow losses (%) 99.4 98.2 97.8 94.3 T t considering shadow reflection and 84.9 84.5 85.2 82.6 Fresnel losses (%) Concentrator aperture diameter 91.9 77.5 51.8 28.6 Dielectric thickness at the center 28.5 24.0 17.0 11.0 Distance from the SS¢ plane to the 34.5 49.0 72.0 101.0 concentrator bottom Concentrator parameters: Refractive index n = 1.483; receiver radius of 1; input bundle SS¢ with a diameter of 80; input bundle AA¢ with a diameter of 9.38; distance from SS¢ to AA¢ of 120; and output bundle yielding maximal concentration. 1.0 0.5 0 00.51.0 2r A /|AA¢| 2r S /|SS¢| 0.95 0.05 0.35 0.65 Figure 8.21 Transmission function of the RX #3 of Figure 8.19. The plotted function gives the power reaching the receiver over the power emitted by the points of the ring [r S , r S + dr S ] (of the circle SS¢) toward the ring [r A , r A + dr A ] (of the circle AA¢). metrical aspect of the RX concentrator is also quite compact, although less than the XR concentrator. Moreover, the distance from SS¢ to the concentrator’s bottom is smaller in an RX than in other classical nonimaging concentrators designed for a finite source as an input bundle. This implies that if a lens is placed at SS¢ as a first stage of concentration, then the aspect ratio of the whole concentrator is more compact in the case of the RX. The total transmission T t is, in general, better for the RX than for an equivalent Compound Elliptical Concentrator (CEC) when the input bundle has a small angular spread, and worse for the RX than for the Flow Line Concentrator, which is one of the few known ideal 3D concentrators (defined by T t = 1). In the case of a large angular spread of the input bundle, the CEC has better total transmission than the RX concentrator. The RX uses much more dielectric material than an equivalent XR concen- trator, but it can be made in a single piece, whereas the XR needs at least two pieces that must be assembled. Then, the RX seems more appropriate than the XR when the cost of the dielectric material is not critical or when the assembly may be a complex process. In both concentrators (RX and XR), the receiver faces in opposite direction as the concentrator’s entry aperture. This is a problem in some applications, such as photovoltaics where the heat sink of the solar cell is in the back side of the cell and may introduce additional losses because of its shadowing. 8.9 THE XX CONCENTRATOR Similarly, we can design a concentrator in which both surfaces are reflective. The design technique is the same, with no shadowing effect taken into account. In general, one of the shadows of the mirrors dramatically degrades the performance of the concentrators for maximal concentration on flat receivers (emitters). This problem can be solved for tubular receivers (see Chapter 14); in other words, the mirrors can be designed so the efficiency reduction due to shadowing is very small. For flat receivers there is also a solution in some cases. This solution is based on using layers of low refractive index material (n 1 ) immersed in a high refractive index layer (n h ) (see Figure 8.22). Such layers appeared also in the design of the 8.9 The XX Concentrator 201 low index (n l ) layer sin -1 (n l /n h ) transmitted reflected high index (n h ) bulk material Figure 8.22 A low index layer immersed in a high index material is a mirror at grazing angles and does not modify the ray trajectories otherwise. CTC (see Chapter 6). The basic idea is that if the layer is thin enough, it does not modify the trajectory of the rays if these hit the interface at an angle smaller than sin -1 (n 1 /n h ), or otherwise the layer behaves like a reflector. Fortunately some XX designs are such that most of the collected rays cross one of the mirror surfaces at small angles when they shouldn’t be reflected accord- ing to the design technique, whereas they hit the mirror surface at great angles when a reflection is necessary. Thus, the low index layer is adequate for such designs. Figure 8.23 shows an example of an XX in which the low index layer tech- nique applies. In the central part of the left mirror the incidence angle is not enough to produce total internal reflection. For this reason it should be mirrored, and this creates some losses by shadowing. These losses can be kept below 5% for most of the designs with maximal concentration and collimated (<±8 degrees) infi- nite source. 