Abstract

The edge-ray principle can be used to tailor a reflector. However, one set of edge rays already fully determines the reflector profile. We present a design method for tailoring compact compound elliptical concentrator (CEC)-type reflectors to a given source and a desired angular power distribution. Two reflected images of the source, one on each side of the source, contribute together with the direct radiation from the source to produce the desired power distribution. We determine the reflector profile by numerically solving a differential equation. No optimization is required. Beyond the angular region in which the power distribution can be strictly controlled, the power drops to zero in a finite decay range. This decay range becomes narrower as the reflector increases in size. We show a reflector for producing a strictly constant irradiance from −43 to 43 deg from a cylindrical source of constant brightness. The reflector extends to a maximum distance of 8 source diameters. No power is radiated beyond ± 50 deg.

© 1994 Optical Society of America

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References

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  1. R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 9, 1902–1908 (1993).
    [Crossref]
  2. J. M. Gordon, H. Ries, “Tailored edge-ray concentrators (TERC’s) as ideal second stages for Fresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
    [Crossref] [PubMed]
  3. T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota I,” Rend. Sedute Accad. R. Dei Lincei IX(1), 185–189 (1900).
  4. T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota II,” Rend. Sedute R. Accad. Dei Lincei IX(1), 237–245 (1900).
  5. G. Salmon, A Treatise on the Higher Plane Curves (Hodges & Smith, Dublin, 1852), Chap. 1.
  6. J. C. Miñano, J. C. Gonzáles, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [Crossref] [PubMed]
  7. W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1859).
  8. J. M. Gordon, A. Rabl, “Nonimaging compound parabolic concentrator-type reflectors with variable extreme directions,” Appl. Opt. 31, 7332–7338 (1992).
    [Crossref] [PubMed]
  9. H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 7–8 (1991).
    [Crossref]
  10. J. Plücker, System der Analytischen Geometrie (Berlin, 1835).
  11. J. L. Coolidge, A History of Geometrical Methods (Clarendon, Oxford, U.K., 1940), Chap. II.II.2.
  12. W. B. Elmer, Optical Design of reflectors2nd ed. (Wiley, New York, 1980), Chaps. 4,6 and 10.

1993 (2)

R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 9, 1902–1908 (1993).
[Crossref]

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

1992 (2)

1900 (2)

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota I,” Rend. Sedute Accad. R. Dei Lincei IX(1), 185–189 (1900).

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota II,” Rend. Sedute R. Accad. Dei Lincei IX(1), 237–245 (1900).

Coolidge, J. L.

J. L. Coolidge, A History of Geometrical Methods (Clarendon, Oxford, U.K., 1940), Chap. II.II.2.

Elmer, W. B.

W. B. Elmer, Optical Design of reflectors2nd ed. (Wiley, New York, 1980), Chaps. 4,6 and 10.

Gonzáles, J. C.

Gordon, J. M.

Levi-Civita, T.

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota I,” Rend. Sedute Accad. R. Dei Lincei IX(1), 185–189 (1900).

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota II,” Rend. Sedute R. Accad. Dei Lincei IX(1), 237–245 (1900).

Miñano, J. C.

Plücker, J.

J. Plücker, System der Analytischen Geometrie (Berlin, 1835).

Rabl, A.

Ries, H.

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

R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 9, 1902–1908 (1993).
[Crossref]

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 7–8 (1991).
[Crossref]

Salmon, G.

G. Salmon, A Treatise on the Higher Plane Curves (Hodges & Smith, Dublin, 1852), Chap. 1.

Smestad, G.

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 7–8 (1991).
[Crossref]

Welford, W. T.

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1859).

Winston, R.

R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 9, 1902–1908 (1993).
[Crossref]

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 7–8 (1991).
[Crossref]

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1859).

Appl. Opt. (3)

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

R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 9, 1902–1908 (1993).
[Crossref]

Rend. Sedute Accad. R. Dei Lincei (1)

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota I,” Rend. Sedute Accad. R. Dei Lincei IX(1), 185–189 (1900).

Rend. Sedute R. Accad. Dei Lincei (1)

T. Levi-Civita, “Complementi al teorema di Malus-Dupin, nota II,” Rend. Sedute R. Accad. Dei Lincei IX(1), 237–245 (1900).

Other (6)

G. Salmon, A Treatise on the Higher Plane Curves (Hodges & Smith, Dublin, 1852), Chap. 1.

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 7–8 (1991).
[Crossref]

J. Plücker, System der Analytischen Geometrie (Berlin, 1835).

J. L. Coolidge, A History of Geometrical Methods (Clarendon, Oxford, U.K., 1940), Chap. II.II.2.

W. B. Elmer, Optical Design of reflectors2nd ed. (Wiley, New York, 1980), Chaps. 4,6 and 10.

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1859).

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Figures (8)

Fig. 1
Fig. 1

â is the unit vector pointing toward the edge of the source, k is a parameterization of the reference line, and û is the unit vector pointing in the direction of the edge ray desired at the reference location t. The reflector contour is specified by R.

Fig. 2
Fig. 2

As the observer proceeds along the reference line, the desired irradiance translates into a desired leading edge. This in turn determines the reflector profile. The arrows indicate the directions of the iterations.

Fig. 3
Fig. 3

The start and the end of the reflector cause a rise zone and a decay zone, respectively, each subtending an angle equal to that of the source.

Fig. 4
Fig. 4

The reflector is designed so that each side illuminates both sides of the target region. Thus an observer sees two reflections in addition to the source S.

Fig. 5
Fig. 5

Power from the source and from the reflections on each side add up to produce an angular power distribution (vertical axis) proportional to 1/ cos2(θ) and thus a constant irradiance on a distant plane from a finite-sized cylindrical source of uniform brightness. Angles A–E correspond to the edge rays marked in Fig. 7.

Fig. 6
Fig. 6

CEC-type reflector profile, which produces a constant irradiance on a distant plane from a cylindrical source of constant brightness and unit diameter.

Fig. 7
Fig. 7

Particular edge rays that correspond to the angles designated in Fig. 5. At the largest angle for which constant irradiance occurs, the radiated power comes from the source’s reflection between R1 and R2.

Fig. 8
Fig. 8

Closest approach distance of edge rays as a function of the angle θ. In this representation, the distance between edge rays at a fixed angle is a measure of the respective contributions to total radiated power.

Equations (4)

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R ( t ) = k ( t ) + D û ( t ) .
d R ( t ) d t [ â R ( t ) û ( t ) ] .
d D d t = d k / d t ( â û ) + D d û / d t â 1 â û .
E = B [ sin ( θ R ) sin ( θ L ) ] ,

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