Abstract

For many tasks in illumination and collection it is necessary that the acceptance angle vary along the reflector. If the acceptance-angle function is known, then the reflector profile can be calculated as a functional of it. The total flux seen by an observer from a source of uniform brightness (radiance) is proportional to the sum of the view factor of the source and its reflection. This allows one to calculate the acceptance-angle function that one needs to produce a certain flux distribution and thereby to construct the reflector profile. We demonstrate the method for several examples, including finite-size sources with reflectors directly joining the source.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
    [CrossRef]
  2. W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989), Chap. 3.
  3. H. C. Hottel, “Radiant energy transmission,” in Heat Transmission, 3rd ed., W. H. McAdams, ed. (McGraw-Hill, New York, 1954), Chap. 4.
  4. W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4.4.
  5. R. Winston, “Cone collectors for finite sources,” Appl. Opt. 17, 688–689 (1978).
    [CrossRef] [PubMed]
  6. M. Collares-Pereira, A. Rabl, R. Winston, “Lens–mirror combinations with maximal concentration,” Appl. Opt. 16, 2677–2683 (1977).
    [CrossRef] [PubMed]
  7. A. Rabl, R. Winston, “Ideal concentrators for finite sources and restricted exit angles,” Appl. Opt. 15, 2880–2883 (1976).
    [CrossRef] [PubMed]
  8. J. M. Gordon, A. Rabl, “Nonimaging CPC-type reflectors with variable extreme directions,” submitted to Appl. Opt.
  9. 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]
  10. R. Winston, “Nonimaging optics: optical design at the thermodynamic limit,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 2–6 (1992).
    [CrossRef]

1993 (1)

1991 (1)

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

1978 (1)

1977 (1)

1976 (1)

Collares-Pereira, M.

Elmer, W. B.

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4.4.

Gordon, J. M.

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]

J. M. Gordon, A. Rabl, “Nonimaging CPC-type reflectors with variable extreme directions,” submitted to Appl. Opt.

Hottel, H. C.

H. C. Hottel, “Radiant energy transmission,” in Heat Transmission, 3rd ed., W. H. McAdams, ed. (McGraw-Hill, New York, 1954), Chap. 4.

Rabl, A.

Ries, H.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989), Chap. 3.

Winston, R.

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

R. Winston, “Cone collectors for finite sources,” Appl. Opt. 17, 688–689 (1978).
[CrossRef] [PubMed]

M. Collares-Pereira, A. Rabl, R. Winston, “Lens–mirror combinations with maximal concentration,” Appl. Opt. 16, 2677–2683 (1977).
[CrossRef] [PubMed]

A. Rabl, R. Winston, “Ideal concentrators for finite sources and restricted exit angles,” Appl. Opt. 15, 2880–2883 (1976).
[CrossRef] [PubMed]

R. Winston, “Nonimaging optics: optical design at the thermodynamic limit,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 2–6 (1992).
[CrossRef]

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989), Chap. 3.

Appl. Opt. (4)

Sci. Am. (1)

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

Other (5)

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989), Chap. 3.

H. C. Hottel, “Radiant energy transmission,” in Heat Transmission, 3rd ed., W. H. McAdams, ed. (McGraw-Hill, New York, 1954), Chap. 4.

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, New York, 1980), Chap. 4.4.

J. M. Gordon, A. Rabl, “Nonimaging CPC-type reflectors with variable extreme directions,” submitted to Appl. Opt.

R. Winston, “Nonimaging optics: optical design at the thermodynamic limit,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1528, 2–6 (1992).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Reflector profile represented in polar coordinates R(ϕ) with the source at the origin. The reflected radiation has an angle θ with the optical axis y and α with the normal to the reflector.

Fig. 2
Fig. 2

Acceptance-angle function that produces a constant irradiance on a distant plane from a narrow one-sided Lambertian two-dimensional strip source; a = 1.

