Nonimaging Optics Winston Episode 7 docx

y component of P 2 ¥ T 2 > 0. Estimate an intermediate point P 12 = (P 1 + P 2 )/2 and its normal N 12 = (N 1 + N 2 )/2. 3. Carry out the 3D ray-tracing on P 12 , associating it with the area of refractive surface between P 1 and P 2 with the direction of impingement U(q 2 ) to obtain the value DA ap12 and thus calculate A ap2 (q 2 ) = A ap1 (q 2 ) +DA ap12 , estimated angular response for the angle q 2 of the portion of lens up to P 2 . 4. Repeat the steps from 2 to 3, iterating on the value of the parameter s (note that DA ap12 increases with s), until obtaining that |1 - A ap (q 2 )/A ap2 (q 2 )|< e, e being a preset margin of error. 5. Carry out the 3D ray-tracing on the point P 12 resulting from the iteration 4, associating it with the area of the refractive surface between P 1 and the point P 2 resulting from the iteration 4, in order to obtain the function DA ap12 (q) and thus calculate A ap2 (q) = A ap1 (q) +DA ap12 (q), estimated angular response of the portion of lens up to P 2 . 6. Increase the value q 3 = q 2 +Dq and repeat the steps from 2 to 5, increasing the subindices by one unit, until for a given angle q n the coordinate z of point P n is negative. 7. Repeat the steps 1–6, iterating on the abscissa of point P 1 until |1 - q n /q MAX |< e¢, e¢ being another preset margin of error. The design is finished. The refractive surface is defined by the set of points cal- culated in the process. If required, it is possible to fit these points by a spline or a polynomial curve, which facilitates handling of the data. The design guarantees that the prescription is adjusted in the whole range 0 < q < q M , but the stepped transition to zero at q = q M is not (i.e., the prescription for q > q M cannot be adjusted) because there are no degrees of freedom to make the outer portion of the lens perform as a Cartesian oval (as done in Section 7.4.2 for the linear case). However, it should be emphasized that the design procedure uses rays imping- ing on the receiver from nearly all possible directions (the whole field of view of the photodiode is covered). This situation is close to optimum in terms of maxi- mizing sensitivity—that is, making the constant A 0 (and k¢) as large as possible. The optimum is equivalent to get isotropic illumination of all the points of the receiver with rays from the specified range q MIN < q < q MAX . In the case that the active surface is not flat, as is common in the case of the surface of LED or IRED emitters, the procedure described is applicable simply by considering the corresponding geometry of the active surface for the ray tracings and directing the refracted ray tangent to that surface (as a generalized concept of the point R) for the calculation of the normal at P k . The method can be easily generalized to include preset rotational sequential surfaces (either refractive or reflective), which deflect the ray trajectories. This is the case shown in Figure 7.9 shows the cross section of a lens designed for a cir- cular photodiode active area of silicon without antireflection coating. The lens has n = 1.49, and an encapsulating material of n¢=1.56 is assumed. The surface separating both media is preset to a sphere. The prescribed A ap (q) function is the linear function of Eq. (7.19). The actual function A ap (q) function is finally calculated by ray tracing (includ- ing Fresnel losses at the air-lens and lens-silicon interfaces) is shown along with the specifications in Figure 7.10a. The procedure, if applied to other angular sen- 7.5 The Finite Disk Source with Rotational Optics 169 sitivity functions as A ap (q) = 1/cos(q) inside for 0 < q < 80° (and null outside that range) lead to another lens (whose profile is not shown here) that also produces the specified sensitivity accurately, as shown in Figure 7.10b. Figure 7.11 shows the results from M. Hernández (2003) of ray trace on several lenses designed for a linear prescribed relative sensitivity with different values of the z-coordinate of the on-axis point of the lens V (obtained from different selec- tions of the z-coordinate of the point A 1 ). All show very good agreement with the linear prescription (perhaps except near q = 0, especially for small V