8.10 THE RXI CONCENTRATOR As we saw before, both the XR and the RX have the active side of the receiver (emitter) facing in the opposite direction of the entry aperture. This creates a prac- tical problem when, for example, we want to use an optoelectronic component. Heat sinks and electrical connections must be in the back side of the component. The RXI concentrator was initially designed to solve this problem. The RXI can also be made in a single dielectric piece but, unlike the RX, the receiver, which is 202 Chapter 8 Simultaneous Multiple Surface Design Method reflector Low index (n l ) layer n h n h reflector receiver Figure 8.23 XX (n 1 = 1.5), refractive index of the low index layer n = 1.33. Half acceptance angle 1.5°, Geometrical Concentration 3283. also immersed in the dielectric, faces the entry aperture (Miñano, González, and Benítez, 1995). There is an important difference between the RXI concentrator and other related concentrators: the RR, the XR, and the RX. All of these concentrators are formed by two optical surfaces. The rays hit one of the surfaces and then the other when they go from the source to the receiver. The RXI also has two optical sur- faces, but unlike the other cases, the rays impinge on its entry aperture, twice. First they suffer a refraction then they are reflected on the second surface and finally they are reflected again (by total internal reflection) on the first surface and sent toward the receiver (see Figure 8.24). The difference with respect to the RX concentrator is that the rays interact twice with the first surface. This fact modi- fies substantially the design procedure. The condition for total internal reflection cannot be fulfilled at every point of the entry aperture, and because of this, a small part of the entry aperture surface should be mirrored. The main aim in the design of the RXI was to obtain (1) a highly compact con- centrator for maximal concentration designs in which M i has a small angular spread; (2) high values of the total transmission T t (like in the XR and the RX); (3) a concentrator that can be made in a single solid piece (like the RX); and (4) the active side of the receiver facing the concentrator’s aperture. This last condi- tion is what differentiates the potential applications of the RX and the RXI. The RXI can be designed in an iterative process of XX and RX designs. Here, the RXI is considered not as a two-surface optical device but as a three-surface 8.10 The RXI Concentrator 203 –100 –80 –60 –40 –20 20 40 60 80 100 0 20–20 –400 refractive surface receiver reflector TIR reflector Figure 8.24 RXI concentrator with maximum concentration (C g3D = 7387, q a =±1° and n = 1.5). one. Two of these surfaces (named 1X and 2) are reflective, and the last one (1R) is refractive. It is also assumed that the rays of the input bundle M i are first refracted on surface 1R and then reflected by surface 2 and finally reflected by surface 1X in this order, no matter what would actually happen. For example, the first change of trajectory of a M i ray is calculated at the point of interception of the ray with the surface 1R, even if this ray has previously intercepted another surface. Henceforth, the surfaces 1R, 2, and 1X will be designated curves 1R, 2, and 1X, respectively, because only meridian rays are considered in the design. If the iterative process is successful, curves 1R and 1X converge to the same curve, which will be designated curve 1. It is assumed that the description of the bundles M i and M o , as well as the refractive index n of the dielectric material, is known for the design. Of course, M i and M o must fulfill E(M i ) = E(M o ). In the examples shown in Figures 8.24, 8.27, and 8.28, M i is a source at infinity, and M o yields maximal concentration (i.e., M o is formed by all the rays reaching the upper face of the receiver). As an example of the design procedure, here are the steps to follow for design- ing an RXI with maximal concentration (or close to it). 1. The iterative process starts with an arbitrary 1X curve subject to the follow- ing restrictions: (a) every ray of the bundle ∂M i intercepts the curve 1X once. The same must happen with the rays of ∂M o when reversed (see Figure 8.25). It is assumed that the forward direction of the rays is from M i and M o . The rays of M i and M o in Figure 8.25 advance rightward toward increasing values of z. (b). The curve 1X is symmetric with respect to the z-axis. 2. Choose a 1R curve coincident with the 1X curve. 