operating every day of the year from sunrise to sunset. The formulas for the target and source étendue are given in Eqs. (10.71) and (10.81), respectively. We assume the total surface area of the target is (10.86) Eq. (10.71) then allows us to calculate the target étendue: (10.87) When the source étendue is set equal to the target étendue, Eqs. (10.81) and (10.87) give the following value for the total surface area of the source: (10.88) For this equal-étendue case, Figure 10.11 depicts the skewness distributions of Eqs. (10.70) and (10.73). The upper limit on étendue that can be transferred from the source to the target by a translationally symmetric concentrator is computed using the integral of Eq. (10.50). For the equal-étendue case, this upper limit on transferred étendue turns out to be (10.89) With reference to Figure 10.11, this étendue limit is equal to that portion of the étendue region contained under the source’s skewness distribution that is inter- sected by the étendue region contained under the target’s skewness distribution. As indicated by Eqs. (10.51) and (10.53), the upper limits on efficiency and con- centration are computed by dividing e max by the total source and target étendue, respectively. For the equal-étendue case, the efficiency and concentration limits have the same value: (10.90) h max max C==90 14.%. e max = 2 832 2 msr A src = 2 029 2 m ep trg s= mr 2 . A tr g = 1 2 m. 10.3 Translational Symmetry 259 2.4 2 1.6 1.2 0.8 0.4 0 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 Translational skewness (unitless) Skewness distribution (m 2 sr) Source Target Figure 10.11 Translational skewness distributions in a unit-refractive-index medium for a Lambertian target and a source having fixed latitudinal cutoffs parallel to the symmetry axis. The angular half width of the source is Q 0 = 23.45°. The source étendue equals that of the target. Using Eqs. (10.51) and (10.53), we can also compute the efficiency and concentra- tion limits for source-to-target étendue ratios other than unity. The resulting effi- ciency limit is plotted as a function of the concentration limit in Figure 10.12. The diamond-shaped marker on this plot indicates the efficiency and concentration limits for the equal-étendue case. It is also useful to plot the efficiency and con- centration limits as a function of the source-to-target étendue ratio itself, as shown in Figure 10.13. Note that the equal-étendue case corresponds to the crossing point of the two curves in this plot. 260 Chapter 10 Consequences of Symmetry 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Concentration (unitless) Efficiency (unitless) Figure 10.12 Plot of the efficiency limit as a function of the concentration limit for trans- lationally symmetric nonimaging devices that transfer flux to a Lambertian target from a source having fixed latitudinal cutoffs parallel to the symmetry axis. The angular half width of the source is Q 0 = 23.45°. The diamond-shaped marker indicates the performance limit for the equal-étendue case. 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 54.5 Source-to-target etendue ratio Efficiency limit Concentration limit Efficiency & concentration (unitless) Figure 10.13 Plot of the efficiency and concentration limits as a function of the source-to- target étendue ratio for translationally symmetric nonimaging devices that transfer flux to a Lambertian target from a source having fixed latitudinal cutoffs parallel to the symme- try axis. The angular half width of the source is Q 0 = 23.45°. The equal-étendue case corre- sponds to the crossing point of the two curves. 10.3.6.2 Flux Transfer to Lambertian Target from Source Having Fixed Longitudinal Cutoffs Parallel to Symmetry Axis with Orthogonal Fixed Latitudinal Cutoffs As a second and final example, we now consider a source of the type analyzed in Section 10.3.5.4. We set the latitudinal half angle equal to the latitudinal half width of incident solar radiation for non-tracking solar concentrators: (10.91) The refractive index is assumed to be unity for both the source and the target. With these choices, this case is representative of a north-south-oriented non- tracking solar concentrator. For such a concentrator, the appropriate value of the longitudinal half angle f 0 depends on the daily hours of operation. We assume that (10.92) which corresponds to daily operation from sunrise to sunset at an equatorial loca- tion. As is apparent from Eq. (10.84), the choice of f 0 affects only the vertical scaling of the skewness distribution. It therefore has no effect on the efficiency and concentration limits as a