z values). The very noticeable difference is that the smaller V z value, the smoother the transition of the sensitivity at q = q M = 60°. Note that, since by étendue conservation all curves in Figure 7.11 fulfill the integral condition (7.20) the useless sensitivity for q > 60° makes that the constant A 0 in Eq. (7.19) increases with V z . Ad nA ap qqq q p () () = = Ú sin 0 2 2 2 170 Chapter 7 Concentrators for Prescribed Irradiance 02468-2-4-6-8 0 2 4 6 8 y(mm) x(mm) Lens profile Receiver (diammeter 3 mm) Preset spherical profile Figure 7.9 Cross-section of a lens designed to get a linear angular sensitivity function in the range 0 £ q £ 60°. (lens refractive index n = 1.49; encapsulant refractive index n¢=1.56) A ap (q)/A A ap (q)/A q (degs) ±5% of prescription 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 (a) q (degs) 20 40 60 80 100 1 0 2 3 4 (b) Figure 7.10 (a) Angular sensitivity A ap (q) of the lens of Figure 7.9 (bold line) obtained by ray tracing and specified curve and 5% tolerance curves (dashed lines). (b) Angular sensi- tivity A ap (q) of the another lens (profile not shown here) for producing 1/cos(q) dependence for 0 < q < 80°. In order to get the maximum possible value of the constant A 0 , the stepped transition to zero at q = q M is needed (i.e., the null prescription for q > q M must also be adjusted). This cannot be done with a single sequential optical surface, as already discussed, but it is possible if two surfaces are used. This is done with the SMS design method presented in Chapter 8. Two complete surfaces are not needed to solve this design problem. For both refractive surfaces, one possible design is indicated in the next steps, which will refer to the points indicated in Figure 7.12: 1. Preset surface S Q from the center. The last point Q T 2 of the present portion of surface S Q will be calculated in step 3. 2. Apply the procedure just described to achieve the prescribed intensity for the calculation of refractive surface S P through the present surface S Q up to point P T , which is the point such that the ray r¢ traced (inversely) from R¢ passing through P T (after the refraction on S R at point Q T1 ) exits the lens toward direc- tion q = q M . 3. Calculate the point Q T2 as the point of S Q on which the ray r from R is refracted toward P T . Note that, up to this point, the intensity prescription has been designed for 0 < q < q T , which is the exit direction of ray r. 7.5 The Finite Disk Source with Rotational Optics 171 0 0.5 1 1.5 2 2.5 3 0 102030405060708090 Vz = 7.5 mm Vz = 6.0 mm Vz = 5.0 mm Vz = 3.0 mm q [grados] A ap (q)/A ph Figure 7.11 Effect of the lens size in the optical performance for the linear prescription of rotational lenses (receiver diameter D = 3mm). R R¢ P T x z Q T 2 S P S R Q T 1 r¢ r q T q M Figure 7.12 The null prescription for q > q M can be achieved if two surfaces are used (see SMS method, Chapter 8). This is the condition for maximum absolute sensitivity. 4. Calculate a new portion of surface S P as the Cartesian oval that makes that the rays r¢ traced (inversely) from R¢ and refracted at S Q at the portion between Q T1 and Q T2 are refracted on the new points of S P toward direction q = q M . 5. Apply the procedure just described to achieve the prescribed intensity for the calculation of refractive surface S Q through the already known portion of surface S P calculated in step 4. 6. Repeat steps 4 and 5 up to convergence onto the line R–R¢. 7.5.1 Comparison with Point Source Designs The point source approximation of Section 7.3 can also be applied to the design problem of a refractive sequential rotational surface for prescribed sensitivity. A comparison shows how important the finite dimension of the source is in a specific example (Hernández, 2003). The comparison of the performance for different lens sizes designed with the point source approximation but ray-traced with the receiver of diameter D = 3mm is shown in Figure 7.13. The linear prescription is well achieved only for large sizes (V Z = 10D). For the size of this lens with practical interest, which is about V z = 3mm, the point size model leads to a lens profile that performs far from the specification, in contrast with the result for V z = 3mm already presented in Figure 7.11. 