Roland Winston

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WATKINS PHYSICS LECTURE October 7, 2009 Beating
Liouvilles theorem How does geometrical optics
know the second law of thermodynamics?
Roland Winston Schools of Engineering Natural
Sciences The University of California, Merced
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Phase space in optics

• Phase space becomes etendue
• Almost the same, etendue DpxDpyDx Dy is the
  4-dimensional Poincare invariant, phase space
  DpDq is 6-dimensional
• Brightness (B) Power/etendue
• Entropy (S) k log(etendue) const.
• Log B -S/k const.
• More carefully, Log (B/n2) -S/k const.

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Statistical Physics by Landau and Lifshitz,
translated by E. Peierls and R. F. Peierls (1958)

• To formulate the basic problem of classical
  statistics
• we must first of all introduce the concept of
  phase space (Dp Dq)
• Which we shall be continually using later on
• Chapter 1, Page 1
• And on page 24,
• In the classical case the entropy is similarly
  defined by the expression
• S log Dp Dq/hs

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Limits to Concentration

• from l max sun 0.5 m
• we measure Tsun 6000 (5670)
• Then from s 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/sin2q

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1/sin2? Law of Maximum Concentration

• The irradiance, of sunlight, I, falls off as 1/r2
  so that at the orbit of earth, I2 is 1/sin2? xI1,
  the irradiance emitted at the suns surface.
• The 2nd 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 refractive index 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

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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?
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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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Chandra
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Radiative transfer between walls in an enclosure
HOTTEL STRINGS Hoyt C. Hottel, 1954,
Radiant-Heat Transmission, Chapter 4 in William
H. McAdams (ed.), Heat Transmission, 3rd ed.
McGRAW-HILL
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Strings 3-walls
F12 (A1 A2 A3)/(2A1) F13 (A1 A3
A2)/(2A1) F23 (A2 A3 A1)/(2A2)
3
1
2
qij AiFij Fii 0
F12 F13 1 F21 F23 1 3 Eqs F31 F32 1
Ai Fij Aj Fji 3 Eqs
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Strings 4-walls
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1
4
2
F12 F13 F14 1 F21 F23 F24 1
F14 (A5 A6) (A2 A3)/(2A1) F23 (A5
A6) (A1 A4)/(2A2)
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Limit to Concentration
3
6
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1
4
2

• F23 (A5 A6) (A1 A4)/(2A2)
• sin(q) as A3 goes to infinity
• This rotates for symmetric systems to sin 2(q)

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the Hottel string method
2D concentrator with acceptance (half) angle ?
string
absorbing surface
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the Hottel string method
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the Hottel string method
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the Hottel string method
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the Hottel string method
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the Hottel string method
stop here, because slope becomes infinite
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the Hottel string method
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The Edge-Ray Principle
C
Edge-ray wave front
?
A
A
B
B
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The Edge-Ray Principle
C
Edge-ray wave front
?
A
A
concentration limit in 2D !
B
B
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the string method
collimator for a tubular light source
étendue conserved ? ideal design!
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Analogy of Fluid Dynamics and Optics
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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 .

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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.

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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.

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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.

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Hmm, gas is so expensive, I must use nonimaging
method !
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How LSC Works
This extremely bright light emitted by the
concentrator can be collected by a Si PV cell
attached to the cell at this edge. Note this
bright emission is from using a white light
source that is about a tenth the power of the
sun. Filled is Dye ( Rhodamine B )
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Thermodynamic Analysis

• Brightness
• Entropy
• Down-shifting process

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Why Quantum Dots (QDs) Tunable
Energy Level
QDs are crystalline semiconductors and degrade
less than organic dyes.
L size of particle m mass of particle
Energy levels are determined by size L
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Why Quantum Dots Tunable Absorption
and Emission

• Mixing Q.D. of different sizes
• Absorbs most of solar spectrum
• Emits at the wavelength preferred by PV Cells

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Advantages of LSC

• can use inexpensive abundant materials to absorb
  the solar light and concentrate it, then a much
  smaller area of the expensive silicon PV is
  needed.
• Concentrates both direct and diffuse radiation
  without tracking (work equally under diffuse
  light), which make them readily adaptable to most
  geographical locations and buildings, such as
  used as smart windows.
• Shift wavelength to match PV cells bandgap.
  Luminescent materials can be chosen that its
  absorption band overlaps with the peak emission
  of the sun and its emission occurs at a
  wavelength range at which PV is efficient. Can
  separate the solar spectrum into two or more
  parts.
• Reduce heat dissipation problems. PV can accept
  cool photons and work efficiently.

