Geometrical optics and blackbody radiation Pablo BenÍTez ab ; Roland Winston a ;Juan C. Miñano b a

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1 This article was downloaded by: [University of California, Merced] On: 6 May 2010 Access details: Access Details: [subscription number ] ublisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Modern Optics ublication details, including instructions for authors and subscription information: Geometrical optics and blackbody radiation ablo BenÍTez ab ; Roland Winston a ;Juan C. Miñano b a School of Natural Sciences and School of Engineering, University of California, Merced, CA 95344, USA b CEDINT, Universidad olitécnica de Madrid, ETSI Telecomunicación, C. Universitaria, Madrid, Spain First published on: 22 September 2007 To cite this Article BenÍTez, ablo, Winston, Roland andmiñano, Juan C.(2008) 'Geometrical optics and blackbody radiation', Journal of Modern Optics, 55: 1, , First published on: 22 September 2007 (ifirst) To link to this Article: DOI: / URL: LEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Modern Optics Vol. 55, No. 1, 10 January 2008, Geometrical optics and blackbody radiation ABLO BENI TEZ*yz, ROLAND WINSTONy and JUAN C. MIN ANOz yschool of Natural Sciences and School of Engineering, University of California, Merced, CA 95344, USA zcedint, Universidad olite cnica de Madrid, ETSI Telecomunicacio n, C. Universitaria, Madrid, Spain (Received 30 November 2006; in final form 23 February 2007) Contrary to the customary way, some of the geometrical optics principles in free-space and two dimensions is deduced from blackbody radiation transfer. This is obtained from three hypotheses: (1) a blackbody radiates power proportional to its perimeter, and a convex blackbody does not radiate to itself, (2) reciprocity in radiation transfer, and (3) energy conservation. The analysis of an abstract relay element will lead to deduction of the reflection law in an unconventional form and new properties of the elliptical mirror as an ideal radiation transfer device. It is well known that the transfer of radiation between blackbodies can be modelled in good approximation with geometrical optics. This is deduced in statistical optics from the wave equation with the condition that the plane waves in which the field can be decomposed are equally weighted and completely uncorrelated (in k-space) [1] and the linear dimensions of the blackbody cavity aperture are much larger than the wavelength. In this paper we will proceed in the reverse way: from three fundamental rules of radiation transfer, we will deduce the fundamental results of geometrical optics in two-dimensional free space. The radiation transfer will occur in a bounded region of the plane as shown in figure 1. The boundary will be formed by line segments. Each line segment in that figure will be a blackbody. We will then define a blackbody as a line segment that will emit radiation in equilibrium according to the following rules. (1) Blackbody B radiates a power _q proportional to its perimeter L, and a convex blackbody does not radiate to itself. (2) The radiation transferred from blackbody B 1 to blackbody, _q B1, must be equal to the radiation transferred from to blackbody B 1, _q B2 B 1 (Reciprocity rinciple). *Corresponding author. pbenitez@etsit.upm.es Journal of Modern Optics ISSN print/issn online # 2008 Taylor & Francis DOI: /

3 100. Benı tez et al. (3) The summation of the power radiated by B to the rest of surrounding black bodies equals the total radiation _q emitted by B (Energy Conservation). From the above hypothesis and with an enclosure composed of two blackbodies, follows the well-known result that the power that a non-convex body radiates to a convex surrounding is proportional to its convex hull. However, the calculation of the radiation transfer between the three and four convex blackbodies in figure 1 only from the hypothesis above is less known and was first referenced by Hottel [2]. The view factors F ij, defined as the fraction of the power radiated by B i that reaches B j, for the three blackbodies in figure 1, are given: F ii ¼ 0, F ij ¼ L i þ L j L k, i 6¼ j, k 6¼ i, k 6¼ j: ð1þ 2L i For the case of four blackbodies in figure 1, also F ii ¼ 0, and writing the view factor F ij for i ¼ 1 and j ¼ 2: F 12 ¼ ½0 Šþ½ 0 Š L 3 L 4, ð2þ 2L 1 where the brackets denote the length of the tense string attached to the two points (then L 1 ½ 0 Š, L 3 ½ 0 0 Š, L 4 ½Š). Equation (2) is usually referred to as the Hottel formula [2]. Let us now consider a new element in our enclosure, labelled as r in figure 2. This element r is not a blackbody, and we will axiomatically set it as an element that B 1 B 1 Figure 1. Radiation transfer between three blackbodies. Radiation transfer between four blackbodies. B 4 r 2 B 1 B 1 r r 1 Figure 2. Relay element r is defined as an element that couples the radiation it receives from adjacent blackbodies B 1 and. Relay elements r 1 and r 2 provide an ideal radiation transfer system, since all radiation from B 1 will reach directly or through the relays, and vice versa.

4 Geometrical optics and blackbody radiation 101 transfers all the radiation that is received from B 1 to and vice versa. We will also consider that the radiation transfer between B 1 or and r can be calculated as if r were a blackbody (i.e. with equation (1)). We do not reveal yet how such a relay could be done, but using equation (1), the radiation power that the relay couples, we can deduce that _q B1 r _q rb2 ) L 1 þ½š ½ 0 Š¼L 2 þ½š ½ 0 Š ) L 1 þ½ 0 Š¼L 2 þ½ 0 Š, ð3þ where the symbol indicates the radiation coupling. Therefore, the points,, 0 and 0 are not free but must fulfill equation (3). In the case B 1 and are straight lines, equation (3) states that both and lie on one ellipse of foci 0 and 0. We can also conceive an ideal device to transfer all the radiation from B 1 to (and vice versa) using two relays r 1 and r 2 as shown in figure 2. Such a device will transfer the radiation either directly from B 1 to or via the relays. The application of equation (3) to both relays leads to L 1 þ½ 0 Š¼L 2 þ½ 0 Š L 1 þ½ 0 Š¼L 2 þ½ 0 Š ) L 1 ¼ L 2, ½ 0 Š¼½ 0 Š: The right equations in equation (4) are necessary conditions for the device of figure 2 to exist. The top right equation corresponds to the energy conservation: to connect all radiation between B 1 to, necessarily they must emit the same amount of heat in equilibrium (i.e. have the same perimeter). Let us advance to design a real device that performs the function of that in figure 2 considering the case in figure 3, which shows a combination of relay element r 1, chosen to couple radiation between B 1 and the auxiliary blackbody X 0, and r 2, chosen to couple and X 0 through it. Let this set r 1 þr 2 perform the function of the relay element r in figure 2. _q B1 ðr 1 þr 2 Þ _q B1 r 1 þ _q B1 r 2 _q B1 r 1 þ _q ðx0 Þr 2 _q r1 r 2, ð5þ ð4þ B 1 r r 1 2 X B 1 r 1 r r N 2 X 1 r k X 2 X k X X k+1 N Figure 3. The combination of relay element r 1 (chosen to couple B 1 and 1 0 ) and r 2 (chosen to couple and 1 0 ) is an example of the relay element r in figure 2. An arbitrary number of connected adjacent relays also performs the relay element r in figure 2.

5 102. Benı tez et al. where the second identity is obtained from _q ðx 0 Þr 2 _q B1 r 2 þ _q r1 r 2. Since r 1 is a relay from B 1 and X 0, _q B1 r 1 _q r1 ðx 0 Þ. Analogously for r 2 as a relay from 0 X and, _q ðx0 Þr 2 _q r2 and thus (5) can be continued as _q B1 r 1 þ _q ðx0 Þr 2 _q r1 r 2 _qr1 ðx 0 Þ þ _q r2 _q r1 r 2 _qr2 þ _q r1 ðx 0 Þ _q r1 r 2 _q r2 þ _q r1 _q ðr1 þr 2 Þ, ð6þ where the fourth identity is obtained since _q ðx 0 Þr 2 _q B1 r 2 þ _q r1 r 2. (Note that the third equality is just a reordering of terms.) Therefore, we have obtained that, effectively, the combination of relays r 1 and r 2 effectively produces _q B1 ðr 1 þr 2 Þ _q ðr1 þr 2 Þ,soitis an example of the relay element r in figure 2. Let us examine the condition of radiation coupling by a relay in equation (3), applied to the relays r 1 and r 2. It leads to L 1 þ½ 0 Š¼½X 0 Šþ½X 0 Š¼L 2 þ½ 0 Š: Therefore, point X in figure 3 is not free either, but fulfills equation (7). In the case B 1 and are straight lines, equation (7) state that, X and lie on a single ellipse of foci 0 and 0. If the relays r 1 and r 2 are further subdivided, the chain of relays r 1,r 2,...,r N of figure 3. Any of these relays r k are set to couple the radiation they receive from the auxiliary blackbody X k 0 to X kþ1 0 and vice versa. Equation (3), which is the condition of radiation coupling by the relay chains, leads to the necessary condition that all the points, and all X k lie on a single ellipse of foci 0 and 0. Figure 4 examines the operation relay link r k performs at the limit of infinitesimal links, i.e. when distðx k, X kþ1 Þ!0. At this limit, points 0 and 0 are located at infinity, and by definition the relay element r k will be an element ð7þ B aux B aux M M θ o θ i X k X k+1 r k Figure 4. When dist(x k, X kþ1 Þ!0, the infinitesimal relay element r k can be defined as the element coupling the radiation it receives from the adjacent half-infinite flat blackbodies B aux and B 0 aux.

