Abstract

Nonimaging reflectors that are an extension of the design principle that was developed for compound parabolic concentrator type devices are proposed for illumination applications. The optical designs presented offer maximal lighting efficiency while they retain sharp angular control of the radiation and highly uniform flux densities on distant target planes. Our results are presented for symmetrical configurations in two dimensions (troughlike reflectors) for flat and for tubular sources. For fields of view of practical interest (half-angle in the 30–60° range), these devices can achieve minimum-to-maximum intensity ratios of ~ 0.7, while they remain compact and incur low reflective losses.

© 1992 Optical Society of America

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Corrections

J. M. Gordon, Peter Kashin, and Ari Rabl, "Nonimaging reflectors for efficient uniform illumination: errata," Appl. Opt. 32, 2303-2303 (1993)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-32-13-2303

References

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  1. D. E. Spencer, L. L. Montgomery, J. F. Fitzgerald, “Macrofocal conics as reflector contours,” J. Opt. Soc. Am. 55, 5–11 (1965).
    [CrossRef]
  2. D. G. Burkhard, D. C. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
    [CrossRef]
  3. D. G. Burkhard, D. L. Shealy, “Specular aspheric surface to obtain a specified irradiance from discrete or continuous line source radiation: design,” Appl. Opt. 14, 1279–1284 (1975).
    [CrossRef] [PubMed]
  4. W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4, pp. 54–75; Chap. 6, pp. 85–123; Chap. 7, pp. 124–127; Chap. 10, pp. 142–186; Chap. 14, pp. 201–214; Part 5, pp. 226–248.
  5. W. B. Elmer, “The optics of reflectors for illumination,” IEEE Trans. Ind. Appl. IA-19, 776 (1983).
    [CrossRef]
  6. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), Chap. 4, pp. 53–76; Chap. 6, pp. 99–114; Chap. 9, pp. 147–163; Chap. 10, pp. 165–199; Chap. 11, pp. 201–221.
  7. A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 5, pp. 125–133; Chap. 6, pp. 147–169; Chap. 7, pp. 170–195; Chap. 8, pp. 196–211.
  8. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), Chap. 7, pp. 170–234.
  9. H. C. Hottel, A. F. Sarofim, Radiation Transfer (McGraw-Hill, New York, 1978), Chap. 2, pp. 25–71.
  10. F. Kreith, M. S. Bohn, Principles of Heat Transfer, 4th ed. (Harper & Row, New York, 1986), Chap. 9, pp. 457–470.
  11. A. Rabl, “Radiation transfer through specular passages,” Int. J. Heat Mass Transfer 20, 323–330 (1977).
    [CrossRef]
  12. M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
    [CrossRef]
  13. A. Rabl, “Solar concentrators with maximal concentration for cylindrical absorbers,” Appl. Opt. 15, 1871–1873 (1976).
    [CrossRef] [PubMed]
  14. H. P. Baum, J. M. Gordon, “Geometric characteristics of ideal nonimaging (CPC) solar collectors with cylindrical absorber,” Sol. Energy 33, 455–458 (1984).
    [CrossRef]
  15. A. Rabl, “Lighting technology: luminaire performance,” Rev. Energ. 424, 482–488 (1990).

1990 (1)

A. Rabl, “Lighting technology: luminaire performance,” Rev. Energ. 424, 482–488 (1990).

1985 (1)

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

1984 (1)

H. P. Baum, J. M. Gordon, “Geometric characteristics of ideal nonimaging (CPC) solar collectors with cylindrical absorber,” Sol. Energy 33, 455–458 (1984).
[CrossRef]

1983 (1)

W. B. Elmer, “The optics of reflectors for illumination,” IEEE Trans. Ind. Appl. IA-19, 776 (1983).
[CrossRef]

1977 (1)

A. Rabl, “Radiation transfer through specular passages,” Int. J. Heat Mass Transfer 20, 323–330 (1977).
[CrossRef]

1976 (1)

1975 (2)

D. G. Burkhard, D. C. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

D. G. Burkhard, D. L. Shealy, “Specular aspheric surface to obtain a specified irradiance from discrete or continuous line source radiation: design,” Appl. Opt. 14, 1279–1284 (1975).
[CrossRef] [PubMed]

1965 (1)

Baum, H. P.

H. P. Baum, J. M. Gordon, “Geometric characteristics of ideal nonimaging (CPC) solar collectors with cylindrical absorber,” Sol. Energy 33, 455–458 (1984).
[CrossRef]

Bohn, M. S.