7.6 THE FINITE TUBULAR SOURCE WITH CYLINDRICAL OPTICS Another particularly useful case is producing a constant irradiance on a distant plane from a cylindrical source of uniform brightness, such as a Lambertian source. As already mentioned, this was worked out by Ries and Winston (1994). In fact, Ong, Gordon, and Rabl (1996) showed that there are four basic types of 172 Chapter 7 Concentrators for Prescribed Irradiance 0 0.5 1 1.5 2 2.5 3 0 10203040506070 Vz 3 mm Vz 5 mm Vz 30 mm Especificación f A ap ( q )/A ph Figure 7.13 Effect of the lens size on the optical performance for the linear prescription of the design obtained with the point source approximation (receiver diameter D = 3mm). solutions for this type of problem. Two classes derive from the fact that the reflec- tor curve can be diverging or converging—that is, the caustics formed can fall behind or in front of the reflector. These types have been referred to as compound hyperbolic concentrator (CHC) or compound elliptical concentrator (CEC). The pos- sibilities are then doubled because the design can be done with the near edge or far edge of the source being always illuminated. The interested reader can find further information in the cited reference. 7.7 FREEFORM OPTICAL DESIGNS FOR POINT SOURCES IN 3D Freeform (without any prescribed symmetry) designs in 3D are not a simple exten- sion from the 2D case. These designs become much more difficult, and conse- quently, they are less developed than their 2D equivalents. In this section we examine overview 3D freeform design methods for point sources—that is, methods that use the point source approximation. This means that the designs will perform as the theory foresees if the optical surfaces are far enough from the source (in terms of source diameter) so it can be considered as a point. At present only one method, which is currently being developed, is able to manage extended sources in 3D geometry. This method is the extension to 3D of the SMS method of Chapter 8 (Benítez et al., 2003). A basic problem in illumination design is that of designing a single surface (reflective or refractive) that transforms a spherical wave front (point source) with a given intensity pattern into an output wave front with a prescribed intensity pattern. Variations of this basic problem are to have a prescribed irradiance pattern at a given surface instead of the output intensity pattern or to have a plane wave front at the input instead of the spherical one. The basic equation governing the solution of this problem is a second order nonlinear partial differential equation of Monge-Ampere type. This was found in 1941 by Komissarov and Boldyrev (1994). Schruben (1972) created the equation governing the design of a luminary reflector that provides a prescribed irradiance pattern on a given plane when the reflector is illuminated by a nonisotropic punc- tual source. During the 1980s and 1990s a strong development of the method was encour- aged by reflector antennas designers. Wescott, Galindo, Graham, Zaporozhets, Mitra, Jervase, and (see References) others contributed to this field of antenna reflector design. The method starts with a procedure purely based in Geometrical Optics. This is the part in which we are more interested for illumination applica- tions. After the Geometrical Optics design, a Physical Optics analysis and syn- thesis procedure is necessary for a fine-tuning of the design. At present there is commercial software for designing these antenna reflectors based on this method (see, for instance, http://www.ticra.dk/). The method is particularly useful for satellite applications. Satellite reflector antennas must provide a given far-field (or intensity) pattern to fit, for instance, a continent contour, in satellite-to-earth broadcasting applications. And this should be done efficiently. In this case a single-shaped reflector is enough to solve the problem. The requirement is equivalent to saying that the amplitude of the field at the aperture is prescribed. In other cases it is required to achieve a prescribed 7.7 Freeform Optical Designs for Point Sources in 3D 173 irradiance pattern at the antenna aperture (in general, this is required to reduce the side-lobes