Abstract

Nonimaging optics is the theory of thermodynamically efficient optics and as such, depends more on thermodynamics than on optics. Historically, nonimaging optics that work as ideal concentrators have been discovered through such heuristic ideas as “edge ray involutes,” “string method,” “simultaneous multiple surface,” and “tailored edge ray concentrator,” without a consistent theoretical definition of what “ideal” means. In this tutorial, we provide a thermodynamic perspective of nonimaging optical designs to shine light on the commonality of all these designing ideas, or what “ideal” nonimaging design means. Hence, in this paper, a condition for the “best” design is proposed based purely on thermodynamic arguments, which we believe have profound consequences. Thermodynamics may also be the most intuitive way for a reader who is new to this subject to understand or study it within a certain framework, instead of learning from sporadic designing methodologies. This way of looking at the problem of efficient concentration and illumination depends on probabilities, the ingredients of entropy, and information theory, while “optics” in the conventional sense recedes into the background. We attempt to link the key concept of nonimaging optics, étendue, with the radiative heat transfer concept of view factor, which may be more familiar to some readers. However, we do not want to limit the readers to a single thermodynamic understanding of this subject. Therefore, two alternative perspectives of nonimaging optics will also be introduced and used throughout the tutorial: the definition of a nonimaging optics design according to the Hilbert integral, and the phase space analysis of the ideal design. The tutorial will be organized as follows: Section 1 highlights the difference between nonimaging and imaging optics, Section 2 describes the thermodynamic understanding of nonimaging optics, Section 3 presents the alternative phase space representation of nonimaging optics, Section 4 describes the most basic nonimaging designs using Hottel’s strings, Section 5 discusses the geometric flow line designing method, and Section 6 summarizes the various concepts of nonimaging optics.

© 2018 Optical Society of America

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References

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  1. W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
    [Crossref]
  2. W. T. Welford and R. Winston, “The ellipsoid paradox in thermodynamics,” J. Stat. Phys. 28, 603–606 (1982).
    [Crossref]
  3. H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. 72, 380–385 (1982).
    [Crossref]
  4. B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
    [Crossref]
  5. M. F. Modest, Radiative Heat Transfer (Academic, 2013).
  6. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, 1972), pp. 204–207.
  7. G. Derrick, “A three-dimensional analogue of the Hottel string construction for radiation transfer,” J. Mod. Opt. 32, 39–60 (1985).
    [Crossref]
  8. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).
  9. J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2015).
  10. L. Jiang and R. Winston, “Flow line asymmetric nonimaging concentrating optics,” Proc. SPIE 9955, 99550I (2016).
    [Crossref]
  11. L. Jiang, “2D Phase space representation example,” https://github.com/wormite/PhaseSpace/ .
  12. T. Sekiguchi and K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
    [Crossref]
  13. E. Fermi, Notes on Thermodynamics and Statistics (Univsersity of Chicago, 1966).
  14. H. Ries and A. Rabl, “Edge-ray principle of nonimaging optics,” J. Opt. Soc. Am. A 11, 2627–2632 (1994).
    [Crossref]
  15. R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245–247 (1970).
    [Crossref]
  16. H. Goldstein, Classical Mechanics (Addison-Wesley, 1950).
  17. L. Jiang, “CPC program for any convex absorber shape,” https://github.com/wormite/AnyCPC2 .
  18. R. Winston and W. T. Welford, “Design of nonimaging concentrators as second stages in tandem with image-forming first-stage concentrators,” Appl. Opt. 19, 347–351 (1980).
    [Crossref]
  19. L. Jiang and R. Winston, “Asymmetric design for Compound Elliptical Concentrators (CEC) and its geometric flux implications,” Proc. SPIE 9572, 957203 (2015).
    [Crossref]
  20. L. Jiang, “CEC example with an absorber of any convex shape,” https://github.com/wormite/AnyCEC .
  21. P. Moon and D. Spencer, “The pharosage vector,” in The Photic Field (MIT, 1981), p. 70.
  22. M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd ACM international conference on Multimedia (1996).
  23. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
    [Crossref]
  24. R. Mehmke, “Über die mathematische Bestimmung der Helligkeit in Rämen mit Tagesbeleuchtung, insbesondere Gemäldesälen mit Deckenlicht,” Zs. f. Math. u. Phys 43, 41 (1893).
  25. P. Moon and D. Spencer, “Theory of the photic field,” J. Franklin Inst. 255, 33–50 (1953).
    [Crossref]
  26. H. Poincaré, New Methods of Celestial Mechanics (AIP, 1992).
  27. R. Winston and W. T. Welford, “Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field: A new derivation of the compound parabolic concentrator,” J. Opt. Soc. Am. 69, 536–539 (1979).
    [Crossref]

