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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 2, Iss. 11 — Nov. 1, 1985
  • pp: 1826–1831

Two-dimensional nonimaging concentrators with inhomogeneous media: a new look

J. C. Miñano  »View Author Affiliations

JOSA A, Vol. 2, Issue 11, pp. 1826-1831 (1985)

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The theory of two-dimensional (2-D) nonimaging concentrators is reviewed. A method for designing 2-D concentrators with inhomogeneous media, in which the refractive-index distribution is an output, is given. This method allows one to choose the shape of the mirrors when they are needed. As a result of the method we obtain a concentrator with maximum theoretical concentration formed by the composition of triangles of three different indices of refraction. The edge-ray principle is proven under several specific assumptions.

© 1985 Optical Society of America

Original Manuscript: October 17, 1984
Manuscript Accepted: March 5, 1985
Published: November 1, 1985

J. C. Miñano, "Two-dimensional nonimaging concentrators with inhomogeneous media: a new look," J. Opt. Soc. Am. A 2, 1826-1831 (1985)

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  1. R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245 (1970). [CrossRef]
  2. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  3. R. Winston, W. T. Welford, “Two-dimensional concentrators for inhomogeneous media,” J. Opt. Soc. Am. 68, 289 (1978). [CrossRef]
  4. W. T. Welford, R. Winston, “Two-dimensional concentrators with refracting optics,” J. Opt. Soc. Am. 69, 917 (1979). [CrossRef]
  5. W. T. Welford, Aberrations of Symmetrical Optical System (Academic, New York, 1974).
  6. J. C. Miñano, “Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A 2, 1821 (1985). [CrossRef]
  7. In the case when the concentrator does not use mirrors, then ω= 0 must be a closed surface, and so the equation u(x, t) = l(x, t) must have as its solution two lines in the x−t plane that connect the boundaries of the entry aperture with those of the collector. It is obvious that Gx= 0 in those lines [Gx= (u− l)/2]. By application of Eq. (4) it is obtained that Gt or Ft must also be equal to zero in those lines. If Ft= 0 (and Gt≠ 0) the value of q for the rays of the upper manifold (q= Ft+ Gt) or the value of q for the rays of the lower manifold (q=Ft−Gt) does not fulfill the second assumption used for proving the edge-ray theorem. Therefore Gx= Gt= 0 in the lines u= l, and so they are level lines of G. In the case when the concentrator uses mirrors we shall see that, since these mirrors must reflect the upper manifold into the lower one and vice versa, they must be placed in level lines of G. Therefore we also conclude that two level lines of G must connect, in the x−t plane, the boundaries of the collector with those of the entry aperture.
  8. R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532 (1979). [CrossRef]
  9. R. Winston, W. T. Welford, “Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field: a new derivation of the compound parabolic concentrator,” J. Opt. Soc. Am. 69, 536 (1979). [CrossRef]
  10. The expression of J in terms of G and F is given byJ=2(Gx2+Gt2Fx2+Ft2)1/2(Fx,Ft)=2(−Gt,Gx).Note that the zero divergence of the vector field J is here reflected in the independence on the order of derivation of the second derivatives of G.
  11. A. Rabl, R. Winston, “Ideal concentrators for finite sources and restricted exit angles,” Appl. Opt. 15, 2880 (1976). [CrossRef] [PubMed]

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