Nonimaging Optics Winston Episode 11 pps

the curve C at some point P on C. We also define the circle c that passes through the point P and is centered on the x-axis. The line tangent to c at point P is designated as T¢. The angle between T and T¢ is designated as a. A constant-angle spiral is defined as any continuous curve C for which the angle a is a constant for all points on the curve. We define s to be the distance of a point P on a constant-angle spiral from the vertex of the cone. We consider the differential change ds in the distance s pro- duced by a differential change df in the angle f. The constant-angle criterion leads to following relationship between ds and df: (11.59) The radial coordinate r can be expressed in the form (11.60) where r 0 and s 0 are the r and s-coordinates, respectively, of the starting point on the spiral. Similarly, the axial coordinate x can be expressed as (11.61) where x 0 is the x-coordinate of the starting point on the spiral. Substitution of Eq. (11.60) into Eq. (11.59) gives (11.62) Integrating both sides of Eq. (11.62), we obtain an expression for s as a function of f: (11.63) where f 0 is the value of the angular coordinate f at the starting point of the spiral. Substitution of Eq. (11.63) into Eq. (11.60) gives the r-coordinate as a function of f: (11.64) Substitution of Eq. (11.63) into Eq. (11.61) gives the x-coordinate as a function of f: (11.65) For the special case of b = 0°, Eqs. (11.64) and (11.65) become (11.66) and (11.67) For the special case of b = 90°, Eqs. (11.64) and (11.65) become (11.68) and (11.69) For the special case of b = 180°, Eqs. (11.64) and (11.65) become xxf ( ) = 0 . rref aff () = () - () 0 0 tan xxrfaff ( ) =+ ( ) - ( ) 00 0 tan . rrf () = 0 xx r ef b abff ( ) =+ ( ) - [] () () - () 0 0 0 1 tan . tan sin rref abff ( ) = () () - () 0 0 tan sin . ss r ef b abff ( ) =+ ( ) - [] () () - () 0 0 0 1 sin , tan sin ds rss 00 +- ( ) ( ) = ( ) sin tan . b afd xx ss=+- ( ) ( ) 00 cos ,b rr ss=+- ( ) ( ) 00 sin ,b ds rd f a= ( ) tan . 11.6 Examples of Globally Optimized Concentrator Designs 289 (11.70) and (11.71) We now combine segments of constant-angle spirals to construct continuous curves that are periodic in the variable f. If the number of periods per 360° is the integer N, then the angular period of the curve is (11.72) Each of the N segments of the curve is assumed to consist of two subsegments having a-values which are equal in magnitude but opposite in sign. The angular width of each subsegment is (11.73) We define the two indices m and n, which refer to the m th subsegment of the n th segment, where m = 0 or 1 and n = 0, 1, , N - 1. The central f-value of the m th subsegment of the n th segment is (11.74) It is also convenient to compute the value of m and n corresponding to any par- ticular value of f between 0° and 360°: (11.75) and (11.76) where the function Int(x) is defined as the greatest integer that is less than or equal to x. Substitution of Eqs. (11.75) and (11.76) into Eq. (11.74) allows us to compute the central f-value of the subsegment in which any given f-value lies (11.77) When f 0 in Eqs. (11.64) and (11.65) is replaced by f cent (f), we find that the periodic segmented spiral curve is specified by the formulas (11.78) and (11.79) where the constant angle a has been multiplied by [2m(f) - 1] to produce the desired alternation of the sign of the spiral angle for adjacent subsegments. Similarly, for the special case of b = 0°, we obtain xx r e am cent f b fbfff ( ) =+ ( ) - {} () - []{} () - () [] 0 0 21 1 tan , tan sin rre am cent f fbfff ( ) = () - []{} () - () [] 0 21tan sin ff f f f cent Int ( ) = Ê Ë Á ˆ ¯ ˜ + È Î Í ˘ ˚ ˙ 1 2 2 1 4D D . m Int Intf f f f f ( ) = Ê Ë Á ˆ ¯ ˜ - Ê Ë Á ˆ ¯ ˜ 22 DD , n Intf f f () = Ê Ë ˆ ¯ D ff mn cent nm , .