2017 (1)

B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
[Crossref]

2016 (1)

L. Jiang and R. Winston, “Flow line asymmetric nonimaging concentrating optics,” Proc. SPIE 9955, 99550I (2016).
[Crossref]

2015 (1)

L. Jiang and R. Winston, “Asymmetric design for Compound Elliptical Concentrators (CEC) and its geometric flux implications,” Proc. SPIE 9572, 957203 (2015).
[Crossref]

1994 (1)

1987 (1)

T. Sekiguchi and K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[Crossref]

1985 (1)

G. Derrick, “A three-dimensional analogue of the Hottel string construction for radiation transfer,” J. Mod. Opt. 32, 39–60 (1985).
[Crossref]

1982 (2)

H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. 72, 380–385 (1982).
[Crossref]

W. T. Welford and R. Winston, “The ellipsoid paradox in thermodynamics,” J. Stat. Phys. 28, 603–606 (1982).
[Crossref]

1980 (2)

R. Winston and W. T. Welford, “Design of nonimaging concentrators as second stages in tandem with image-forming first-stage concentrators,” Appl. Opt. 19, 347–351 (1980).
[Crossref]

W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
[Crossref]

1979 (1)

1970 (1)

1953 (1)

P. Moon and D. Spencer, “Theory of the photic field,” J. Franklin Inst. 255, 33–50 (1953).
[Crossref]

1939 (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

1893 (1)

R. Mehmke, “Über die mathematische Bestimmung der Helligkeit in Rämen mit Tagesbeleuchtung, insbesondere Gemäldesälen mit Deckenlicht,” Zs. f. Math. u. Phys 43, 41 (1893).

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2015).

Derrick, G.

G. Derrick, “A three-dimensional analogue of the Hottel string construction for radiation transfer,” J. Mod. Opt. 32, 39–60 (1985).
[Crossref]

Fermi, E.

E. Fermi, Notes on Thermodynamics and Statistics (Univsersity of Chicago, 1966).

Gershun, A.

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1950).

Hanrahan, P.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd ACM international conference on Multimedia (1996).

Howell, J. R.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, 1972), pp. 204–207.

Jiang, L.

B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
[Crossref]

L. Jiang and R. Winston, “Flow line asymmetric nonimaging concentrating optics,” Proc. SPIE 9955, 99550I (2016).
[Crossref]

L. Jiang and R. Winston, “Asymmetric design for Compound Elliptical Concentrators (CEC) and its geometric flux implications,” Proc. SPIE 9572, 957203 (2015).
[Crossref]

Levoy, M.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd ACM international conference on Multimedia (1996).

Mehmke, R.

R. Mehmke, “Über die mathematische Bestimmung der Helligkeit in Rämen mit Tagesbeleuchtung, insbesondere Gemäldesälen mit Deckenlicht,” Zs. f. Math. u. Phys 43, 41 (1893).

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (Academic, 2013).

Moon, P.

P. Moon and D. Spencer, “Theory of the photic field,” J. Franklin Inst. 255, 33–50 (1953).
[Crossref]

P. Moon and D. Spencer, “The pharosage vector,” in The Photic Field (MIT, 1981), p. 70.

Poincaré, H.

H. Poincaré, New Methods of Celestial Mechanics (AIP, 1992).

Rabl, A.

Ries, H.

Sekiguchi, T.

T. Sekiguchi and K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[Crossref]

Siegel, R.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, 1972), pp. 204–207.

Sinclair, D.

W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
[Crossref]

Spencer, D.

P. Moon and D. Spencer, “Theory of the photic field,” J. Franklin Inst. 255, 33–50 (1953).
[Crossref]

P. Moon and D. Spencer, “The pharosage vector,” in The Photic Field (MIT, 1981), p. 70.

Welford, W.

W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
[Crossref]

Welford, W. T.

Widyolar, B.

B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
[Crossref]

Winston, R.