3. Trace the ray trajectory r 1 after being refracted at the point R of the curve 1R and the ray trajectory of r 2 before being reflected at the point I of the curve 1X (see Figure 8.25). r 1 is the edge ray of ∂M i whose point of interception with the curve 1R has the most negative coordinate x and whose direction cosine p 204 Chapter 8 Simultaneous Multiple Surface Design Method R X curve 1X = curve 1R curve 2 receiver symmetry axis 20 15 10 5 0 –5 –5–10 0 5 ∂M i- rays r 2 I ∂M o - rays Figure 8.25 Curve 2 obtained in the first iteration of an RXI design. Curve 1X (coincident with curve 1R in this design step) is the one proposed at the beginning of the design pro- cedure. The RXI is designed for a source at infinity (subtending an angle of ±5°) as input bundle and for maximal concentration. (with respect to the x-axis) at this point is the most negative one when com- pared with other edge rays reaching this point (this point is the point R in Figure 8.25). When evaluating the sign of the direction cosine, it is important to keep in mind the convention about the forward direction of the rays. r 2 is the ray of ∂M o whose point of interception with the curve 1X has the most neg- ative coordinate x (point I in Figure 8.25). The trajectories of r 1 after refrac- tion on 1R and that of r 2 before reflection on 1X cross at the point X (if not, start the process again with a new 1R curve at step 1). The point X will be a point of the curve 2. 4. Now make up the subset ∂M i- by choosing, at every point of the curve 1R, the ray of ∂M i with the most negative value of p and form ∂M o- in a similar way with the rays of ∂M i when they intercept the 1X curve. The ray r 1 belongs to ∂M i- , and r 2 belongs to ∂M o- . 5. Calculate the reflector curve 2 that passes through X and couples ∂M i- and ∂M o- (Figure 8.25). The ray r 1 will become r 2 after reflection at X. Observe that this problem is qualitatively identical to a Cartesian oval problem. In order to solve it, it is necessary to have no more than one ray of ∂M i- and no more than one ray of ∂M o- at any point on the curve 2. If this condition is not fulfilled, the iterative process should be restarted again with another 1R curve (go to step 1). This condition is not fulfilled when, for instance, the ∂M i- rays form a caustic after refraction on 1R or when the ∂M o- rays form a caustic before reflection on 1X. These cases can be easily recognized because a loop or a dis- continuity of the slope appears in the shape of the curve 2. When that occurs it is advisable to use a flatter curve 1R at step 1. Curve 2 must be calculated from the point X up to the symmetry axis. 6. Trace the rays of ∂M i+ (rays of ∂M i not belonging to ∂M i- ) after refraction at curve 1R and then after reflection at curve 2. Calculate now the new 1X curve that passes through the point I and reflects the ∂M i+ rays into the ∂M o+ rays (these are the ∂M o rays not belonging to ∂M o- ; see Figure 8.26). The calcula- tions needed to get the new 1X curve are again a Cartesian oval problem and 8.10 The RXI Concentrator 205 –5–10 0 5 20 15 10 5 0 –5 curve 1R curve 1X R X curve 2 symmetry axis ∂ M i- rays r 1 ∂ M o - rays r 2 Figure 8.26 Curve 1X obtained at step 6 of the first iteration of the RXI design. the same considerations done in step 5 apply to this step. The curve has to be calculated up to the symmetry axis. 7. If the old and new 1X curves are close enough, then the 2D design is finished; if not, let the new 1R curve be equal to the last calculated 1X curve, and con- tinue the process in step 5. 8. Analyze the condition for total internal reflection (for the edge rays) on curve 1 (at this step 1X and 1R have converged to the same curve). Usually, this con- dition is not fulfilled in the central part of the curve, and thus this part should be mirrored (see Figure 8.24). The preceding design procedure does not always converge, and the analysis of the conditions for convergence is a complex task. Obviously, convergence depends on the selection of the 1X curve in step 1. In general, it is advisable to choose a smooth curve in this step. When the starting curve 1X is very close to the receiver, then the process may converge in an RXI concentrator that has part or its entire receiver outside the dielectric. In this case the receiver is virtual. Figures 8.24, 8.27 and 8.28 show three different RXI designs for maximal con- centration and for a source at infinity (subtending angles ±1°, ±3°, and ±5°, respec- tively). Figure 8.29 shows three RXI concentrators that have also been designed for maximal concentration but whose input bundle is a finite source formed by the rays issuing from every point of AA¢ toward every point of BB¢. 