emissions) besides the specified far-field pattern. In these cases, two shaped reflectors are enough to solve the problem, and not only the output ampli- tude is controlled, but also the phase distribution at this aperture. This second problem is very similar to the first, although it may look different. The single reflect or designs for satellite applications do not differ strongly from a parabola shape because the desired intensity pattern is highly collimated in general. This fact has allowed developing several approximate methods to solve the Monge Ampere that worked well within these conditions. Beginning in the 1980s until the present, the subject has been of interest to mathematicians like Oliker, Caffarelli, Kochengin, Guan, Glimm, and Newman (see References). Conditions of existence and uniqueness of the solutions have been found as well as new design procedures have been proposed. For instance, Glimm and Oliker (2003) have shown recently that the problem can also be solved as a variational problem in the framework of a Monge-Kantorovich mass transfer problem, which allows solving the problem numerically by techniques from linear programming. The designs are not limited to reflectors but extend also to refrac- tive surfaces. Already in the present decade, the subject has come back to the illu- mination field by Ries and Muschaweck (2002). In this reference, multigrid numerical techniques are efficiently used to solve the Monge-Ampere equation. The solutions are classified into four types depending on the location of the centers of curvature of the output wavefronts to design: In two of these types, the surfaces of curvature centers (each one corresponding to one of the two families of curva- ture lines) are at one “side” of the optical surface, whereas in the remaining types the surfaces of curvature centers are at both sides of the optical surface. 7.7.1 Formulation of the Problem We shall restrict the explanations to the problem of designing a single optical surface (reflective or refractive) that transforms a given intensity pattern of the source into another prescribed intensity pattern (Minˆano and Benítez, 2002). Let rˆ be a unit vector characterizing an emitting direction of the source. This unit vector can be determined with two parameters, u and v. These two parameters can be, for instance, the two angular coordinates (q, f) of the spherical coordinates. In this case, rˆ is given by (see Figure 7.14) (7.21) ˆ cos sin , sin sin , cosr = () fq fq q 174 Chapter 7 Concentrators for Prescribed Irradiance x y z f q r a b s Figure 7.14 Definition of the unit vectors rˆ and s. Let the unit vector sˆ define an outgoing direction of the rays after deflection on the optical surface. Using spherical coordinates (a, b), sˆ can be written as (7.22) Because rˆ and sˆ are unit vectors, then (7.23) Nevertheless rˆ u is not necessarily normal to rˆ v and s u is not necessarily normal to s v . The differential of solid angle dW r subtended by the rays in a differential dudv can be written as (7.24) The second equality of Eq. (7.24) assumes that we have chosen the parameters u and v such that the vector rˆ u ¥ rˆ v points in the same direction as rˆ. A similar equa- tion applies for the vector sˆ and the solid angle dW s . 7.7.2 Basic Equations 7.7.2.1 Laws of Reflection and Refraction According to Herzberger (1958), if we have two surfaces defined by the vectors a ළ and a ළ ¢ that are crossed by a one-parameter beam of rays and such that u is the parameter, we have (7.25) where the unit vectors sˆ¢ and sˆ are pointing in the ray directions at each one of the surfaces, E is the optical path length from the surface defined by a ළ ¢ to the surface defined by a ළ , and n¢, n are, respectively, the refractive indices at each one of the surfaces. We can obtain both the equation of reflection and the equation from Eq. (7.25). Assume that the one-parameter bundle of rays is passing through the coordi- nate origin. The surface defined by the vector a ළ ¢ is just a point and thus a ළ ¢ u = 0. Let r ළ be a vector defining a reflective surface. r ළ is the vector a ළ of Eq. (7.25). Now consider a two-parameter bundle of rays passing through the coordinate origin. The two parameters are u and