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Conclusions

• LSC technology is unique
• Work well in diffuse light
• Can be easily integrated in existing building
  designs,
• Can be spectrally tuned to enable more light
  absorption by long-lived phosphors,
• Enable power generation during cloudy conditions,
• Can be portable, without tracking system
• Efficiency can be arrived at 30,
• Cost can be arrived at 10/m2 or lower.

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Outline

• Aplanatism
• Two-mirror aplanatic design
• Luneburg method
• Problems
• Novel aplanatic design
• Method
• Result
• Summary

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Aplanatism

• Why imaging optics is rarely efficient for light
  concentration?
• Aberrations hurt étendue packing
• Exception Aplanatic systems
• Aplanatism is the condition of freedom from
  spherical aberration and coma
• A necessary condition for image-forming system to
  have the theoretical maximum concentration is
  that the image formation should be aplanatic
• Condition for Aplanatism Abbe Sine Condition
• Challenge realize aplanatism in a practical way

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Abbe Sine Condition for Distant Object

• Condition rays parallel to the optical axis
  intersect rays converging to the focus on the
  surface of a sphere, called Abbe Sphere.
• Two requirements
• On axis image of infinity
• Converging from Abbe Sphere
• It is remarkable that an on-axis condition which
  can be simply formulated ensures good off-axis
  performance.

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Two-mirror System

• In general at least two surfaces required to
  satisfy the two requirements
• Two-mirror solution
• Luneburg, R. K. Mathematical Theory of Optics
  (University of California Press, Berkeley, 1964)
• Analytical solutions by Schawarzchild and
  Lynden-Bell

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Luneburg Method

• Approximate mirror surfaces by small line
  segments
• Start from center (optical axis) toward edge
• Initial segments are vertical and their positions
  are two free parameters
• Whole mirrors can be built up with iteration
• Precision controlled by step size

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Luneburg Method

• Approximate mirror surfaces by small line
  segments
• Start from center (optical axis) toward edge
• Initial segments are vertical and their positions
  are two free parameters
• Whole mirrors can be built up with iteration
• Precision controlled by step size

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Luneburg Method

• Symmetry can be applied to obtain the whole
  profile

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Problem of Two-mirror Aplanat

• Front mirror blocks incident light

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Problem of Two-mirror Aplanat

• Front mirror blocks incident light

• Fully developed two- mirror system may receive
  no light at all, making the system useless!

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If there is a one-way mirror
Apparently the problem is solved if there is a
one-way mirror.
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TIR as oneway Mirror

• Total Internal Reflection (TIR)
• gt qc
• Critical angle, qc arcsin(n0/n1)

• Almost one-way mirror except
• Reflected rays incidence angle must exceed the
  critical angle
• Transmitted rays direction is changed due to
  refraction

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Design Proposal

• Filling the space between the two mirror surfaces
  with a refractive medium

• Reflective coating the rear surface

• Treat the front surface as a one-way mirror

• Discarding or reflective coating the portions
  where TIR fails.

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Obstacle

• How to deal with the disturbance due to
  refraction at the front surface?

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Luneburg Method ?

• Luneburg method fails because it violates
  causality.
• Ray refraction depends on the slope which is not
  determined until future steps in the iteration.

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Solution

• The problem of causality is solved with a new
  prescription which includes two keys
• A self-consistent initial configuration
• Building up the two surfaces from edge to center
  (optical axis)

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Initial Configuration

• Choose initial point of front surface on abbe
  sphere.
• Ray must be reflected back on itself by the rear
  surface and towards the center of the Abbe sphere
  by the front surface.
• Slope of the front surface at the point must
  satisfy both refraction and reflection, therefore
  It is determinate
• Angle q and position of the initial point of the
  rear surface are free parameters

((n0/n1)cos(?))/sin(?)
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Construction
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Construction
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Results

• Max. Converging Angle
• 60 degrees
• Aspect (height diameter)
• 13
• Central obscuration
• 4

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Off-axis PerformanceCompared with Parabolic
Mirror

• Same angular aperture

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Summary

• With a one-way mirror concept and a new
  construction method, we solved the light blocking
  problem in two-mirror aplanatic systems.
• Result is compact and fast aplanatic systems.
• Applications include light concentration,
  illumination, and imaging.