Thermodynamically efficient NONIMAGING 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 Limits to Concentration

•from λ max sun ~ 0.5 μ

we measure Τsun ~ 6000° (5670°) • Then from σ T4 - solar surface flux~ 58.6 W/mm2 – The solar constant ~ 1.35 mW/mm2 – The second law of thermodynamics – C max ~ 44,000 – Coincidentally, C max = 1/sin2θ 1/sin2θ Law of Maximum Concentration

• The irradiance, of sunlight, I, falls off as 1/r2 so that at the orbit of 2 earth, I2 is 1/sin θ xI1, the irradiance emitted at the sun’s surface. nd • The 2 Law of Thermodynamics forbids concentrating I2 to levels greater than I1, since this would correspond to a brightness temperature greater than that of the sun. • In a medium of n, one is allowed an additional factor of n2 so that the equation can be generalized for an absorber immersed in a refractive medium as

Nonimaging Optics 3 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? 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 `

Chandra B3 B3 Q’ P’

B2 B2

B1 B1

B4 Q

P (a) (b)

Radiative transfer between walls in an enclosure Strings 3-walls

F12 = (A1 + A2 – A3)/(2A1)

3 F13 = (A1 + A3 – A2)/(2A1) 1 F23 = (A2 + A3 – A1)/(2A2)

2 qij = AiFij F12 + F13 = 1 F21 + F23 = 1 3 Eqs Fii = 0 F31 + F32 = 1

Ai Fij = Aj Fji 3 Eqs Strings 4-walls 3

6 1 5 4

2 F12 + F13 + F14 = 1 F21 + F23 + F24 = 1 F14 = [(A5 + A6) – (A2 + A3)]/(2A1) F23 = [(A5 + A6) – (A1 + A4)]/(2A2) 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) the string method

r de sli ϑ

2D concentrator with acceptance (half) angle ϑ

string

absorbing surface the string method the string method the string method the string method the string method

stop here, because slope becomes infinite the string method ' Α ϑ ϑ Β sin + sin / Β ' A' ' A Β BB = = = A ' C B AC Α AA = + = A Α → ' BB' B' Β (tilted parabola sections) Concentrator (CPC) Compound Parabolic Compound Parabolic A C

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d E The Edge- Principle A’ Nonimaging Optics Fundamentals ϑ

ϑ /2) = B’B /2) = π sin / ' = A’A sin BB = ' = B’B sin( AA → concentration limit in 2D ! 2D étendue A 2D étendue C

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d E The Edge-Ray Principle A’ Nonimaging Optics Fundamentals 2D cylindrical optics: example:nonimaging collimator for a tubular optics source basics: the string method ϑ 2πR/sinϑ er étendue lid s conserved Æ ideal design!

tubular light source kind of “involute” R of the circle

21 Availability of Solar Flux over a range 1 – 105 Suns Conc.= 1 - 4 Fixed Heating&Cooling, PV

Conc.= 4 -150 1 axis tracking Power generation, and seasonal Heating&Cooling, Low CPV Conc.= 500 - 2 axis tracking Power generation 10,000 (dish&tower) High CPV Conc.= 20,000 – 2 axis tracking Solar Furnace, 100,000 Materials, Lasers, Space Propulsion, Experiments Analogy of Fluid Dynamics and Optics

fluid dynamics optics

phase space general (twice the dimensions of ordinary space )

positions positions momenta directions of light rays multiplied by the index of of the medium

incompressible fluid volume in “phase space” is conserved

Nonimaging Optics 23 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 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 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 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 Nonimaging Optics 28 BRIGHTER THAN THE SUN an experiment on the roof of the U of C HEP Building Roof top Physics Ultra High Flux Experiment

3D Rendering of Our New Design

PMMA cover

Secondary

Solar cell on heat spreader Primary mirror Heat sink Features of Our New Design ¾ Light impinging on the primary mirror is not focused onto the cell, but onto the secondary mirror ¾ This results in a uniform cell illumination with an average concentration of 500 suns

Secondary mirror

Focal ring on secondary Light radially distributed mirror along cell

Primary mirror Dimensions (in mm) PALO ALTO WATER SolFocus Array Optical Performance Comparison of Various CPV Designs (1) 8

7 Theoretical limit (n=1.5; 60° exit angle) Theoretical limit (n=1; 60° exit angle) Dielectric TIR Aplanat (circular) 6 XR (circular) Two aplanatic (air filled) + prism 5 Two aplanatic (glass filled) mirrors [degrees] Fresnel without secondary

angle AR ... Aspect ratio (depth/aperture diameter) ‐ 4 half

3 Acceptance 2 AR=0.3 AR=0.6 AR=0.3 1 AR=0.3

AR=1.9 0 0 200 400 600 800 1,000 1,200 1,400 Geometrical concentration

39 With Apologies to Benny Goodman It don’t mean a thing If it doesn’t have Sin θ=n/√C

Sarah Kurtz and Jerry Olson, Dan David Laureates 2007