v. Then, application of Eq. (7.25) gives (see Figure 7.15) asnasn E uu u ◊- ¢ ◊ ¢¢ = ˆˆ d r r dudv r r rdudv ruv uv W= ¥ = ¥ () ◊ ˆˆ ˆˆˆ ˆˆˆ ˆˆ ˆˆ ˆˆˆ ˆˆˆˆ rrr ffrr sss ssss uv uv 2 2 10 10 =◊= fi◊ =◊ = =◊= fi◊ =◊ = ˆ cos sin , sin sin , coss = () ab ab b 7.7 Freeform Optical Designs for Point Sources in 3D 175 x y z u = constant v = constant s r r u r v Figure 7.15 Vector rˆ is impinging on the reflector where it is reflected as vector sˆ. (7.26) where it has been taken into account that E u = r u ; r is the modulus of r ළ —that is, r = r ළ . Eq. (7.27) is derived from this definition of the modulus. (7.27) Combining Eqs. (7.26) and (7.27) we get the reflection law in the form that we are going to use (note that the vectors rˆ u and rˆ v are not a unit vectors) (7.28) We can apply Eq. (7.25) to a refractive surface to obtain the refraction law in a similar way as we got the reflection law. The result is (7.29) that is, for our purposes, both laws can be summarized in Eq. (7.29) taking n = 1 in the case of reflection. 7.7.2.2 Power Conservation Let E(sˆ) be the desired output radiant intensity (for instance, in Watt/stereora- dian), and let I(rˆ) be the intensity emitted by the source. Energy conservation can be written as (7.30) Expressing the vectors rˆysˆ as functions of the two parameters (u, v), then we have that (7.31) Eq. (7.31) is the form that we will use for the energy conservation. Using Eq. (7.24), Eq. (7.31) can be written as (7.32) Where the sign ± takes into account that the trihedron sˆ - sˆ u - sˆ v may have two possible orientations. We have chosen rˆ, rˆ u , rˆ v to be in the positive orientation (rˆ·rˆ u ¥ rˆ v > 0), but we don’t have the freedom to choose the orientation of sˆ - sˆ u - sˆ v . 7.7.2.3 Malus-Dupin Theorem The dependence of sˆ with (u, v) is not totally free. This is due to the Malus-Dupin theorem, which states that a normal congruence remains like this after being deflected by a mirror or a lens surface. For our particular case (a single reflective or refractive surface and a punctual source) the Malus-Dupin theorem is nothing else than the equality of the crossed derivatives of the function describing the optical surface—that is, r uv = r vu (see Eq. (7.26)). (7.33) rs rs uv vu ◊-◊= ˆˆ 0 Es s s s Ir r r r uv uv ˆˆ ˆ ˆ ˆˆ ˆ ˆ () ¥ () ◊=± () ¥ () ◊ Ess s Ir r r uv uv ˆˆ ˆ ˆˆ ˆ () ¥= () ¥ Esd Ird r s ˆˆ ( ) = ( ) WW r r rs nrs r r rs nrs uu vv = ◊ -◊ = ◊ -◊ ˆˆ ˆˆ ˆˆ ˆˆ r r rs rs r r rs rs uu vv = ◊ -◊ = ◊ -◊ ˆˆ ˆˆ ˆˆ ˆˆ 11 rrrr rrrr r rrrr uu uvv v ==+ =+ ˆˆˆ ˆˆ rs r rs r uu vv ◊= ◊= ˆ ˆˆ 176 Chapter 7 Concentrators for Prescribed Irradiance which can also be written as (7.34) 7.7.3 Mathematical Statement of the Problem Eqs. (7.29), (7.32), and (7.34) form a system of equations with unknown vari- ables r(u, v) and sˆ(u, v). Variable r can be eliminated with Eqs. (7.29) and (7.34), resulting (7.35) The system is now formed by Eqs. (7.35) and (7.32), where the unknown function is sˆ(u, v)—that is, we have to find a mapping of the unit sphere into itself satisfy- ing Eqs. (7.35) and (7.32). We can take (q, f) in Eq. (7.21) as the parameters u, v and take a(u, v), b(u, v) (Eq. (7.22)) as the unknown functions of this equation system. Eliminating sˆ and its derivatives from the equation system Eqs. (7.29), (7.32), and (7.41) leads to a single, second order partial differential equation of the Monge Ampere type, which can be found, for instance, in Schruben (1972). In this case the unknown is the function r(u, v). 7.7.4 Dual Optical Surfaces The previous development allows us to introduce easily the concept of dual optical surfaces (Miñano and Benítez, 2002). As seen in the previous section, the mathe- matical problem can be summarized in (7.36) assume that this equation system is solved—that we know the function sˆ(u, v) sat- isfying Eq. (7.36) with the contour conditions. The calculation of the optical surface can be done with Eq. (7.28)—that is, by integration of r(u, v). Note that the system of Eq. (7.36) is the same that we would have if (a) rˆ is the output unit vector. (b) sˆ is a unit vector departing from the source. (c) I(rˆ) is the required intensity distribution and E(sˆ) is the source intensity distribution. In this case, the optical surface would be given by the function s(u, v) by means of (7.37) Assume that we have two functions sˆ(u, v) yrˆ(u, v) satisfying the system of Eq. (7.36), then we have two functions r(u, v) ys(u, v) fulfilling Eq. (7.28) y Eq. (7.37) and generating two optical systems that we call duals. Functions r(u, v) and s(u, v) fulfill (7.38) and so —-◊ () =- ◊+ ◊ -◊ ◊+ ◊ -◊ Ê Ë ˆ ¯ =-— () uv uuvv uv nrs rssr nrs rs sr nrs rs , , ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ln lnM s s sr nrs s s sr nrs uu vv = ◊ -◊ = ◊ -◊ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ rr ss rr ss nrs rs Es s s s Ir r r r uv vuuvvu uv uv ¥ () ◊¥ () -¥ () ◊¥ () =◊-◊ () () ¥ () ◊=± () ¥ () ◊ rr ss rr ss nrs rs uv vuuvvu ¥ ( ) ◊¥ ( ) -¥ ( ) ◊¥ ( ) =◊-◊ ( ) ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ rr s rr s rr s rr s uv uvvu vu ˆˆ ˆ ˆ ˆˆ ˆ ˆ ◊+ ◊- ◊- ◊ =0 7.7 Freeform Optical Designs for Point Sources in 3D 177 (7.39) If one of the systems is known, the other can be easily calculated with Eq. (7.39). One of the systems produces a pattern E(sˆ) when the point source radiates as given by I(rˆ), and the other (dual) system produces the pattern I(rˆ) when the source is radiating as E(sˆ). REFERENCES Aoki, K., Miyahara, N., Makino, S., Urasaki, S., and Katagi, T. (1999). Design method for offset shaped dual-reflector antennas with an elliptical aperture of low cross-polarisation characteristics. IEE Proc. Microw. Antennas Propag. 146, 60–64. Benítez, P., Miñano, J. C., Blen, J., Mohedano, R., Chaves, J., Dross, O., Hernán- dez, M., and Falicoff, W. (2004). “Simultaneous multiple surface optical design method in three demensions”, Opt. Eng., vol. 43, no. 7. Benítez, P., Miñano, J. C., Hernández, M., Hirohashi, K., Toguchi, S., and Sakai, M. (2000). “Novel nonimaging lens for photodiode receivers with a prescribed angular response and maximum integrated sensitivity”, Optical Wireless Com- munications III, Eric J. Korevaar, Editor Vol. 4214 pp. 94–103. Boston MA. Boldyrev, N. G. (1932). About calculation of Asymmetrical Specular reflectors. Svetotekhnika 7, 7–8. Brown, K. A., and Prata, A. (1994). A design procedure for classical offset dual reflector antennas with circular apertures. IEEE Trans. on Antennas and Propagation, Vol. 42, 8, 1145–1153. Caffarelli, L., Kochengin, S., and Oliker, V. I. (1999). On the numerical solution of the problem of reflector design with given far-field scattering data. Contem- porary Mathematics, Vol. 226, 13–32. Elmer, W. B. (1980). The Optical Design of Reflectors, 2nd ed. Wiley, New York. Galindo, V. (1964). Design of dual-reflector antennas with arbitrary phase and amplitude distributions. IEEE Trans. Antennas Propagat, 403–408. Galindo Israel, V., Imbriale, W. A., and Mittra, R. (1987). On the theory of the synthesis of single and dual offset shaped reflector antennas. IEEE Trans. Antennas Propagat., Vol. AP-35, 887–896. Galindo-Israel, V., Imbriale, W. A., Mittra, R., and Shogen, K. (1991). IEEE Trans. Antennas Propagat., Vol. 39, 620–626. Glimm, T., and Oliker, V. I. (2003). Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem. Journal of Mathematical Sci- ences, Vol. 117, 3, 4096–4108. Hernández, M. (2003). PhD. dissertation, UPM, Madrid. Herzberger, M. (1958). Modern Geometrical Optics. Interscience, New York. Jervase, J. A., Mittra, R., Galindo-Israel, V., and Imbriale, W. (1989). Interpola- tion solutions for the problem of synthesis of dual-shaped offset reflector antennas. Microwave Opt. Technol. Lett., Vol. 2, 43–47. Kildal, P. S. (1984). Comments on “Synthesis of offset dual shaped subreflector antennas for control of cassegrain aperture distributions.” IEEE Trans. Anten- nas Propagat., Vol. AP-32, 1142–1145. nrsrs-◊ () = ˆˆ const 178 Chapter 7 Concentrators for Prescribed Irradiance [...]... zR 0.5 1 3 5 7 28,880 94.43 98. 67 0.05 0.2 57 7,221 94.62 95.64 0.15 0.2 87 802.9 90.22 92.38 0.65 0.313 289.5 90.03 93.9 1.0 0.356 148 80 .74 88.18 1.5 0.348 13.56 35 -5.42 4.56 8.5 10.08 2. 57 4 5.88 2. 