An optical lens for focusing two pairs of points (Q1113354)

In the plane a lens is constructed which focuses all the rays from a given point \(X_ i\) on the x-axis at another given point \(Y_ i\) on the x-axis, for \(i=1,2\). The lens is of the form \[ \{F_ 1(y)\leq x\leq F_ 2(y),\quad | y| \leq y_ 0\} \] with \(F_ 1(0)\) and \(F_ 2(0)\) given. The lens is constructed such that the rays do not cross the x- axis, and it is shown that for \(X_ 1<X_ 2\) and \(Y_ 1<Y_ 2\) a unique solution with analytic \(F_ 1\) and \(F_ 2\) exists. Afterwards the limit \(X_ 1\to X_ 2\), \(Y_ 1\to Y_ 2\) is considered and it is shown that the limiting lens is in general non-symmetric, which contrasts with an earlier result of the authors [Arch. Ration. