Abstract

The goal of the optical design of luminaires and other radiation distributors is to attain the desired illumination on the target with a given source while minimizing losses. Whereas the required design procedure is well known for situations in which the source can be approximated as a point or as a line, the development of a general analytical design method for extended sources began only recently. One can obtain a solution for extended sources by establishing a one-to-one correspondence between target points and edge rays. Here the possible solutions in two dimensions (cylindrical sources) are identified, based on only one reflection for the edge rays. The solutions depend on whether the “image” on the reflector is bound by rays from the near or the far edge of the source. For each case there are two solutions that could be called converging and diverging by analogy with imaging optics. Counting the topological choices for the boundaries of the “image” we obtain a complete classification of the building blocks from which luminaires can be designed. One can construct interesting hybrid configurations by combining these building blocks. Thus one can gain a great deal of flexibility for tailoring designs to specific requirements. The differential equation for the reflector is shown to have an analytical solution. Explicit results are presented for symmetric configurations with the target at infinity.

© 1994 Optical Society of America

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References

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  1. W. B. Elmer, The Optical Design of Reflectors (Wiley, New York, 1980;TLA Lighting Consultants, Inc., Salem, Mass., 1989). In the 1989 edition, see in particular p. 92 on the extinction curves and Section 6.3, p. 95ff on the design for the point of line sources.
  2. J. M. Gordon, P. Kashin, A. Rabl, “Nonimaging reflectors for efficient uniform illumination,” Appl. Opt. 31, 6027–6035 (1992).
    [Crossref] [PubMed]
  3. R. Winston, H. Ries, “Nonimaging reflectors as functionals of the desired irradiance,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
    [Crossref]
  4. A. Rabl, “Edge-ray method for analysis of radiation transfer among specular reflectors,” Appl. Opt. 33, 1248–1259 (1994).
    [Crossref] [PubMed]
  5. A. Rabl, “Non-imaging optics reflector designs,” first annual rep. contract RP8012-14 with Electric Power Research Institute (Centre d'Energétique, Ecole des Mines, 60 boulevard St. Michel, 75272 Paris Cedex 06, France, 1992).
  6. J. M. Gordon, A. Rabl, “Nonimaging compound parabolic concentrator-type reflectors with variable extreme direction,” Appl. Opt. 31, 7332–7338 (1992).
    [Crossref] [PubMed]
  7. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981). Chap. 9, pp. 281– 318.
  8. E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.
  9. H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
    [Crossref]
  10. S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

1994 (2)

1993 (1)

1992 (2)

Cess, R. D.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

Elmer, W. B.

W. B. Elmer, The Optical Design of Reflectors (Wiley, New York, 1980;TLA Lighting Consultants, Inc., Salem, Mass., 1989). In the 1989 edition, see in particular p. 92 on the extinction curves and Section 6.3, p. 95ff on the design for the point of line sources.

Gordon, J. M.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981). Chap. 9, pp. 281– 318.

Kashin, P.

Rabl, A.

Ries, H.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981). Chap. 9, pp. 281– 318.

Sparrow, E. M.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

Winston, R.

Wolfram, S.

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

Other (5)

A. Rabl, “Non-imaging optics reflector designs,” first annual rep. contract RP8012-14 with Electric Power Research Institute (Centre d'Energétique, Ecole des Mines, 60 boulevard St. Michel, 75272 Paris Cedex 06, France, 1992).

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981). Chap. 9, pp. 281– 318.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

W. B. Elmer, The Optical Design of Reflectors (Wiley, New York, 1980;TLA Lighting Consultants, Inc., Salem, Mass., 1989). In the 1989 edition, see in particular p. 92 on the extinction curves and Section 6.3, p. 95ff on the design for the point of line sources.

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Figures (9)

Fig. 1
Fig. 1

Two approaches to the design of TED's. Target point T receives the radiation both directly from the source S and from the “image” (the active part of ℛ) on the reflector ℛ. One edge of the “image” is fixed at the edge of ℛ. (a) Image bounded by the ray from O, the far edge of the source. (b) Image bounded by the ray from O, the near edge of the source.

Fig. 2
Fig. 2

Four TED types if one edge of the “image” is on the edge of the reflector (symmetric designs with the target at infinity): O, the edge of the source chosen as the design basis; O′, the other edge of the source; R0, the starting point; R1, the end point. The devices shown produce a constant flux map for |θ| < |θ1| with the θ1 values indicated. (a) Far edge, diverging, θ1 = 30°; (b) near edge, diverging, θ1 = 26.9°; (c) far edge, converging, θ1 = 30°; (d) near edge, converging, θ1 = 25°.

Fig. 3
Fig. 3

Two TED types if both sides of the “image” are bounded by edge rays from the source (as solved by Ries and Winston9). Arrows indicate how the “image” moves with target point T. (a) Diverging TED (compound hyperbolic concentrator type); (b) converging TED (compound elliptical concentrator type).

Fig. 4
Fig. 4

Definition of coordinates and angles; θ and ϕ are positive in the clockwise direction.

Fig. 5
Fig. 5

Derivation of the differential equation [Eq. (2)] for a reflector.