=+ + Ê Ë ˆ ¯ 1 2 1 4 D df f = D 2 . Df = 360deg . N xxrfaff ( ) =- ( ) - ( ) 00 0 tan . rrf () = 0 290 Chapter 11 Global Optimization of High-Performance Concentrators (11.80) and (11.81) For b = 90°, we find that (11.82) and (11.83) For b = 180°, we have (11.84) and (11.85) By converting r 0 , x 0 , a, and b into the continuous functions r 0 (q), x 0 (q), a(q), and b(q) of the parameter q, Eqs. (11.78) through (11.85) then provide us with the desired specification of the entire reflector surface. The specific forms of the functions r 0 (q) and x 0 (q) are equivalent to the radial and axial coordinates for the parameterization scheme described in the previous subsection for rotationally symmetric reflectors. The cone-angle function b(q) is defined as the slope angle with respect to the x-axis of the local surface normal of the parametric curve defined by r 0 (q) and x 0 (q). We employ a finite-dimensional cubic-spline parameter- ization to describe a as a function of q. We now present a design example in which the parameters used to control the shape deviation, truncation, acceptance angle, and axial source position shift were the same 17 parameters used in the previous subsection. In addition, 10 degrees of freedom (knots) were allocated to the description of a as a function of q, for a total of 27 design degrees of freedom in the optimization. The lower and upper parameter ranges for the original 17 variables were the same as used in the pre- vious subsection. The lower and upper parameter ranges for the knots used to define a as a function of q were 0° and 35°, respectively. When the shape devia- tions of the star lobes as a function of f are sufficiently small, the ray tracing can be simplified by modeling the f-dependent shape perturbations purely as slope perturbations imposed on a rotationally symmetric surface. This approximation was utilized in the ray tracing to obtain the results presented below. Global optimization was used to determine the form of 3D OSC that maxi- mizes the flux transferred from a 10-mm-diameter spherical Lambertian source into an emergent conical beam subtending a 30° half angle. The constraint that the source and the target must have equal étendue was enforced, which leads to an aperture-to-source area ratio of 1/sin 2 (30°) = 4. Thus, a target disk diameter of 40mm was used. The reflector surface was assumed to be loss-free and specular. The resulting shape of the 3D OSC design is shown in Figures 11.13 and 11.14. In contrast to the axisymmetric solution discussed in the previous subsection, the source does not protrude behind the back of the reflector, and there exists a small clearance gap between the source and the reflector. The performance of this solu- tion was computed using 200,000 rays, which provides accuracy better than 0.1% in efficiency and concentration. The design produces a value of 84.9% for both flux- xxr m cent faffff ( ) =- ( ) - [] {} - ( ) [] 00 21tan . rrf () = 0 xxf ( ) = 0 . rre m cent f af fff () = () - []{} - () [] 0 21tan xxr m cent faffff ( ) =+ ( ) - [] {} - ( ) [] 00 21tan . rrf () = 0 11.6 Examples of Globally Optimized Concentrator Designs 291 292 Chapter 11 Global Optimization of High-Performance Concentrators -10-5 0 5 1015202530354045505560 x-coordinate (mm) -25 -20 -15 -10 -5 0 5 10 15 20 25 )mm( etanidrooc- y 3D OSC Spherical source Figure 11.13 Shape profile of star concentrator optimized to transfer flux from a sphere to a 30° disk. Figure 11.14 3D cut-away view of star concentrator. transfer efficiency and concentration. The variation of the star-lobe angle along the arc-length of the reflector is depicted in Figure 11.15. The skewness distribu- tions for the sphere, the disk, and the rays output within the required 30° half angle are depicted in Figure 11.16. Note that the output skewness distribution extends over the full skewness range of the disk target’s skewness distribution. The symmetry-breaking star cross section has allowed the original skewness distribution of the source to be transformed into an output skewness distribution 11.6 Examples of Globally Optimized Concentrator Designs 293 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized arc length (unitless) 0 5 10 15 20 25 30 35 )ge d( elgna ebol-ratS Figure 11.15 Star-lobe angle versus normalized arc length along reflector profile for 3D OSC optimized to transfer flux from a sphere to a 30° disk. -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Skewness (mm) 0 20 40 60 80 100 120 140 160 180 200 ) m m/1( sd/Ed Disk Sphere 3D OSC Figure 11.16 Skewness distributions for spherical source, disk target, and flux transferred by 3D OSC to a disk within a 30° half angle. providing a superior match to that of the target. Figure 11.17 depicts the efficiency versus concentration curve for the 3D OSC in comparison with the curve for the truncated 30° involute CPC and the theoretical limiting curve for rotationally symmetrical systems. As previously discussed, the efficiency versus concentration curves for the two concentrators were generated by varying the acceptance half angle of the disk target, while keeping all other source, target, and concentrator characteristics fixed. The large range of concentration values over which the efficiency of the optimized design exceeds the limit for axisymmetric designs is an indication of the robustness of the design relative to variations in either efficiency or concentration from the nominal design point of equal étendue. In Figure 11.18 we compare the far-field intensity profile produced by the 3D OSC with that produced by the truncated 30° involute CPC. It is of interest to note that the 3D OSC provides significantly improved far-field intensity uniformity relative to the involute CPC design. 11.6.4 Nonaxisymmetric Concentrator for a Cylindrical Source and a Disk Target We now consider the problem of designing a 3D OSC that maximizes the flux trans- fer from a homogeneous cylindrical source to a coaxial disk target of equal étendue (Shatz, Bortz, Ries, and Winston, 1997). We choose a height-to-radius ratio H/R = 10, which is typical of an incandescent filament. In this case the skewness mis- match between the source and target is strongly pronounced. The theoretical per- formance limit on efficiency for rotationally symmetric optics is 46.8% for the equal-étendue case, which means that this problem is more strongly affected by skewness mismatch than the case involving the spherical source. We seek a star concentrator solution and state the design problem as one of determining the form of a 3D OSC that maximizes the flux transferred from a cylindrical Lambertian source into an emergent conical beam subtending a 30° half angle. As in the previous subsection, we use a target disk diameter of 40mm. 294 Chapter 11 Global Optimization of High-Performance Concentrators 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Concentration (unitless) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 )sseltinu( ycneiciffE Skewness limit 3D OSC Operating point, 3D OSC Trunc. involute CPC (TICPC) Operating point, TICPC Figure 11.17 Efficiency versus concentration for 3D OSC optimized to transfer flux from a sphere to a disk within a 30° half angle. The equal-étendue requirement, along with the chosen height-to-radius ratio, leads to a source height and radius of 22.36mm and 2.236mm, respectively. The optimization was performed using the same parametrization scheme, number of optimization parameters, and parameter ranges as in the last subsection. The resulting optimized shape of the 3D OSC design is shown in Figures 11.19 and 11.20. Note that the end of the source coincides precisely with the beginning of the reflector. The performance of this design was computed using 200,000 rays, 11.6 Examples of Globally Optimized Concentrator Designs 295 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Emission angle relative to symmetry axis (deg) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 )r s/ W ( ytisnetn i dl e if-r aF 3D OSC Trunc. involute CPC (TICPC) Design half angle Figure 11.18 Far-field intensity versus emission angle for 3D OSC optimized to transfer flux from a sphere to a 30° disk. -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 x-coordinate (mm) -25 -20 -15 -10 -5 0 5 10 15 20 25 )mm( etanidrooc-y 3D OSC Cylindrical source Figure 11.19 Shape profile of star concentrator, optimized to transfer flux from a cylinder to a 30° disk. 296 Chapter 11 Global Optimization of High-Performance Concentrators Figure 11.20 3D cutaway view of star concentrator. 