Abstract

We establish a fundamental bound on the field of view over which strictly uniform far-field irradiance can be achieved in symmetric two-dimensional (2D, troughlike) and three-dimensional (3D, conelike) illumination systems. Earlier results derived for particular 2D devices are shown to be special cases of the general formula. For a rotationally symmetric 3D luminaire with a Lambertian disk light source and a prescribed uniform core region half-angle θc, no more than tan2c) can be projected within a uniform core region. Hence the efficiency with which such illuminators can produce uniform flux is severely limited for many problems of practical interest. Guided by the tailored edge-ray device formalism for the design of 2D luminaires, we develop a 3D reflector that produces extremely uniform far-field illuminance.

© 1998 Optical Society of America

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References

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  1. R. Winston, H. Ries, “Nonimaging reflectors as functionals of the desired irradiance,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
    [CrossRef]
  2. A. Rabl, J. M. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Appl. Opt. 33, 6012–6021 (1994).
    [CrossRef] [PubMed]
  3. H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
    [CrossRef]
  4. P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
    [CrossRef]
  5. P. T. Ong, J. M. Gordon, A. Rabl, “Tailoring lighting reflectors to prescribed illuminance distributions: compact partial-involute designs,” Appl. Opt. 34, 7877–7887 (1995).
    [CrossRef] [PubMed]
  6. A. Rabl, P. T. Ong, J. M. Gordon, W. Cai, “Iterative algorithm for reflector design for non-isotropic sources,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 16–23 (1995).
    [CrossRef]
  7. P. T. Ong, J. M. Gordon, A. Rabl, “Tailored edge-ray designs for illumination with tubular sources,” Appl. Opt. 35, 4361–4371 (1996).
    [CrossRef] [PubMed]
  8. C. L. Wyatt, Radiometric System Design (Macmillan, New York, 1987).

1996 (1)

1995 (2)

P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
[CrossRef]

P. T. Ong, J. M. Gordon, A. Rabl, “Tailoring lighting reflectors to prescribed illuminance distributions: compact partial-involute designs,” Appl. Opt. 34, 7877–7887 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (1)

Cai, W.

P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
[CrossRef]

A. Rabl, P. T. Ong, J. M. Gordon, W. Cai, “Iterative algorithm for reflector design for non-isotropic sources,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 16–23 (1995).
[CrossRef]

Gordon, J. M.

P. T. Ong, J. M. Gordon, A. Rabl, “Tailored edge-ray designs for illumination with tubular sources,” Appl. Opt. 35, 4361–4371 (1996).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, “Tailoring lighting reflectors to prescribed illuminance distributions: compact partial-involute designs,” Appl. Opt. 34, 7877–7887 (1995).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
[CrossRef]

A. Rabl, J. M. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Appl. Opt. 33, 6012–6021 (1994).
[CrossRef] [PubMed]

A. Rabl, P. T. Ong, J. M. Gordon, W. Cai, “Iterative algorithm for reflector design for non-isotropic sources,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 16–23 (1995).
[CrossRef]

Ong, P. T.

P. T. Ong, J. M. Gordon, A. Rabl, “Tailored edge-ray designs for illumination with tubular sources,” Appl. Opt. 35, 4361–4371 (1996).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, “Tailoring lighting reflectors to prescribed illuminance distributions: compact partial-involute designs,” Appl. Opt. 34, 7877–7887 (1995).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
[CrossRef]

A. Rabl, P. T. Ong, J. M. Gordon, W. Cai, “Iterative algorithm for reflector design for non-isotropic sources,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 16–23 (1995).
[CrossRef]

Rabl, A.

P. T. Ong, J. M. Gordon, A. Rabl, “Tailored edge-ray designs for illumination with tubular sources,” Appl. Opt. 35, 4361–4371 (1996).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, “Tailoring lighting reflectors to prescribed illuminance distributions: compact partial-involute designs,” Appl. Opt. 34, 7877–7887 (1995).
[CrossRef] [PubMed]

P. T. Ong, J. M. Gordon, A. Rabl, W. Cai, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34, 1726–1737 (1995).
[CrossRef]

A. Rabl, J. M. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Appl. Opt. 33, 6012–6021 (1994).
[CrossRef] [PubMed]

A. Rabl, P. T. Ong, J. M. Gordon, W. Cai, “Iterative algorithm for reflector design for non-isotropic sources,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 16–23 (1995).
[CrossRef]

Ries, H.

Winston, R.

Wyatt, C. L.

C. L. Wyatt, Radiometric System Design (Macmillan, New York, 1987).

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Figures (6)

Fig. 1
Fig. 1

Schematic drawing of a luminaire that produces uniform far-field irradiance. Producing a strictly uniform core region of half-angle θ c necessitates projecting a substantial fraction of emitted power into a spillover region out to half-angle θmax.

Fig. 2
Fig. 2

Cross section of a luminaire for a flat one-sided source (OO′) with A = 1; i.e., the projected area of the source equals its total radiating area. At far-field view angle θ the irradiance has contributions directly from the source (thick gray line) and from the image of the source in the reflector (thick black line).

Fig. 3
Fig. 3

Cross section of a 2D luminaire for a tubular source with a partial involute5 (reflector section adjacent to tube), where the source projected area at θ = 0 is less than its actual radiating area (A < 1).

Fig. 4
Fig. 4

Roughly elliptical radiating area comprising the circular source and its image in the reflector.

Fig. 5
Fig. 5

Scale drawing of the cross section of the 3D TED luminaire that produces uniform far-field illuminance within a core region of half-angle 30°.

Fig. 6
Fig. 6

Plot of far-field illuminance normalized to its maximum value against view angle for the luminaire of Fig. 5. The wiggles are characteristic of the calculational accuracy of our ray-trace simulation.

Equations (14)

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L θ = A   cos - d θ ,
F = 0 θ c   L θ d θ ,     2 D   luminaires ,
F = 2   0 θ c   L θ sin θ d θ     3 D luminaires .
F = A   tan θ c ,     2 D   luminaires ,
F = A   tan 2 θ c ,     3D luminaires .
θ c tan - 1 1 / A ,     2 D   luminaires ,
θ c tan - 1 1 / A ,     3 D   luminaires .
F = A   tan θ c / sin ψ max ,     2 D   luminaires ,
F = A   tan 2 θ c / sin 2 ψ max ,     3 D   luminaires .
θ c tan - 1 sin ψ max / A ,     2 D   luminaires ,
θ c tan - 1 sin ψ max / A ,     3 D   luminaires .
L θ = cos - p θ ,
L 3 D θ = const . D L D T = cos - 3 θ .
cos - p θ cos - p θ = cos - 3 θ ,

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