6 Geometrical optics and blackbody radiation 103 that transfers all the radiation that is received from blackbody B aux to B 0 aux. These two auxiliary blackbodies are half-infinite straight lines. From equation (3) it is deduced that points X k and X kþ1 not only freely located but fulfill X k M ¼ X kþ1 M 0 ) i ¼ o : ð8þ We have deduced that the normal to segment X k X kþ1 must be parallel to the bisectrix of the half-infinite straight lines B aux to B 0 aux. Thus, equation (8) can be identified then as a reflection law for the relay element r k, which has been deduced from the radiation transfer hypotheses without explicit reference to rays or mirrors. A practical way to do the relay infinitesimal element r k is by means of a specular flat reflector joining points X k and X kþ1. This means that we have deduced that relay element r in figure 2 can be done with an elliptical reflector (whose overall profile was shown in figure 3). Moreover, we have also deduced from the initial hypothesis that the ideal device shown in figure 2 can be done with r 1 and r 2 as two elliptical mirrors, which is a well known CEC-type non-imaging concentrator [3]. Finally, let us show that the concepts that we have introduced here have also application for the design. Consider first the configuration of one blackbody B and two relays r 1 and r 2 shown in figure 5. Relay r 1 produces the transfer of radiation impinging on it from B to r 2, and r 2 will re-relay that radiation to B (and vice versa). We have seen that relays r 1 and r 2 can be built as circular reflectors with centres 1 and 2 (which is the particular case of the elliptical reflector of coincident foci), and these two reflectors will send all the rays from B back to B after exactly two reflections. Figure 5 shows another device that is analogous to that of figure 5 but using three relays. Then relay r 1 produces the transfer of radiation impinging on it from B to r 2, and r 2 will re-relay that radiation to r 3 and r 3 to B (and vice versa). As seen, to perform the relay function, r 1,r 2 and r 3 can be elliptical reflectors (r 1 with foci 3 and r 2 3 r 1 r r 1 r 3 1 B 2 4 B 1 Figure 5. All the radiation from B comes back to B after two successive relays on r 1 and r 2. (or on r 2 and r 1 ). This implies a novel ideal optical device form by a reflector with two arcs of circumferences. All the radiation from B towards r 1 (r 3 ) comes back to B from r 3 (r 1 ) after an intermediate relay on r 2, and it is deduced that the radiation from B to r 2 is sent back to B. This implies a novel ideal optical device formed by a reflector with three elliptical sectors.

7 104. Benı tez et al. 4,r 2 with foci 1 and 4, and r 3 with foci 1 and 2 ). The relays are guaranteeing that all the radiation emitted from B to r 1 is received back at B through r 3 and (vice versa) after exactly three reflections. Therefore, we deduce also that the radiation transferred from B to r 2 must be sent back to B after a single reflection (i.e. with no reflection on r 1 or r 3 ). Note that this is a property of reflector r 2 that we have deduced, which is additional to its relay function. This property was already known, since this elliptical reflector was suggested as an external cavity for light trapping in photovoltaic applications [4]. Acknowledgement ablo Benı tez and Juan C. Mi~nano thank the Spanish Ministerio de Educacio n for their support under project TEC References [1] W.H. Carter and E. Wolf, J. Opt. Soc. Am (1975). [2] H. Hottel, in Heat Transmission, edited by W.H. McAdams (McGraw-Hill, New York, 1954). [3] R. Winston, J.C. Mi~nano and. Benı tez, Nonimaging Optics (Elsevier, Amsterdam, 2005). [4] J.C. Mi~nano, in hysical Limitations to hotovoltaic Solar Energy Conversion, edited by A. Luque and G.L. Arau jo (Hilger, Bristol, 1990), pp