206 Chapter 8 Simultaneous Multiple Surface Design Method 10 20 30 0 –10 –20 –30 10 0 –10 receiver Figure 8.27 The same as in Figure 8.24 but with an angle q i = 3°. 8.11 THREE-DIMENSIONAL RAY TRACING OF SOME RXI CONCENTRATORS This section gives results of the 3D analysis of RXI obtained by rotational sym- metry around the z-axis. Two types of RXI concentrators have been designed and analyzed. The input bundle of the first type of RXI concentrator is a source at infin- ity that comprises all the rays impinging on the concentrator aperture (surface 8.11 Three-Dimensional Ray Tracing of Some RXI Concentrators 207 receiver 10 15 –5 –10 –15 5 0 –5 0 5 Figure 8.28 The same as in Figure 8.24 but with an angle q i = 5°. #1 #3 #2 012010080604020 –40 –20 0 20 40 reflector receiver A¢ A B ¢ B Figure 8.29 RXI concentrators for the input bundle formed by the rays issuing from any point of AA¢ toward any point of BB¢. The output bundle contains all the rays that reach the receiver. generated by curve 1) whose angle with the axis of symmetry is not greater than a given value q i . The input bundle for the other type of RXI concentrators com- prises the rays issuing from the disk with diameter AA¢ toward the disk with diam- eter BB¢ (see Figure 8.29). The output bundle for both types of RXI concentrators comprises all the rays reaching the receiver. This is a flat disk centered at the sym- metry axis and orthogonal to it. Figure 8.30 shows the transmission-angle curves for several RXIs with sources at infinity and with different acceptance angles. Three of these RXIs are those whose cross section appears in Figures 8.24, 8.27, and 8.28. These curves do not take into account the shadowing of the front metallic area nor the one produced by rays intercepting the back side of the receiver. Both shadowing effects are taken into account in the calculations of the total transmission appearing in Tables 8.4 and 8.5—that is, for the calculations of the total transmissions in both tables, it is considered that a ray is lost if it intercepts the outer face of the mirrored region in the aperture surface or if it intercepts the back face of the receiver. Table 8.4 also gives other results of the 3D analysis and some geometrical characteristics of the rotational RXI concentrators designed for a source at infin- ity. The geometrical concentration C G is defined C G = A e /A r , where A e is the area of the projection of the concentrator entry aperture on a plane orthogonal to the 208 Chapter 8 Simultaneous Multiple Surface Design Method 3° 5° 1° 0.5 ° incidence angle (degrees) 0 1 2 3 4 5 6 100 0 Transmission (%) 80 60 40 20 Figure 8.30 Transmission-angle curves for several 3D RXIc’s for sources at the infinity with angle q a . The transmission includes the shadowing of the upper mirror. Table 8.4. Geometric Characteristics and 3D Ray-Tracing Results of Selected RXI Concentrators for Maximum Concentration and Infinite Source. Refractive index n = 1.5, length unit is equal to receiver radius. Acceptance Angle q 1 (degrees) Parameters ±0.5 ±1 ±3 ±5 Geometric concentration C G 29,546 7,387 821.5 296.2 Total transmission, T t (q i ) (%) 96.9 97.3 94.5 86.9 Total transmission without shadow T t (q i ) (%) 97.4 97.7 96.6 96.7 Thickness/entry aperture diameter 0.278 0.279 0.289 0.332 Dielectric thickness 95.6 47.9 16.6 11.4 Entry aperture diameter 343.8 171.9 57.3 34.4 Diameter of the front metallic reflector 23.2 11.2 8.2 10.8 Receiver to concentrator bottom distance 76.0 38.6 11.6 6.5 [...]