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An optical lens for focusing two pairs of points

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Abstract

We construct an optical lens in the (x, y)-plane which focuses two pairs of points, i.e., all the rays from a given point X i are focused by the lens at a given point y i , for i = 1, 2. The points X 1, X 2, Y 1, Y 2 lie on the x-axis and the lens has the form

$$\left\{ {\gamma _{\text{1}} {\text{ }} + {\text{ }}f_{\text{1}} {\text{(}}y{\text{) }}\underline \leqslant {\text{ }}x{\text{ }}\underline \leqslant {\text{ }}\gamma _{\text{2}} {\text{ }} + {\text{ }}f_{\text{2}} {\text{(}}y{\text{)}},{\text{ }}\left| y \right|{\text{ }}\underline \leqslant {\text{ }}y_{\text{0}} } \right\}$$

where γ 1, γ 2 are given, and f i (0) = 0, f i (−y) = f i (y). We then let X 2X 1, Y 2Y 1 and investigate the limiting lens. We show that this limit is generally not a symmetric lens, i.e., f 1 + f 2 ≇ 0.

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References

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Friedman, A., McLeod, B. An optical lens for focusing two pairs of points. Arch. Rational Mech. Anal. 101, 57–83 (1988). https://doi.org/10.1007/BF00281783

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Keywords

  • Neural Network
  • Complex System
  • Nonlinear Dynamics
  • Electromagnetism
  • Optical Lens