Abstract

Freeform optical surfaces embedded in three-dimensional space, without any symmetry, are tailored so as to solve the archetypal problem of illumination design: redistribute the radiation of a given small light source onto a given reference surface, thus achieving a desired irradiance distribution on that surface. The shape of the optical surface is found by solving a set of partial nonlinear differential equations. For most cases, a few topologically distinct solutions exist, given suitable boundary conditions.

© 2002 Optical Society of America

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References

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  1. J. M. Gordon, H. Ries, “Tailored edge-ray concentrators as ideal second stages for Fresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
    [CrossRef] [PubMed]
  2. H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
    [CrossRef]
  3. H. Ries, J. Gordon, “Double-tailored imaging concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer V, R. Winston, ed., Proc. SPIE3781, 129–134 (1999).
    [CrossRef]
  4. J. M. Gordon, H. Ries, “Tailord edge-ray concentrators (TERC’s): Approaching the thermodynamic limit with new ideal second stages for Fresnel reflectors,” presented at The Biennial Congress of the International Solar Energy Society, Budapest, August 19–23, 1993.
  5. R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
    [CrossRef]
  6. L. D. Landau, E. M. Lifschitz, Klassische Feldtheorie. Vol. 2 of Lehrbuch der Theoretischen Physik, G. Heber, Ed. (Akademie Verlag, Berlin, 1904).
  7. H. Ries, W. Spirkl, “Caustic and its use in designing optimal absorber shapes for 2D concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 2–9 (1995).
    [CrossRef]
  8. R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
    [CrossRef]
  9. G. Monge, Application de l’analyse à la géométrie, 5th ed. Corrected and annotated by M. Liouville (Bachelier, Paris, 1850).
  10. V. D. Komissarov, N. G. Boldyrev, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).
  11. J. S. Schruben, “Formulation of a reflector design problem for a lighting fixture,” J. Opt. Soc. Am. 62, 1498–1501 (1972).
    [CrossRef]
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).
  13. P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).
  14. W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 191–193 (1998).
  15. W. A. Parkyn, “Illuminating Lens designed via extrinsic differential geometry,” U.S. patent5,924,788, July20, 1999.
  16. LightTools 3.1, 2001, Optical Research Associates, www.opticalres.com , 3280 East Foothill Boulevard, Pasadena, California 91107.

1996 (1)

R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
[CrossRef]

1995 (1)

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

1994 (1)

1993 (1)

1972 (1)

1941 (1)

V. D. Komissarov, N. G. Boldyrev, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Boldyrev, N. G.

V. D. Komissarov, N. G. Boldyrev, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Dagan, E.

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).

Friedman, R. P.

R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
[CrossRef]

Gordon, J.

H. Ries, J. Gordon, “Double-tailored imaging concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer V, R. Winston, ed., Proc. SPIE3781, 129–134 (1999).
[CrossRef]

Gordon, J. M.

R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
[CrossRef]

J. M. Gordon, H. Ries, “Tailored edge-ray concentrators as ideal second stages for Fresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
[CrossRef] [PubMed]

J. M. Gordon, H. Ries, “Tailord edge-ray concentrators (TERC’s): Approaching the thermodynamic limit with new ideal second stages for Fresnel reflectors,” presented at The Biennial Congress of the International Solar Energy Society, Budapest, August 19–23, 1993.

Karni, J.

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

Komissarov, V. D.

V. D. Komissarov, N. G. Boldyrev, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Landau, L. D.

L. D. Landau, E. M. Lifschitz, Klassische Feldtheorie. Vol. 2 of Lehrbuch der Theoretischen Physik, G. Heber, Ed. (Akademie Verlag, Berlin, 1904).

Lifschitz, E. M.

L. D. Landau, E. M. Lifschitz, Klassische Feldtheorie. Vol. 2 of Lehrbuch der Theoretischen Physik, G. Heber, Ed. (Akademie Verlag, Berlin, 1904).

Monge, G.

G. Monge, Application de l’analyse à la géométrie, 5th ed. Corrected and annotated by M. Liouville (Bachelier, Paris, 1850).

Moon, P.

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

Parkyn, W. A.

W. A. Parkyn, “Illuminating Lens designed via extrinsic differential geometry,” U.S. patent5,924,788, July20, 1999.

W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 191–193 (1998).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).

Ries, H.

R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
[CrossRef]

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
[CrossRef]

J. M. Gordon, H. Ries, “Tailored edge-ray concentrators as ideal second stages for Fresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
[CrossRef] [PubMed]

H. Ries, W. Spirkl, “Caustic and its use in designing optimal absorber shapes for 2D concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 2–9 (1995).
[CrossRef]

J. M. Gordon, H. Ries, “Tailord edge-ray concentrators (TERC’s): Approaching the thermodynamic limit with new ideal second stages for Fresnel reflectors,” presented at The Biennial Congress of the International Solar Energy Society, Budapest, August 19–23, 1993.