... concentrators In Nonimaging Optics: Maximum Efficiency Light Transfer III (Winston, R., ed.) Proceedings of SPIE, Vol 25 38, 30–41 Gleckman, P., O’Gallagher, J., and Winston, R (1 989 ) Approaching the irradiance of the sun through nonimaging optics Optics News, 33–36 Luneburg, R K (1964) Mathematical Theory of Optics University of California Press, Berkeley Miñano, J C., and González, J C (1991) Design of nonimaging. .. RXI: A high-gain, compact, nonimaging concentrator Applied Optics, Vol 34, 34, 785 0– 785 6 Ning, X., Winston, R., and O’Gallagher, J (1 987 ) Dielectric totally internally reflecting concentrators Appl Opt., Vol 26, 300 Schulz, G (1 983 ) Achromatic and sharp real imaging of a point by a single aspheric lens Appl Opt., Vol 22, 3242–32 48 Schulz, G (1 988 ) Aspheric surfaces In Progress in Optics (Wolf, E., ed.)... The Optics of Rays, Wavefronts, and Caustics Academic Press, New York US Letters Patent 6,639,733 High Efficiency Non-Imaging Optics Welford, W T., and Winston, R (1 989 ) High Collection Nonimaging Optics Academic Press, New York Welford, W T., O’Gallagher, J., and Winston, R (1 987 ) Axially symmetric nonimaging flux concentrators with the maximum theoretical concentration ratio J Opt Soc Am A 4, 66– 68 Winston, ... to concentrator bottom distance 1 96.0 97 .8 0.252 15.4 61.2 7.9 11.3 2 93.2 96 .8 0.256 11.2 43.6 7.2 8. 2 3 86 .7 95.1 0.226 6.1 26 .8 5.6 4.0 symmetry axis, and Ar is the receiver area Table 8. 5 gives the results of the 3D analysis of the rotational RXI concentrators whose cross section appears in Figure 8. 29 8. 12 COMPARISON OF THE SMS CONCENTRATORS WITH OTHER NONIMAGING CONCENTRATORS AND WITH IMAGE FORMING... Section 8. 5 REFERENCES Chaves, J., and Collares-Pereira, M (1999) New ideas for new solar concentrators In Nonimaging Optics: Maximum Efficiency Light Transfer V (Winston, R., ed.) Proceedings of SPIE, Vol 3 781 , 174– 182 Friedman, R P., and Gordon, J M (1996) Optical designs for ultra high-flux infrared and solar energy collection: Monolithic dielectric tailored edge-ray concentrators Appl Opt 35, 6 684 –6691... nonimaging lenses and lensmirror combinations In Nonimaging Optics: Maximum Efficiency Light Transfer (Winston, R., and Holman, R L., eds.) Proc SPIE, Vol 15 28, 104–116 Miñano, J C., and González, J C (1992) New method of design of nonimaging concentrators Appl Opt 31, 3051–3060 Miñano, J C., Benítez, P., and González, J C (1995) RX: A nonimaging concentrator Applied Optics, Vol 34, 13, 2226–2235 Miñano, J C.,... 66– 68 Winston, R (1 980 ) Cavity enhancement by controlled directional scattering Appl Opt 19, 195–197 2 18 Chapter 8 Simultaneous Multiple Surface Design Method Wolf, E (19 48) On the designing of aspheric surfaces Proc Physical Soc., Vol LXI, 494–503 Wolf, E., and Preddy, W S (1947) On the determination of aspheric profiles Proc Physical Soc., Vol LIX, 704–711 9 IMAGING APPLICATIONS OF NONIMAGING CONCENTRATORS... CPC-type concentrator The highest levels of solar concentration have been achieved with combinations of parabolic mirrors and nonimaging concentrators, similar to the aforementioned one (except that the nonimaging concentrator is not a CEC) (Gleckman, O’Gallagher, and Winston, 1 989 ) Such combinations are thicker than the equivalent SMS concentrators Their optical performance may be better or worse depending... point R (see Figure 8. 34c) Again, the condition of constant optical path length determines the solution Calculate the following stage of the refractive surface, knowing that the rays e(+), after being reflected on the mirror, are refracted toward point R¢ (see Figure 8. 34d) Repeat steps 5 and 6 until the mirror and the refractive surface intersect the symmetry axis (see Figures 8. 34d and 8. 34e) The design... designed with refractive index n = 1.5, geometrical concentration 114.6x¢, acceptance angle a = ±0.75°, and primary rim angle f = 80 ° Cg / Cmax 1 XRIF 0 .8 Parabola + DTIRC 0.6 0.4 Limit of non-simultaneous Lambertian designs 0.2 0 20 30 40 50 60 70 80 90 Rim angle f (degrees) Figure 8. 37 Comparison of the XRIF concentrator with the conventional parabolic mirrorDTIRC secondary combination References 217 Note . 98. 2 98. 3 95.7 T t considering shadow losses (%) 99.4 98. 2 97 .8 94.3 T t considering shadow reflection and 84 .9 84 .5 85 .2 82 .6 Fresnel losses (%) Concentrator aperture diameter 91.9 77.5 51 .8. Transfer III (Winston, R., ed.). Proceedings of SPIE, Vol. 25 38, 30–41. Gleckman, P., O’Gallagher, J., and Winston, R. (1 989 ). Approaching the irradiance of the sun through nonimaging optics. Optics. Efficiency Non-Imaging Optics. Welford, W. T., and Winston, R. (1 989 ). High Collection Nonimaging Optics. Academic Press, New York. Welford, W. T., O’Gallagher, J., and Winston, R. (1 987 ). Axially symmetric