57 3 5.91 2.33 3 3 .77 negative when qa is small, meaning that the dielectric and the receiver are fully inside the volume enclosed by the mirror surface and the plane of the entry aperture 8 .7 THE RX CONCENTRATOR... Vol 29, 14, 1 275 –1 276 Winston, R., and Ries, H (1993) Nonimaging reflectors as functionals of the desired irradiance JOSA A, Vol 10(9), 1902–1908 Xu-Jia Wang (1996) On the design of reflector antenna Inverse Problems 12(2), 351– 375 8 SIMULTANEOUS MULTIPLE SURFACE DESIGN METHOD 8.1 INTRODUCTION In this chapter we examine the Simultaneous Multiple Surface (SMS) (U.S Letters Patent, 6,639 ,73 3, “High Efficiency... Publications, Vol 57, 155–169 Oliker, V I (2003) Mathematical aspects of design of beam shaping surfaces in geometrical optics In Trends in Nonlinear Analysis (Kirkilionis, M., Kromker, S., Rannacher, R., and Tomi, F., eds.) Springer-Verlag, New York, pp 192–224 Ong, P T., Gordon, J M., and Rabl, A (1996) Tailored edge-ray designs for illumination with tubular sources Applied Optics 35, 4361–4 371 Pengfei Guan... 10 20 3,600 97. 5 0.04 101.5 1.161 20 32.53 72 .35 Geometrical concentration, Cg3D Total transmission, T(qa) (%) Cut-off angular spread Dq (degrees) Thickness at the center Length/entry aperture diameter, f Exit aperture radius, Ro Exit aperture to receiver distance zR - zX Entry aperture to receiver distance zR - zN 2 225 98.0 0.065 25.65 1.16 5 8.09 17. 95 36 96.9 0.5 9.5 1.161 2.5 3.98 7. 84 12.25 96.6... antennas Electronic Letters, Vol 29, 20, 173 5– 173 7 Westcott, B S., and Zaporozhets, A A (1994) Single reflector synthesis using an analytical gradient procedure Electronics Letters, Vol 30, 18, 1462–1463 Westcott, B S., and Zaporozhets, A A (1995) Dual-reflector synthesis based on analytical gradient-iteration procedures IEE Proc., Vol 142, 2, 129–135 180 Chapter 7 Concentrators for Prescribed Irradiance... Problems 14, 661– 678 Komissarov, V D (1941) The foundations of calculating specular prismatic fittings Trudy VEI 43, 6–61 Lee, J J., Parad, L I., and Chu, R S (1 979 ) A shaped offset-fed dual reflector antenna IEEE Trans Antennas Propagat., Vol 27, 2, 165– 171 Miñano, J C., and Benítez, P (2002) Design of reflectors and dioptrics for prescribed intensity and irradiance pattern Light Prescrptions LLC, internal... (1998) On a Monge-Ampere equation arising in geometrical optics J Differential Geometry 48, 205–223 Ries, H., and Muschaweck, J (2002) Tailored freeform optical surfaces J Opt Soc Am A 19, 590–595 Ries, H., and Winston, R (1994) Tailored edge-ray reflectors for illumination J Opt Soc Am A., Vol 11, 4 1260–1264 Rubiños-López, J O., and García-Pino, A (19 97) A ray-by-ray algorithm for shaping dual-offset reflector... parameters), whereas the regions ∂Mi and ∂Mo are one-parametric: This reduction of the number of parameters clearly simplifies the design problem 8.3 DESIGN OF A NONIMAGING LENS: THE RR CONCENTRATOR The simplest example to start with is the design of a nonimaging lens, also called RR concentrator Figure 8.3 shows an example of these lenses The source extends from S to S¢ and the receiver from R to R¢ The... call these surfaces generalized Cartesian ovals (see, for instance, Luneburg, 1964, and Stavroudis, 1 972 ) The problem of determining a generalized Cartesian can be solved simply requiring the constant path length between the incident and the emergent wavefront (see Wolf, 1948, and Wolf and Preddy, 19 47, for an example of a generalized Cartesian Oval of refraction) A Cartesian oval problem is that of... N X Y M R P x Q z rc rh R' S' Figure 8.6 The remaining points of the lens are obtained with the point-by-point method, departing from the Cartesian ovals NM and XY 8.3 Design of a Nonimaging Lens: The RR Concentrator 1 87 jectory of rg inside the lens can be easily calculated (with the refraction law), since the profile MN is known Then, point P, where the ray rg leaves the lens, can be calculated by . r uu vv ◊= ◊= ˆ ˆˆ 176 Chapter 7 Concentrators for Prescribed Irradiance which can also be written as (7. 34) 7. 7.3 Mathematical Statement of the Problem Eqs. (7. 29), (7. 32), and (7. 34) form a system. sˆ. (7. 26) where it has been taken into account that E u = r u ; r is the modulus of r ළ —that is, r = r ළ . Eq. (7. 27) is derived from this definition of the modulus. (7. 27) Combining Eqs. (7. 26). of ray r. 7. 5 The Finite Disk Source with Rotational Optics 171 0 0.5 1 1.5 2 2.5 3 0 10203040506 070 8090 Vz = 7. 5 mm Vz = 6.0 mm Vz = 5.0 mm Vz = 3.0 mm q [grados] A ap (q)/A ph Figure 7. 11 Effect