H. Ries, J. Gordon, “Double-tailored imaging concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer V, R. Winston, ed., Proc. SPIE3781, 129–134 (1999).
[CrossRef]

Schruben, J. S.

Spencer, D. E.

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

Spirkl, W.

H. Ries, W. Spirkl, “Caustic and its use in designing optimal absorber shapes for 2D concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 2–9 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).

Winston, R.

Zaibel, R.

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Sol. Energy (1)

R. P. Friedman, J. M. Gordon, H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56, 607–615 (1996).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

R. Zaibel, E. Dagan, J. Karni, H. Ries, “An astigmatic corrected target-aligned heliostat for high concentration,” Sol. Energy Mater. Sol. Cells 37, 191–202 (1995).
[CrossRef]

Trudy VEI (1)

V. D. Komissarov, N. G. Boldyrev, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Other (10)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, New York, 1992).

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 191–193 (1998).

W. A. Parkyn, “Illuminating Lens designed via extrinsic differential geometry,” U.S. patent5,924,788, July20, 1999.

LightTools 3.1, 2001, Optical Research Associates, www.opticalres.com , 3280 East Foothill Boulevard, Pasadena, California 91107.

L. D. Landau, E. M. Lifschitz, Klassische Feldtheorie. Vol. 2 of Lehrbuch der Theoretischen Physik, G. Heber, Ed. (Akademie Verlag, Berlin, 1904).

H. Ries, W. Spirkl, “Caustic and its use in designing optimal absorber shapes for 2D concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 2–9 (1995).
[CrossRef]

H. Ries, J. Gordon, “Double-tailored imaging concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer V, R. Winston, ed., Proc. SPIE3781, 129–134 (1999).
[CrossRef]

J. M. Gordon, H. Ries, “Tailord edge-ray concentrators (TERC’s): Approaching the thermodynamic limit with new ideal second stages for Fresnel reflectors,” presented at The Biennial Congress of the International Solar Energy Society, Budapest, August 19–23, 1993.

G. Monge, Application de l’analyse à la géométrie, 5th ed. Corrected and annotated by M. Liouville (Bachelier, Paris, 1850).

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Figures (7)

Fig. 1
Fig. 1

Sketch of source location s, a point p on the optical surface, and points t on the target surface. N denotes the unit normal to the optical surface.

Fig. 2
Fig. 2

Curvature of the optical surface, illustrated here as a reflector, and the curvature of the incoming wave front combine to determine the curvature of the outgoing wave front.

Fig. 3
Fig. 3

Sketch of the setting: The lens is made to collect light from an isotropic point source and redistribute it onto a target perpendicular to the axis. Length units are arbitrary; the drawing is not to scale.

Fig. 4
Fig. 4

Shape of the freeform lens that casts the letters OEC on a square background, as it would appear under oblique illumination. A slight bulging is visible, which remotely resembles the letter shape. Note that the rim lies on a cone as required by the boundary condition.

Fig. 5
Fig. 5

Shape of the freeform lens in an isosag representation. Lines indicate equal sag. Spacing is 0.01 unit. The ragged rim is an artifact of the representation. The rim of the lens was made to lie on a cone oriented with its axis toward the center of the desired distribution.

Fig. 6
Fig. 6

Irradiance distribution produced by the freeform tailored lens based on Monte Carlo ray tracing of 9×106 rays. Intersection of the rays with a plane at the position of the nominal target was binned by 129×129 pixels. Intensity variations are entirely compatible with shot noise.

Fig. 7
Fig. 7

Horizontal cross section through the irradiance distribution in Fig. 5 at vertical position 0. Irradiance is in arbitrary units. Note the contrast of 0/1/3. Noise is completely within statistical bounds.

Equations (12)

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[1+n2-2n(Out·In)]1/2N=Out-nIn.
Kij=Nitji, j=1, 2.
f1=p-1c1N,f2=p-1c2N.
E(p)=E(p0)|p0-f1||p0-f2||p-f1||p-f2|.
[1+n2-2n(Out·In)]K¯o
=P¯Out·K¯Out·J¯Out-nP¯In·K¯In·J¯In.
JIn ij=tiIntjO,JOut ij=tiOuttjO.
P¯In=J¯In=N·In001;
P¯Out=J¯Out=N·Out001.
2(1+cos(2θ))K¯o=100cos(θ)-1·(K¯In-K¯Out)·cos(θ)001.
E(t)=I(In)(p-s)·(p-s)×|p-g1||p-g2||t-g1||t-g2|Nt·Out.
N·curl(N)=0.

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