Archive for Rational Mechanics and Analysis

, Volume 101, Issue 1, pp 57–83

An optical lens for focusing two pairs of points

  • Avner Friedman
  • Bryce McLeod
Article

DOI: 10.1007/BF00281783

Cite this article as:
Friedman, A. & McLeod, B. Arch. Rational Mech. Anal. (1988) 101: 57. doi:10.1007/BF00281783

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 Xiare focused by the lens at a given point yi, for i = 1, 2. The points X1, X2, Y1, Y2 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 γ1, γ2 are given, and fi(0) = 0, fi(−y) = fi(y). We then let X2X1, Y2Y1 and investigate the limiting lens. We show that this limit is generally not a symmetric lens, i.e., f1 + f2 ≇ 0.

Copyright information

© Springer-Verlag GmbH & Co 1988

Authors and Affiliations

  • Avner Friedman
    • 1
    • 2
  • Bryce McLeod
    • 1
    • 2
  1. 1.University of MinnesotaMinneapolis
  2. 2.Oxford UniversityEngland