## 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

where *γ*
_{1}, *γ*
_{2} are given, and *f*
_{
i
}
*(0) = 0, f*
_{
i
}
*(−y) = f*
_{
i
}
*(y)*. We then let *X*
_{2} → *X*
_{1}, *Y*
_{2} → *Y*
_{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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### Cite this article

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