0.00.10.20.30.40.50.60.70.80.91.0 Normalized arc length (unitless) 0 5 10 15 20 25 30 )ged( elgna ebol-ratS Figure 11.21 Star-lobe angle versus normalized arc length along reflector profile, for 3D OSC optimized to transfer flux from a cylinder to a 30° disk. which provided accuracy better than 0.1% in dilution and efficiency. The solution provides a 75.1% value of both efficiency and concentration. The variation of the star-lobe angle along the arc-length of the reflector is depicted in Figure 11.21. As was the case for the 3D OSC with the spherical source, we observe a generally decreasing behavior of the angle as arc length is increased. The skewness dis- tributions for the cylinder, disk target, and flux transferred to the target within the 30° acceptance half angle are depicted in Figure 11.22. As with the 3D OSC for the spherical source, the output skewness distribution extends the full range of the disk’s skewness distribution. Figure 11.23 depicts the efficiency versus 11.6 Examples of Globally Optimized Concentrator Designs 297 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Skewness (mm) 0 50 100 150 200 250 300 ) mm/1( s d /Ed Disk Cylinder 3D OSC Figure 11.22 Skewness distributions for cylindrical source, disk target, and flux trans- ferred by 3D OSC to disk within its 30° acceptance half angle. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Concentration (unitless) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 )sseltinu( ycneiciffE Skewness limit 3D OSC Operating point, 3D OSC Figure 11.23 Efficiency versus concentration for 3D OSC optimized to transfer flux from a cylinder to a 30° disk. concentration curve for the 3D OSC design in comparison with the theoretical limiting curve for rotationally symmetrical systems. The efficiency versus concen- tration curve was generated by computing the efficiency and concentration for different values of the target’s acceptance half angle, while holding all other system characteristics constant. It is of interest to note that the 3D OSC provides efficiency superior to the performance limit for axisymmetric systems over a large range of concentration values. Figure 11.24 depicts the efficiency versus TSER for the 3D OSC in relation to both the étendue limit, which is the upper limit for nonaxisymmetric optics, as well as the skewness limit, which is the upper limit for axisymmetric optics. 11.6.5 Nontracking Solar Concentrator with Broken Translational Symmetry We now consider an example of a globally optimized design that uses micro- structure ridges to overcome the performance limits imposed by translational symmetry (Bortz, Shatz, and Winston, 1997). The problem to be solved is that of designing a north-south-oriented nontracking solar concentrator in a material of unit refractive index, as discussed in Section 10.3.6.2. The latitudinal half angle of the solar radiation is 23.45°. The target is assumed to be a 20-mm-diameter cylindrical tube. We consider the equal-étendue case, with longitudinal half angle (11.86) Since the sun’s angular position changes by 15° every hour, this half angle corre- sponds to a total daily operation interval of 6hr, 40min. We recall that the trans- f 0 50=∞. 298 Chapter 11 Global Optimization of High-Performance Concentrators 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 TSER (unitless) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 )sseltinu( ycne i ciffE Etendue limit Skewness limit 3D OSC Design TSER Figure 11.24 Efficiency versus target-to-source étendue ratio (TSER) for 3D OSC optimized to transfer flux from a cylinder to a 30° disk. [...]... involute CPC is depicted in Fig 11. 26 as a dashed line A ray trace through the baseline concentrator is shown in Fig 11. 27 The rays visible to the right of the aperture in Figs 11. 26 and 11. 27 represent rays that have been rejected by the concentrator As expected from the higher efficiency of the optimized concentrator, fewer rejected rays are visible in Fig 11. 26 than in Fig 11. 