function of étendue. The formulas for the target and source étendue are given in Eqs. (10.71) and (10.85). As in Section 10.3.6.1, we assume the total surface area of the target is (10.93) so that the target étendue is (10.94) When the source and target étendue are equal, Eqs. (10.85) and (10.94) give the following value for the total surface area of the source: (10.95) For this equal-étendue case, the skewness distributions of Eqs. (10.70) and (10.84) are as depicted in Figure 10.14. The upper limit on étendue that can be trans- ferred from the source to the target by a translationally symmetric concentrator is computed using the integral of Eq. (10.50). For the equal-étendue case, this upper limit on transferred étendue turns out to be (10.96) The upper limits on efficiency and concentration are computed by dividing e max by the total source and target étendue, respectively. For the equal-étendue case, the efficiency and concentration limits have the same value: (10.97) As in Section 10.3.6.1, we can also compute efficiency and concentration limits for source-to-target étendue ratios other than unity. The resulting efficiency limit as a function of the concentration limit is shown in Figure 10.15. As before, the diamond-shaped marker on this plot indicates the efficiency and concentration limits for the equal-étendue case. Figure 10.16 provides plots of the efficiency and concentration limits as a function of the source-to-target étendue ratio. Concen- h max max C==49 30.%. e max sr= 1 549 2 m A src = 2 029 2 m ep tr g sr= m 2 . A trg = 1 2 m, f 0 90=∞, q 0 23 45=∞ 10.3 Translational Symmetry 261 262 Chapter 10 Consequences of Symmetry 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 Translational skewness (unitless) Skewness distribution (m 2 sr) Source Target Figure 10.14 Translational skewness distributions in a unit-refractive-index medium for a Lambertian target and a source having fixed longitudinal cutoffs parallel to symmetry axis with orthogonal fixed latitudinal cutoffs. The latitudinal angular half width of the source is q 0 = 23.45°. The source étendue equals that of the target. 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Concentration (unitless) Efficiency (unitless) Figure 10.15 Plot of the efficiency limit as a function of the concentration limit for trans- lationally symmetric nonimaging devices that transfer flux to a Lambertian target from a source having fixed longitudinal cutoffs parallel to symmetry axis with orthogonal fixed lat- itudinal cutoffs. The latitudinal angular half width of the source is q 0 = 23.45°. The diamond- shaped marker indicates the performance limit for the equal-étendue case. tration limit in this figure never exceeds the 49.30% value of Eq. (10.97). This is because the half width of the skewness distribution of the source is always sin(q 0 ) = 0.3979, independent of the value of the source-to-target étendue ratio. Thus, no matter what value of the source-to-target étendue ratio is used, no étendue can be transferred from the source to the target by a translationally symmetric non- imaging device for skewness values satisfying 0.3979 <|S z |£1. References 263 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 54.5 Source-to-target étendue ratio Efficiency limit Concentration limit Efficiency & concentration (unitless) Figure 10.16 Plot of the efficiency and concentration limits as a function of the source-to- target étendue ratio for translationally symmetric nonimaging devices that transfer flux to a Lambertian target from a source having fixed longitudinal cutoffs parallel to symmetry axis with orthogonal fixed latitudinal cutoffs. The latitudinal angular half width of the source is q 0 = 23.45°. The equal-étendue case corresponds to the crossing point of the two curves. REFERENCES Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer Verlag, New York. Bortz, J., Shatz, N., and Ries, H. (1997). Consequences of étendue and skewness conservation for nonimaging devices with inhomogeneous sources and targets. Proceedings of SPIE, Vol. 3139, 59–75. Bortz, J., Shatz, N., and Winston, R. (2001). Performance limitations of transla- tionally symmetric nonimaging devices. Proceedings of SPIE, Vol. 4446, 201–220. Ries, H., Shatz, N., Bortz, J., and Spirkl, W. (1997). Performance limitations of rotationally symmetric nonimaging devices. J. Opt. Soc. Am. A., Vol. 14, 10, 2855–2862. Shatz, N., and Bortz, J. (1995). An inverse engineering perspective on nonimag- ing optical design. Proceedings of SPIE, Vol. 2538, 136–156. [...]... 