Fig. 6
Fig. 6

Far-edge-diverging TED. The origin O of the r, ϕ coordinates at the left edge of source OO′ (thick gray line). The right reflector extends from R0 to R1. The “image” (thick black line) seen from the θ direction extends from R0 to R.

Fig. 7
Fig. 7

Near-edge-diverging TED. Origin O of the r, ϕ coordinates at the right-hand edge of source OO′. The right reflector extends from R0 to R1. The “image” (thick black line) seen from the θ direction extends from R0 to R. (a) Geometry of the design that is to produce constant illuminance on a distant target up to θ1 = 26.9°. (b) Flux map: illuminance of the target (normalized by maximum) versus tan(θ). (The wiggles are due to numerical inaccuracies in the ray-trace calculation.)

Fig. 8
Fig. 8

Solution with a contiguous image and source for a flux map proportional to 1 + |tan(θ)|. Source width s = 1. The solution is for θ1 = 30° and θ2 = 43.9°. (a) Reflector (solid lines) and source (thick line between -0.5 and 0.5). (b) Flux map: illuminance of target (normalized by maximum) versus tan(θ).

Fig. 9
Fig. 9

Hybrid design: The inner portion is designed with the source and image contiguous. In the outer portion the source and image are disjointed. The reflector is shown by solid lines and the source by the thick line between −0.5 and 0.5. The dashed lines show some edge rays. The flux map is constant for |θ| < θ1 = 36°, decreasing to zero at θ2 = 52°.

Tables (1)

Tables Icon

Table 1 Classification of TED Types Based on One-Reflection Edge Rays in Two Dimensions

Equations (31)

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2 α = ϕ θ .
d r r d ϕ = tan ( α ) .
L ( θ ) = I ( θ ) cos 2 ( θ ) .
L ( θ ) = r ( ϕ ) sin ( ϕ θ ) r 0 sin ( ϕ 0 θ ) + s cos ( θ ) ,
p ( θ ) = r ( ϕ ) sin ( ϕ θ ) ,
p ( θ ) = L ( θ ) + r 0 sin ( ϕ 0 θ ) s cos ( θ ) .
d α d θ = sin ( α ) cos ( α ) d log [ p ( θ ) ] d θ sin 2 ( α ) .
f dir ( θ ) = { 1 for | θ | < | θ O | [ tan ( θ O ) tan ( θ ) ] y 0 / s for | θ O | < | θ | < | θ O | , 0 for | θ O | < | θ |
u = 1 / tan ( α )
d u d θ + u d log p ( θ ) d θ = 1 .
d [ up ( θ ) ] d θ = p ( θ ) ,
u ( θ ) = 1 p ( θ ) θ m θ p ( θ ) d θ .
p 1 ( θ ) = r 0 sin ( ϕ 0 θ ) + s cos ( θ ) s cos 2 ( θ ) .
θ m 1 θ p 1 ( θ ) d θ = r 0 [ cos ( ϕ 0 θ ) cos ( ϕ 0 θ m 1 ) ] + s [ sin ( θ ) sin ( θ m 1 ) tan ( θ ) + tan ( θ m 1 ) ] .
tan ( ϕ 0 / 2 ) = r 0 sin ( ϕ 0 ) r 0 [ cos ( ϕ 0 ) cos ( ϕ 0 θ m 1 ) ] + s [ tan ( θ m 1 ) sin ( θ m 1 ) ] ,
p 2 ( θ ) = r 0 sin ( ϕ 0 θ ) s cos 2 ( θ ) + y 0 × [ tan ( θ O ) cos ( θ ) sin ( θ ) ] .
θ m 2 θ p 2 ( θ ) d θ = r 0 [ cos ( ϕ 0 θ ) cos ( ϕ 0 θ m 2 ) ] s [ tan ( θ ) tan ( θ m 2 ) ] + y 0 { tan ( θ O ) [ sin ( θ ) sin ( θ m 2 ) ] + cos ( θ ) cos ( θ m 2 ) } ,
tan ( 79.17 ° ) = p 2 ( θ = 10.17 ° ) θ m 2 θ = 10.17 ° p 2 ( θ ) d θ ,
r ( ϕ = π / 2 ) / s = g / s = p 2 ( θ = θ 1 ) / cos ( θ 1 )
tan ( π 4 θ 1 2 ) = p 2 ( θ = θ 1 ) θ m 2 θ 1 p 2 ( θ ) d θ
p 2 ( θ = θ 1 ) θ m 2 θ 1 p 2 ( θ ) d θ = p 1 ( θ = θ 1 ) θ m 1 θ 1 p 1 ( θ ) d θ
tan ( ϕ 0 / 2 ) = p 1 ( θ = θ 1 ) θ m 1 θ 1 p 1 ( θ ) d θ
r 0 2 = y 0 2 + ( x 0 0.5 s ) 2 ,
ϕ 0 = π θ 0 ,
tan ( θ O ) = ( x 0 0.5 s ) / y 0 ,
tan ( θ O ) = ( x 0 + 0.5 s ) / y 0 .
S
T
A
S
T

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