Abstract

Some form of the edge-ray principle is used to design most nonimaging systems. (Only systems based on geometrical optics are considered.) For proving certain statements of this principle, the optical system is considered to be surrounded by an enclosure that any ray emerging from the system must intersect. A phase space, of topology Sphere × Disk, corresponding to the intersection of rays with the enclosure is introduced, and the system gives rise to a mapping f among points of this space. The proofs hold only if f is continuous, which is not the case for all real systems. Discontinuities in f may be caused by (1) tangential incidence of a ray with a surface, (2) incidence where the radius of curvature of a surface is zero, (3) transition from refraction to total internal reflection, and (4) intersection of different types of optical surface. Although the condition of the continuity of f is used in the proofs, even among systems in which continuity is absent it is difficult in practice to find counterexamples to the edge-ray principle formulated.

© 1994 Optical Society of America

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References

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  1. M. Bassett, W. T. Welford, R. Winston, “Nonimaging optics for flux concentrators,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 163–226.
  2. D. Carroll, “Alpha Solarco’s high-concentration photovoltaic array development program,” in 20th IEEE Photovoltaic Specialists’ Conference (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1138–1143.
    [Crossref]
  3. P. A. Davies, “Design of single-surface spherical lenses as secondary concentrators for photovoltaic cells,” Pure Appl. Opt. 2, 315–324 (1993).
    [Crossref]
  4. C. Miñano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
    [Crossref]
  5. J. C. Miñano, J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [Crossref] [PubMed]
  6. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), p. 54.
  7. R. S. Mackay, J. D. Meiss, Hamiltonian Dynamical Systems (reprint selection) (Institute of Physics, Bristol, UK, 1987).
  8. Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25(2), 137–189 (1970).
    [Crossref]

1993 (1)

P. A. Davies, “Design of single-surface spherical lenses as secondary concentrators for photovoltaic cells,” Pure Appl. Opt. 2, 315–324 (1993).
[Crossref]

1992 (1)

1986 (1)

1970 (1)

Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25(2), 137–189 (1970).
[Crossref]

Bassett, M.

M. Bassett, W. T. Welford, R. Winston, “Nonimaging optics for flux concentrators,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 163–226.

Carroll, D.

D. Carroll, “Alpha Solarco’s high-concentration photovoltaic array development program,” in 20th IEEE Photovoltaic Specialists’ Conference (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1138–1143.
[Crossref]

Davies, P. A.

P. A. Davies, “Design of single-surface spherical lenses as secondary concentrators for photovoltaic cells,” Pure Appl. Opt. 2, 315–324 (1993).
[Crossref]

González, J. C.

Mackay, R. S.

R. S. Mackay, J. D. Meiss, Hamiltonian Dynamical Systems (reprint selection) (Institute of Physics, Bristol, UK, 1987).

Meiss, J. D.

R. S. Mackay, J. D. Meiss, Hamiltonian Dynamical Systems (reprint selection) (Institute of Physics, Bristol, UK, 1987).

Miñano, C.

Miñano, J. C.

Sinai, Ya. G.

Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25(2), 137–189 (1970).
[Crossref]

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), p. 54.

M. Bassett, W. T. Welford, R. Winston, “Nonimaging optics for flux concentrators,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 163–226.

Winston, R.

M. Bassett, W. T. Welford, R. Winston, “Nonimaging optics for flux concentrators,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 163–226.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), p. 54.

Appl. Opt. (1)

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

Pure Appl. Opt. (1)

P. A. Davies, “Design of single-surface spherical lenses as secondary concentrators for photovoltaic cells,” Pure Appl. Opt. 2, 315–324 (1993).
[Crossref]

Russ. Math. Surv. (1)

Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25(2), 137–189 (1970).
[Crossref]

Other (4)

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), p. 54.

R. S. Mackay, J. D. Meiss, Hamiltonian Dynamical Systems (reprint selection) (Institute of Physics, Bristol, UK, 1987).

M. Bassett, W. T. Welford, R. Winston, “Nonimaging optics for flux concentrators,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 163–226.

D. Carroll, “Alpha Solarco’s high-concentration photovoltaic array development program,” in 20th IEEE Photovoltaic Specialists’ Conference (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1138–1143.
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Optical system placed inside an enclosure. The source and the target belong to the surface of the enclosure.

Fig. 2
Fig. 2

Example based on sets in the plane in which, although the boundary of Y falls inside Z, neither nor C[int(Y)] is contained in the interior of Z. This situation is possible when (Z) is not a connected set. Thus, in lemma 2, (Z) is required to be connected.

Fig. 3
Fig. 3

Example of an optical system that can be interpreted superficially as a counterexample to the edge-ray principle. The lines and shaded areas inside the rectangle in the phase-space diagram represent rays that fail to reach the target s represents the position on the source, measured from the left-hand end, and θ represents the angle to the horizontal of the emitted rays. The outline of the rectangle represents the edge rays, all of which reach the interior of the target except for four (marked by circles) that encounter the corners of the prism, where they cannot be traced further by geometrical optics.

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