Abstract

The edge-ray principle of nonimaging optics states that nonimaging devices can be designed by the mapping of edge rays from the source to the edge of the target. However, in most nonimaging reflectors, including the compound parabolic concentrator (CPC), at least part of the radiation undergoes multiple reflections, some rays even appear to be reflected infinitely many times, and closer examination reveals that some edge rays of the source are not mapped onto the edge of the target even though the CPC is indeed ideal in two dimensions. Using a topological approach, we refine the formulation of the edge-ray principle to ensure its validity for all configurations. We present two different versions of the general principle. The first involves the boundaries of the different zones corresponding to a different number of reflections. The second version is stated in terms of only a single reflection, but it involves the addition of an auxiliary region of phase space. We discuss the use of the edge-ray principle as a design procedure for nonimaging devices. The CPC is used to illustrate all steps of the argument.

© 1994 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989).
  2. J. C. Minano, J. C. Gonzáles, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [Crossref] [PubMed]
  3. N. Fraidenraich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
    [Crossref] [PubMed]
  4. P. A. Davies, “Edge-ray principle of nonimaging optics,” J. Opt. Soc. Am. A 11, 1256–1259 (1994).
    [Crossref]
  5. J. C. Minano, “Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
    [Crossref]
  6. J. C. Minano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
    [Crossref]
  7. J. C. Minano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1886).
    [Crossref]
  8. A. Rabl, “Edge-ray method for analysis of radiation transfer among specular reflectors,” Appl. Opt. 33, 1248–1259 (1994).
    [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, 7–14 (1991).
    [Crossref]
  10. J. M. Gordon, H. Ries, “Tailored edge-ray concentrators as ideal second stages for Fresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
    [Crossref] [PubMed]
  11. R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
    [Crossref]
  12. H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
    [Crossref]

1994 (3)

1993 (3)

1992 (1)

1985 (2)

J. C. Minano, “Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
[Crossref]

J. C. Minano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
[Crossref]

1886 (1)

Davies, P. A.

Fraidenraich, N.

Gonzáles, J. C.

Gordon, J. M.

Minano, J. C.

Rabl, A.

Ries, H.

Salcedo, I. H.

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, 7–14 (1991).
[Crossref]

Welford, W. T.

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

Winston, R.

H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
[Crossref]

R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
[Crossref]

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

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, 7–14 (1991).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. A (5)

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

J. C. Minano, “Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
[Crossref]

Other (2)

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, 7–14 (1991).
[Crossref]

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

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Figures (7)

Fig. 1
Fig. 1

The CPC is designed so that all rays entering the aperture under an angle from −θ to +θ are reflected to the left-hand or the right-hand edge of the target.

Fig. 2
Fig. 2

Phase space of source and target. The region marked T0S0 is the overlap. The reflector has to map regions of the source marked S>0 onto corresponding target regions T>0.

Fig. 3
Fig. 3

Effective map of the boundary of the regions T and S for the CPC. Not all the boundary of S is mapped onto the boundary of T. A part is mapped onto a sequence of arches inside T, which converge to h. Similarly, the boundary of T originates in a sequence of arches inside S, which converge to g.

Fig. 4
Fig. 4

Close-up view of the map of the boundary of S and T for a CPC. The curves separate regions of rays undergoing different numbers of reflections marked R2, R3, R4, and R5. The reflections converge to h. The effective map is discontinuous at the boundaries.

Fig. 5
Fig. 5

If R denotes the region of rays that will be reflected by a reflector, then R* is the region of rays that have been reflected. The intersection R>1 contains the rays that have been multiply reflected. The part of the rim marked by an arrow is responsible for the discontinuities of the effective map.

Fig. 6
Fig. 6

Effective map of a region undergoing n reflections from source to target. The points on the boundary marked Sa to Sd and the boundary pieces that they delimit are mapped onto the corresponding target region boundary points Ta to Td, respectively, and the points in between.

Fig. 7
Fig. 7

Design of the CPC according to the general edge-ray principle in terms of the primary map. The points h–d–f–e–g–h are mapped into the points h–j–a = b–c–g–h by one reflection. The primary map is continuous on the border g–d between the source and the auxiliary region. This border is mapped onto the arc g–i–j.

Equations (12)

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E ( X Y ) = E ( X ) + E ( Y )             for X Y = .
m ( x ) x x R .
M ( x ) = m n ( x )             if m n + 1 ( x ) = m n ( x ) .
S = i S i with S i S j = for i j , T = i T i with T i T j = for i j , T i = m ( S i ) i .
D n , k D i , j =             for ( n , k ) ( i , j ) .
n = 1 k = 1 n D n , k k = 1 D , k ( T > 0 A ) .
E ( D n , k ) = E ( S n ) = E n k n ,             E ( D , k ) = E ( S ) = E k ,
n = 1 k = 1 n E n + k = 1 E E ( A ) + E ( T > 0 ) .
A = n = 1 m n ( S > 0 ) \ T .
m ( S > 0 A ) = n = 1 m n ( S > 0 ) = T > 0 A .
n = 1 n E n E ( A ) + E ( T > 0 ) .
n = n = 1 n E n / n = 1 E n 1 + E ( A ) / E ( S > 0 ) .

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