B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
[Crossref]

L. Jiang and R. Winston, “Flow line asymmetric nonimaging concentrating optics,” Proc. SPIE 9955, 99550I (2016).
[Crossref]

L. Jiang and R. Winston, “Asymmetric design for Compound Elliptical Concentrators (CEC) and its geometric flux implications,” Proc. SPIE 9572, 957203 (2015).
[Crossref]

W. T. Welford and R. Winston, “The ellipsoid paradox in thermodynamics,” J. Stat. Phys. 28, 603–606 (1982).
[Crossref]

W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
[Crossref]

R. Winston and W. T. Welford, “Design of nonimaging concentrators as second stages in tandem with image-forming first-stage concentrators,” Appl. Opt. 19, 347–351 (1980).
[Crossref]

R. Winston and W. T. Welford, “Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field: A new derivation of the compound parabolic concentrator,” J. Opt. Soc. Am. 69, 536–539 (1979).
[Crossref]

R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245–247 (1970).
[Crossref]

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Wolf, K. B.

T. Sekiguchi and K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[Crossref]

Am. J. Phys. (1)

T. Sekiguchi and K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[Crossref]

Appl. Opt. (1)

J. Franklin Inst. (1)

P. Moon and D. Spencer, “Theory of the photic field,” J. Franklin Inst. 255, 33–50 (1953).
[Crossref]

J. Math. Phys. (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

J. Mod. Opt. (1)

G. Derrick, “A three-dimensional analogue of the Hottel string construction for radiation transfer,” J. Mod. Opt. 32, 39–60 (1985).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Photonics Energy (1)

B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7, 28002 (2017).
[Crossref]

J. Stat. Phys. (1)

W. T. Welford and R. Winston, “The ellipsoid paradox in thermodynamics,” J. Stat. Phys. 28, 603–606 (1982).
[Crossref]

Phys. Today (1)

W. Welford, R. Winston, and D. Sinclair, “The optics of nonimaging concentrators: light and solar energy,” Phys. Today 33(6), 56–57 (1980).
[Crossref]

Proc. SPIE (2)

L. Jiang and R. Winston, “Flow line asymmetric nonimaging concentrating optics,” Proc. SPIE 9955, 99550I (2016).
[Crossref]

L. Jiang and R. Winston, “Asymmetric design for Compound Elliptical Concentrators (CEC) and its geometric flux implications,” Proc. SPIE 9572, 957203 (2015).
[Crossref]

Zs. f. Math. u. Phys (1)

R. Mehmke, “Über die mathematische Bestimmung der Helligkeit in Rämen mit Tagesbeleuchtung, insbesondere Gemäldesälen mit Deckenlicht,” Zs. f. Math. u. Phys 43, 41 (1893).

Other (12)

H. Poincaré, New Methods of Celestial Mechanics (AIP, 1992).

L. Jiang, “CEC example with an absorber of any convex shape,” https://github.com/wormite/AnyCEC .

P. Moon and D. Spencer, “The pharosage vector,” in The Photic Field (MIT, 1981), p. 70.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd ACM international conference on Multimedia (1996).

L. Jiang, “2D Phase space representation example,” https://github.com/wormite/PhaseSpace/ .

E. Fermi, Notes on Thermodynamics and Statistics (Univsersity of Chicago, 1966).

M. F. Modest, Radiative Heat Transfer (Academic, 2013).

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, 1972), pp. 204–207.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2015).

H. Goldstein, Classical Mechanics (Addison-Wesley, 1950).

L. Jiang, “CPC program for any convex absorber shape,” https://github.com/wormite/AnyCPC2 .

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Figures (29)

Figure 1
Figure 1 Comparison between imaging (left) and nonimaging (right) optics.
Figure 2
Figure 2 Elliptical paradox where the imaging optics fails.
Figure 3
Figure 3 Typical setup of a concentrator.
Figure 4
Figure 4 Hottel’s strings can be used to solve for P21.
Figure 5
Figure 5 Radiative heat transfer between areas 1 and 2.
Figure 6
Figure 6 Discrete representation of the phase space.
Figure 7
Figure 7 Dense representation of the phase space.
Figure 8
Figure 8 (a) Configuration of an aperture of a concentrator. (b) The phase space representation of an aperture.
Figure 9
Figure 9 (a) Phase space representation of the absorber achieving maximum concentration. (b) The phase space example that does not achieve maximum concentration.
Figure 10
Figure 10 (a) 3D configuration of light coming into an aperture A subtending angle θ. (b) The L/M space representation of the incoming rays.
Figure 11
Figure 11 Setup of edge rays or wavefronts.
Figure 12
Figure 12 Using simple straight line reflectors to form the concentrator.
Figure 13
Figure 13 Mapping points to edge rays.
Figure 14
Figure 14 Phase space of a CPC.
Figure 15
Figure 15 Difference between nonimaging and imaging optics.
Figure 16
Figure 16 Ray 1 cannot (red square) interchange its position with a ray (blue square) that is not originally inside the boundary.
Figure 17
Figure 17 In order for the ray 1 to interchange its phase space position, it will have to merge with a small phase space that was originally not on the boundary. This violates the Liouville theorem.
Figure 18
Figure 18 Merging of the edge ray with an internal ray in their phase space representations. Similarly, it violates the Liouville theorem.
Figure 19
Figure 19 CPC design for a tubular absorber.
Figure 20
Figure 20 Asymmetric CEC setup.
Figure 21
Figure 21 Flow line bisects the two extreme angles.
Figure 22
Figure 22 Flow line of a spherical source.
Figure 23
Figure 23 Flow line example of an orange cone.
Figure 24
Figure 24 “Mountain top” shape as the source for flow line that generates CPC.
Figure 25
Figure 25 Flow line plot of an asymmetric compound elliptical concentrator.
Figure 26
Figure 26 Close-up flow line plot of the asymmetric CEC.
Figure 27
Figure 27 Hilbert integral as a high-level understanding of nonimaging optics.
Figure 28
Figure 28 Hilbert integral configured without refractive optics.
Figure 29
Figure 29 Phase space volume remains constant.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