F. Kreith, M. S. Bohn, Principles of Heat Transfer, 4th ed. (Harper & Row, New York, 1986), Chap. 9, pp. 457–470.

Burkhard, D. G.

D. G. Burkhard, D. C. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

D. G. Burkhard, D. L. Shealy, “Specular aspheric surface to obtain a specified irradiance from discrete or continuous line source radiation: design,” Appl. Opt. 14, 1279–1284 (1975).
[CrossRef] [PubMed]

Carvalho, M. J.

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

Collares-Pereira, M.

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

Elmer, W. B.

W. B. Elmer, “The optics of reflectors for illumination,” IEEE Trans. Ind. Appl. IA-19, 776 (1983).
[CrossRef]

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4, pp. 54–75; Chap. 6, pp. 85–123; Chap. 7, pp. 124–127; Chap. 10, pp. 142–186; Chap. 14, pp. 201–214; Part 5, pp. 226–248.

Fitzgerald, J. F.

Gordon, J. M.

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

H. P. Baum, J. M. Gordon, “Geometric characteristics of ideal nonimaging (CPC) solar collectors with cylindrical absorber,” Sol. Energy 33, 455–458 (1984).
[CrossRef]

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, Radiation Transfer (McGraw-Hill, New York, 1978), Chap. 2, pp. 25–71.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), Chap. 7, pp. 170–234.

Kreith, F.

F. Kreith, M. S. Bohn, Principles of Heat Transfer, 4th ed. (Harper & Row, New York, 1986), Chap. 9, pp. 457–470.

Montgomery, L. L.

Rabl, A.

A. Rabl, “Lighting technology: luminaire performance,” Rev. Energ. 424, 482–488 (1990).

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

A. Rabl, “Radiation transfer through specular passages,” Int. J. Heat Mass Transfer 20, 323–330 (1977).
[CrossRef]

A. Rabl, “Solar concentrators with maximal concentration for cylindrical absorbers,” Appl. Opt. 15, 1871–1873 (1976).
[CrossRef] [PubMed]

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 5, pp. 125–133; Chap. 6, pp. 147–169; Chap. 7, pp. 170–195; Chap. 8, pp. 196–211.

Sarofim, A. F.

H. C. Hottel, A. F. Sarofim, Radiation Transfer (McGraw-Hill, New York, 1978), Chap. 2, pp. 25–71.

Shealy, D. C.

D. G. Burkhard, D. C. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

Shealy, D. L.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), Chap. 7, pp. 170–234.

Spencer, D. E.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), Chap. 4, pp. 53–76; Chap. 6, pp. 99–114; Chap. 9, pp. 147–163; Chap. 10, pp. 165–199; Chap. 11, pp. 201–221.

Winston, R.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), Chap. 4, pp. 53–76; Chap. 6, pp. 99–114; Chap. 9, pp. 147–163; Chap. 10, pp. 165–199; Chap. 11, pp. 201–221.

Appl. Opt. (2)

IEEE Trans. Ind. Appl. (1)

W. B. Elmer, “The optics of reflectors for illumination,” IEEE Trans. Ind. Appl. IA-19, 776 (1983).
[CrossRef]

Int. J. Heat Mass Transfer (1)

A. Rabl, “Radiation transfer through specular passages,” Int. J. Heat Mass Transfer 20, 323–330 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Rev. Energ. (1)

A. Rabl, “Lighting technology: luminaire performance,” Rev. Energ. 424, 482–488 (1990).

Sol. Energy (3)

M. J. Carvalho, M. Collares-Pereira, J. M. Gordon, A. Rabl, “Truncation of CPC solar collectors and its effect on energy collection,” Sol. Energy 35, 393–399 (1985).
[CrossRef]

H. P. Baum, J. M. Gordon, “Geometric characteristics of ideal nonimaging (CPC) solar collectors with cylindrical absorber,” Sol. Energy 33, 455–458 (1984).
[CrossRef]

D. G. Burkhard, D. C. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

Other (6)

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4, pp. 54–75; Chap. 6, pp. 85–123; Chap. 7, pp. 124–127; Chap. 10, pp. 142–186; Chap. 14, pp. 201–214; Part 5, pp. 226–248.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), Chap. 4, pp. 53–76; Chap. 6, pp. 99–114; Chap. 9, pp. 147–163; Chap. 10, pp. 165–199; Chap. 11, pp. 201–221.