20 OSC (97.21%) + Total transmission (%) 100 97 10 OSC (96.36%) + 20 CPC (96.74%) 10 CPC (95.68%) 95 5 10 15 20 25 30 35 Acceptance angle (deg) 40 45 50 Figure 11.4 Total transmission within the acceptance angle as a function of the acceptance angle for the 3D OSC and the 3D CPC 100 Transmission (%) 80 60 40 20 0 0 2 4 6 8 Angle of incidence (deg) 10 12 10 OSC 10 CPC Figure 11.5 Plot of transmission... 150 125 100 75 50 25 0 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Skewness s (mm) Skewness distribution for disk Skewness distribution for sphere Computed skewness dist, all output rays Figure 11.11 Skewness distributions for spherical source, disk target, and all rays transferred to the disk by the 3D OSR 200 175 de/ds (1/mm) 150 125 100 75 50 25 0 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Skewness... dxsrc Truncation Acceptance angle Spacing-control parameter Axial source position shift Units Unitless Degrees Unitless mm 0.00 20.0 0.4 -2.0 0.05 30.0 1.0 2.0 20 10 0 10 20 10 0 10 20 30 40 50 Figure 11.8 3D OSR designed to transfer ux from a 10- mm-diameter sphere to an equalộtendue disk having an acceptance half angle of 30 of symmetry of the OSR Similarly, the coordinates of the last deviation knot... angle of incidence for the 3D OSC and the 3D CPC with a 10 acceptance angle 279 280 Chapter 11 Global Optimization of High-Performance Concentrators 100 Transmission (%) 95 90 85 80 75 0 0.1 0.2 0.3 0.4 0.5 Skew invariant (cm) 0.6 0.7 0.8 0.9 10 OSC 10 OPC Figure 11.6 Transmission versus the skew invariant for the 3D OSC and the 3D CPC with a 10 acceptance angle that the 3D OSC has a sharper transmission... Radial coordinate (cm) 5 0 5 0 5 10 15 20 Axial coordinate (cm) 25 30 35 30 35 OSC profile Disk absorber Figure 11.2 The 3D OSC with a 10 acceptance half angle Difference in radial coordinate (mm) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0 5 10 15 20 Axial coordinate (cm) 25 Figure 11.3 Radial shape difference between the 3D OSC and the 3D CPC as a function of the axial coordinate for a 10 acceptance angle The difference... between 0.9 and 1.1 For the case of a 10 acceptance half angle, the global optimization procedure produced the 3D OSC design depicted in Figure 11.2 The shape difference between the 3D OSC and the 3D CPC is shown in Figure 11.3 Total transmission within the acceptance angle is shown in Figure 11.4 For the 10 case, the 3D OSC transmits 96.36% of the energy into the 10 beam, as compared to 95.68% for the... 75.3% It is apparent from the efciency-versus-concentration curve that an efciency limit of 100 % can only be achieved for concentrations less than or equal to about 40% It can be shown that the exact value of this concentration limit is 4/p 2 100 % ê 40.5% It can also be shown that a concentration limit of 100 % can only be achieved for efciency values less than or equal to exactly 25% As an example,... the upper limits on ux-transfer performance for nonimaging optical systems designed for use with inhomogeneous sources and targets We consider the general case of nonsymmetric optical systems, for which performance limitations due to the skew invariant do not apply The special cases of performance limits for rotationally and translationally symmetric optics were considered in the previous chapter 11.4.1... solution topology, such as {xn } = arg min( f ) R1ặR N (11.37) In our experience, the solution topology for a broad range of nonimaging optical design problems has been found to be multimodal and Lipschitz continuous Consequently, the use of local optimization techniques to accomplish nonimaging optical system design has limited application Over the past two decades, a branch of optimization has emerged,... Global Optimization of High-Performance Concentrators 1.1 1 0.9 Efficiency 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 Projected half angle (degrees) Performance limit 30-deg OSR 30-deg truncated involute CPC 51.76-deg untruncated inv CPC Design projected half angle Figure 11 .10 Efciency versus projected half angle for 3D OSR and two edge-ray designs tion as a function of the targets . are given in Eqs. (10. 71) and (10. 81), respectively. We assume the total surface area of the target is (10. 86) Eq. (10. 71) then allows us to calculate the target étendue: (10. 87) When the source. Eqs. (10. 81) and (10. 87) give the following value for the total surface area of the source: (10. 88) For this equal-étendue case, Figure 10. 11 depicts the skewness distributions of Eqs. (10. 70). and source étendue are given in Eqs. (10. 71) and (10. 85). As in Section 10. 3.6.1, we assume the total surface area of the target is (10. 93) so that the target étendue is (10. 94) When the source and target