Thermodynamically efficient NONIMAGING OPTICS Dan David Symposium UC MERCED September 26, 2008

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1 Thermodynamically efficient NONIMAGING OPTICS Dan David Symposium UC MERCED September 26, 2008 Nonimaging optics departs from the methods of traditional optical design by instead developing techniques for maximizing the collecting power of illumination elements and systems. Nonimaging designs exceed the concentration attainable with focusing techniques by factors of four or more and approach the theoretical limit (ideal concentrators). Roland Winston Schools of Engineering & Natural Science University of California, Merced

2 Limits to Concentration from λ max sun ~ 0.5 μ we measure Τsun ~ 6000 (5670 ) Then from σ T 4 - solar surface flux~ 58.6 W/mm 2 The solar constant ~ 1.35 mw/mm 2 The second law of thermodynamics C max ~ 44,000 Coincidentally, C max = 1/sin 2 θ

3 1/sin 2 θ Law of Maximum Concentration The irradiance, of sunlight, I, falls off as 1/r 2 so that at the orbit of earth, I 2 is 1/sin 2 θ xi 1, the irradiance emitted at the sun s surface. The 2 nd Law of Thermodynamics forbids concentrating I 2 to levels greater than I 1, since this would correspond to a brightness temperature greater than that of the sun. In a medium of refractive index n, one is allowed an additional factor of n 2 so that the equation can be generalized for an absorber immersed in a refractive medium as Nonimaging Optics 3

4 During a seminar at the Raman Institute (Bangalore) in 2000, Prof. V. Radhakrishnan asked me: How does geometrical optics know the second law of thermodynamics?

5 First and Second Law of Thermodynamics NIO is the theory of maximal efficiency radiative transfer It is axiomatic and algorithmic based As such, the subject depends much more on thermodynamics than on optics `

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7 Chandra

8 B 3 P B 3 Q B 1 B 2 B 1 B 2 B 4 Q (a) (b) P Radiative transfer between walls in an enclosure

9 Strings 3-walls F12 = (A1 + A2 A3)/(2A1) 1 3 F13 = (A1 + A3 A2)/(2A1) F23 = (A2 + A3 A1)/(2A2) 2 qij = AiFij Fii = 0 F12 + F13 = 1 F21 + F23 = 1 F31 + F32 = 1 Ai Fij = Aj Fji 3 Eqs 3 Eqs

10 Strings 4-walls F12 + F13 + F14 = 1 F21 + F23 + F24 = 1 F14 = [(A5 + A6) (A2 + A3)]/(2A1) F23 = [(A5 + A6) (A1 + A4)]/(2A2)

11 Limit to Concentration F23 = [(A5 + A6) (A1 + A4)]/(2A2) = sin(θ) as A3 goes to infinity This rotates for symmetric systems To sin 2 (θ)

12 the string method slider ϑ 2D concentrator with acceptance (half) angle ϑ string absorbing surface

13 the string method

14 the string method

15 the string method

16 the string method

17 the string method stop here, because slope becomes infinite

18 the string method

19 Nonimaging Optics Fundamentals Edge-ray wave front C The Edge-Ray Principle ϑ A A Β'Α + ΑC = B' A = B A BB' = AC = Β' Β + Β Α' AA' sinϑ AA' = BB' / sin ϑ Compound Parabolic Concentrator (CPC) (tilted parabola sections) B B

20 Nonimaging Optics Fundamentals Edge-ray wave front C The Edge-Ray Principle ϑ A A 2D étendue = A A sin ϑ AA ' = BB' sinϑ concentration limit in 2D! / B B 2D étendue = B B sin(π/2) = B B

21 2D cylindrical optics: nonimaging optics basics: the string example: collimator for a tubular light source 2πR/sinϑ slider method ϑ étendue conserved ideal design! tubular light source R kind of involute of the circle 21

22 Availability of Solar Flux over a range Conc.= 1-4 Conc.= Conc.= ,000 Conc.= 20, , Suns Fixed 1 axis tracking and seasonal 2 axis tracking (dish&tower) 2 axis tracking Heating&Cooling, PV Power generation, Heating&Cooling, Low CPV Power generation High CPV Solar Furnace, Materials, Lasers, Space Propulsion, Experiments

23 Analogy of Fluid Dynamics and Optics fluid dynamics optics phase space (twice the dimensions of ordinary space ) general etendue positions momenta incompressible fluid positions directions of light rays multiplied by the index of refraction of the medium volume in phase space is conserved Nonimaging Optics 23

24 Imaging in Phase Space Example: points on a line. An imaging system is required to map those points on another line, called the image, without scrambling the points. In phase space Each point becomes a vertical line and the system is required to faithfully map line onto line. Nonimaging Optics 24

25 Edge-ray Principle Consider only the boundary or edge of all the rays. All we require is that the boundary is transported from the source to the target. The interior rays will come along. They cannot leak out because were they to cross the boundary they would first become the boundary, and it is the boundary that is being transported. Nonimaging Optics 25

26 Edge-ray Principle It is very much like transporting a container of an incompressible fluid, say water. The volume of container of rays is unchanged in the process. conservation of phase space volume. The fact that elements inside the container mix or the container itself is deformed is of no consequence. Nonimaging Optics 26

27 Edge-ray Principle To carry the analogy a bit further, suppose one were faced with the task of transporting a vessel (the volume in phase-space) filled with alphabet blocks spelling out a message. Then one would have to take care not to shake the container and thereby scramble the blocks. But if one merely needs to transport the blocks without regard to the message, the task is much easier. Nonimaging Optics 27

28 Nonimaging Optics 28

29 BRIGHTER THAN THE SUN an experiment on the roof of the U of C HEP Building Roof top Physics

30 Ultra High Flux Experiment

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35 3D Rendering of Our New Design PMMA cover Secondary mirror Solar cell on heat spreader Primary mirror Heat sink

36 Features of Our New Design Light impinging on the primary mirror is not focused onto the cell, but onto the secondary mirror This results in a uniform cell illumination with an average concentration of 500 suns Secondary mirror Light radially distributed along cell Focal ring on secondary mirror Primary mirror

37 Dimensions (in mm)

38 PALO ALTO WATER SolFocus Array

39 8 Optical Performance Comparison of Various CPV Designs (1) 7 Theoretical limit (n=1.5; 60 exit angle) Theoretical limit (n=1; 60 exit angle) Acceptance half angle [degrees] AR=0.3 AR=0.6 Dielectric TIR Aplanat (circular) XR (circular) Two aplanatic (air filled) mirrors + prism Two aplanatic (glass filled) mirrors Fresnel lens without secondary AR... Aspect ratio (depth/aperture diameter) AR=0.3 1 AR=0.3 0 AR= ,000 1,200 1,400 Geometrical concentration 39

40 With Apologies to Benny Goodman It don t mean a thing If it doesn t have Sin θ=n/ C

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42 Sarah Kurtz and Jerry Olson, Dan David Laureates 2007