Pij=radiation fromitojtotal radiation emitted fromi.
Q13=A1σT14P13,
σT4A1P13=σT4A3P31,A1P13=A3P31.
q13=Q13A3=A1P13A3σT14=P31σT14.
q13=σT14if and only ifP31=1.
Q12=Q13,σT14A1P12=σT14A1P13,A1P12=A1P13.
A1P12=A2P21.
C=A2A3=P13P211P21.
P21=(AD¯+BC¯)(AC¯+BD¯)2CD¯.
P21=2(AD¯AC¯)2CD¯=DE¯CD¯=sin(θ),Cmax=1P21=1sin(θ).
Cmax,3D=1P21=1sin2(θ).
P12=1πA1A1A2cosβ1cosβ2r2(dA1dA2).
E12=A1A2cosβ1  cosβ2r2(dA1dA2).
E12=πA1P12.
E12=πP12A1=πP21A2=E21.
Eij=πPijA1=πPjiAj=Eji.
E12=E13=πA3.
dpxdx=constantétendue.
Cmax=1sin(θ).
dpxdpydxdy=étendue.
dE=dxdydpxdpy.
étendue=dpxdpydxdy.
étendue=dpxdpydxdy=dpxdpydxdy=Adpxdpy=n2AdLdM.
CA+AB=AB+BB
AB=AB.
CA=BB.
AC=sinθ·AA,
AABB=1sinθ.
AA·sin(θ)=BB·1,C=AABB=1/sin(θ).
ab+bc=ac+cc,
ab+bc=ac+cc.
2cc=ab+ababab.
C=bbcc=A2A3=2A2A4+A5A6A7=1P21=Cmax.
J=(dpydpz,dpxdpz,dpxdpy).
E=dpxdpydxdy+dpxdpzdxdz+dpzdpydzdy=J·ds,
J·ds=|J|sin(θ1+θ22)ds=2sin(θ2θ12)sin(θ1+θ22)A2.
J·ds=2sin(90°)sin(90°)A3=2A3.
C=A2A3=1sin(θ2θ12)sin(θ1+θ22).
I(1,2)=P1P2nk·ds,
Iα(AB)=ABnsinϕds,
Iα(AB)=n  sinϕLAB,
Iα(AB)Iβ(AB).
Iα(AB)=[Pα¯B][PαA],
E=[Pα¯B]+[PβA][PαA][Pβ¯B].
[Pα¯B]+[BA]α¯+[APα¯]=[PαA]+[AB]α+[BPα],[PβA]+[AB]β+[BPβ]=[Pβ¯B]+[BA]β¯+[APβ¯],
{([PβA][PαA])([Pβ¯B][Pα¯B])}{([APβ¯][APα¯])([BPβ][BPα])}=[AB]α[AB]β+[BA]β¯[BA]α¯.
Construct  pxx˙+pyy˙L(x,y,x˙,y˙)=HdH=pxdx˙+dpxx˙+pydy˙+dpyy˙[Lxdx+Lydy+Lx˙dx˙+Ly˙dy˙]=px˙dxpy˙dy+x˙dpx+y˙dpy.
V(t+Δt)V(t)=Surface(t)vΔt·ds,
Surface(t)vΔt·ds=·vdτ,
V(t+Δt)V(t)=0;or,V(t)=constant.