27 A threedimensional depiction... concentration outweigh its advantages It is the task of nonimaging optics to mitigate these penalties—that is, to concentrate the solar flux at the least possible “cost” in tracking tolerance, precision of components, and so on In effect the optics of solar flux concentration has usually been nonimaging, ” since the image quality is rarely an issue But what nonimaging optics has come to signify is the systematic... Optica 25, 2826–2830 Sun, Y., Winston, J., O’Gallagher, J., and Snail, K A (2002) Statistical optics and radiance measurement in the diffraction limit Optics Communications 206, 243–251 Walther, A (1973) Radiometry and coherence J Opt Soc Am A 63, 1622–1623 Winston, R., and Litteljohn, R G (1997) Measuring the instrument function of radiometers J Opt Soc Am A 14, 3099–3101 Winston, R., Sun, Y., and Littlejohn,... Born, M., and Wolf, E (1999) Principles of Optics, 7th Ed Cambridge University Press Littlejohn, R G., and Winston, R (1993) Corrections to classical radiometry J Opt Soc Am A 10, 2024–2037 Littlejohn, R G., and Winston, R (1995) Generalized radiance and measurement J Opt Soc Am A 12, 2736–2743 Mandel, L., and Wolf, E (1995) Optical Coherence and Quantum Optics Cambridge University Press Mather, J... Target Transferred Figure 11. 31 Source, target, and transferred skewness distributions for the optimized nontranslationally symmetric concentrator f0 = 90∞ (11. 90) This half angle is representative of dawn-to-dusk operation of the concentrator The computed flux-transfer efficiency and concentration for this second optimized design were found to be hoptim = Coptim = 68.4%, (11. 91) which represents a... (1994) Optical properties of nonimaging concentrators with corrugated reflectors Proceedings of SPIE, Vol 2255, 595–602 Rykowski, R., and Wooley, B (1997) Source modeling for illumination design Proceedings of SPIE, Vol 3130, 204–208 Shatz, N., and Bortz, J (1995) An inverse engineering perspective on nonimaging optical design Proceedings of SPIE, Vol 2538, 136–156 304 Chapter 11 Global Optimization of... formalism in optics, see Littlejohn and Winston, 1995) It is appropriate to associate Q with the signal We then derived an analytical form for the instrument function for a simple radiometer in one space dimension One difficulty in using Eq (12.2) is that it may not be easy to compute the instrument function Although the one-dimensional calculation in Littlejohn and 307 12.3 Measuring Radiance Winston (1995)... hoptim = Coptim = 72.7%, (11. 89) which represents a 47.4% performance improvement relative to the baseline translationally symmetric concentrator The performance of this optimized design is indicated by the square marker on the efficiency versus concentration plot shown in Fig 11. 25 The shape profile of the optimized collector, with and without traced rays, is shown in Fig 11. 26 For comparison, the profile... shown in Fig 11. 28 To illustrate the microstructure geometry, the relative size of the symmetry-breaking ridges has been magnified by a large factor in this figure The tilt angle of the 300 Chapter 11 Global Optimization of High-Performance Concentrators 1.2 Efficiency (unitless) 1 Optimized concentrator 0.8 0.6 Baseline 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Concentration (unitless) 1 1.2 Figure 11. 25 Plot of... thought of as a form of diffuser As such, it has the effect of intro- 301 11. 6 Examples of Globally Optimized Concentrator Designs y-coordinate (mm) 100 50 0 –50 –100 0 50 100 150 x-coordinate (mm) 200 250 300 Figure 11. 27 Shape profile of the baseline translationally symmetric concentrator (involute CPC), with traced rays Figure 11. 28 Three-dimensional depiction of the optimized non-translationally-symmetric . f: (11. 64) Substitution of Eq. (11. 63) into Eq. (11. 61) gives the x-coordinate as a function of f: (11. 65) For the special case of b = 0°, Eqs. (11. 64) and (11. 65) become (11. 66) and (11. 67) For the special. . rrf () = 0 290 Chapter 11 Global Optimization of High-Performance Concentrators (11. 80) and (11. 81) For b = 90°, we find that (11. 82) and (11. 83) For b = 180°, we have (11. 84) and (11. 85) By converting. Substitution of Eqs. (11. 75) and (11. 76) into Eq. (11. 74) allows us to compute the central f-value of the subsegment in which any given f-value lies (11. 77) When f 0 in Eqs. (11. 64) and (11. 65) is replaced