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 5, pp. 125–133; Chap. 6, pp. 147–169; Chap. 7, pp. 170–195; Chap. 8, pp. 196–211.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), Chap. 7, pp. 170–234.

H. C. Hottel, A. F. Sarofim, Radiation Transfer (McGraw-Hill, New York, 1978), Chap. 2, pp. 25–71.

F. Kreith, M. S. Bohn, Principles of Heat Transfer, 4th ed. (Harper & Row, New York, 1986), Chap. 9, pp. 457–470.

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Figures (10)

Fig. 1
Fig. 1

Several CPC configurations with field of view 2θa: (a) flat one-sided source; (b) cylindrical source; T1T2 indicates possible truncation.

Fig. 2
Fig. 2

Luminaire–target geometry symmetric about the vertical axis. Height H of the luminaire above the target plane is large relative to luminaire dimensions: x = H tan(θ) is the position on the target and xmax = H tan(θa) is the end point of the target.

Fig. 3
Fig. 3

Example of a CPC with an effective flat source that was generated by an involute CPC (concentration ratio of 1) for the actual tubular source.

Fig. 4
Fig. 4

General arrangement of source S, target T, and aperture A of the optical device with angular limits θl and θr for integration in Eq. (7).

Fig. 5
Fig. 5

CPC with a gap between the source and the reflector. The left-hand reflector A1G1 is the arc of the parabola with focus at S2 and the axis parallel to extreme ray A2S1. The right-hand reflector A2G2 is the arc of the parabola with focus at S1 and the axis parallel to extreme ray A1S2. Truncation to aperture T1T2, with extreme accepted ray θt, is also shown.

Fig. 6
Fig. 6

Examples of angular acceptance functions f(θ) versus θ: solid line, CPC without gaps (g/a = 0); dashed curve, CPC with gaps (g/a = 1.7), both untruncated, with θa = 50°.

Fig. 7
Fig. 7

CPC with a gap between the tubular source and the reflector. One designs the reflector for the ice-cream-cone shaped virtual source V1V2V3V1. Truncation to aperture T1T2, with extreme accepted ray θt, is also shown.

Fig. 8
Fig. 8

Normalized flux map on the target plane for untruncated CPC’s with a flat source and several gap sizes to illustrate identification of optimal gap size and sensitivity of illuminance uniformity to gap size. The abscissa is the normalized target position x/xmax (by symmetry only x ≥ 0 is shown).

Fig. 9
Fig. 9

Normalized flux map on the target plane for CPC’s with a flat source that, at fixed θa, have the relative gap size g/a that maximizes uniformity. The abscissa is a normalized target position x/xmax (by symmetry only x ≥ 0 is shown). The truncated devices are characterized by truncation angle θt: (a) θa = 30°; (b) θa = 40°; (c) θa = 50°; (d) θa = 60°.

Fig. 10
Fig. 10

Normalized flux map on the target plane for CPC’s with a tubular source that, at fixed θa, have the relative gap size g/r that maximizes uniformity. The abscissa is a normalized target position x/xmax (by symmetry only x ≥ 0 is shown). The truncated devices are characterized by truncation angle θt: (a) θa = 30°; (b) θa = 40°; (c) θa = 50°; (d) θa = 60°.

Tables (4)

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Table 1 Optimal Untruncated Luminaires for a Flat Source

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Table 2 Optimal Truncated Luminaires for a Flat Sourcea

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Table 3 Optimal Truncated Luminaires for a Tubular Source

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Table 4 Optimal Untruncated Luminaires for a Tubular Sourcea

Equations (11)

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

E d T A d T = E S A S F S - d T .
A d T F d T - S = A S F S - d T
E d T = E S F d T - S .
F d T - S = F d T - A f ,
f = { 1 if d T is within the field of view 0 otherwise .
E P = { E S F d T - A if d T is within the field of view 0 otherwise .
E P = 1 2 E S θ l θ r cos ( θ ) d θ             in two dimensions ,
E P 1 2 E S cos ( θ ) Δ θ             with Δ θ = Δ a cos ( θ ) r in two dimensions ,
E P 1 2 E S cos ( θ ) Δ θ             with             Δ θ = Δ A cos ( θ ) r 2 .
E P 1 2 E S Δ a H cos 3 ( θ )             in 2 dimensions as r .
E ( x ) = E P = Δ a 2 H E S f ( θ ) cos 3 θ             with x = H tan θ .

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