Fig. 3
Fig. 3

Reflector profile that produces a constant irradiance on a distant plane from a one-sided two-dimensional Lambertian strip source at the origin, R(ϕ = π/2) = 1, a = 1. CEC (inner curve) and CHC-type solutions (outer truncated curve) are shown.

Fig. 4
Fig. 4

Reflector designed to produce a reflected image adjacent to the source. The combined intensity radiated in the direction −θ is determined by the separation of the two edge rays: R sin(2α).

Fig. 5
Fig. 5

Acceptance-angle function that produces a constant irradiance on a distant plane from a finite one-sided Lambertian strip source. There is only a CHC-type solution.

Fig. 6
Fig. 6

The reflector profile that produces a constant irradiance on a distant plane from a finite one-sided Lambertian strip source of width two units. Note that there is only a CHC-type solution, and it is truncated.

Fig. 7
Fig. 7

Deviation of the reflector depicted in Fig. 6 from a true V trough.

Fig. 8
Fig. 8

Desired irradiance distribution on a distant plane perpendicular to the optical plane divided by the irradiance produced along the axis by the source alone. The dashed curves show the irradiance of a truncated device.

Fig. 9
Fig. 9

Angular power distribution corresponding to the irradiance distribution shown in Fig. 8. The dashed curves refer to a truncated device.

Fig. 10
Fig. 10

Acceptance-angle function corresponding to the desired irradiance distribution plotted in Fig. 8.

Fig. 11
Fig. 11

The reflector profile that produces the desired irradiance shown in Fig. 8 on a distant plane from a finite one-sided Lambertian strip source of width two units. Note that there is only a CHC-type solution, and it is truncated.

Fig. 12
Fig. 12

Slope of the reflector as a function of vertical distance from the source.

Fig. 13
Fig. 13

Deviation of the reflector depicted in Fig. 11 from a true V trough.

Fig. 14
Fig. 14

Effect of truncation indicated by the angle up to which the truncated device matches the desired power distribution, plotted as a function of the vertical length of the reflector.

Equations (22)

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

d R 0 D .
[ d log ( R ) ] / d ϕ = tan ( α ) .
α = ( ϕ - θ ) / 2.
log [ R ( ϕ ) / R 0 ] = 0 α ( ϕ ) tan [ α ( ϕ ) ] d ϕ
R ( ϕ ) = R 0 exp { 0 α ( ϕ ) tan [ α ( ϕ ) ] d ϕ } .
P o ( θ ) = P s ( θ ) + M P s ( ϕ ) .
M m = d ϕ d θ .
M s = d μ 1 d μ 2 = sin ( ϕ ) sin ( θ ) ,
M t = M s M m = d cos ( ϕ ) d cos ( θ ) .
ϕ max + θ max = π .
cos θ + | cos ( ϕ ) d ϕ d θ | = a cos 2 ( θ ) .
sin ( ϕ ) = 1 - tan ( θ ) - sin ( θ ) .
R sin ( 2 α ) = P o ( θ ) .
d α d θ = sin ( 2 α ) 2 d log [ P o ( θ ) ] d θ - sin 2 ( α ) .
P T = R ( θ T ) 2 α T π sin γ d γ = R ( θ T ) [ 1 + cos ( 2 α T ) ] .
R ( θ T ) = P o ( θ T ) sin ( 2 α T ) .
1 + cos ( 2 α T ) sin ( 2 α T ) = 1 P o ( θ T ) θ max θ T P o ( ψ ) d ψ = B ( θ T ) .
B ( 0 ) = 1 P o ( 0 ) θ max 0 P o ( ψ ) d ψ = 1 .
2 α = arccos [ ( B 2 - 1 ) / ( B 2 + + 1 ) ] .
ϕ ( θ ) = θ + 2 α .
R ( θ ) = P o ( θ ) [ ( B 2 + 1 ) / 2 B ] .
R ( ϕ ) = R o 1 1 + cos ( ϕ ) .

Metrics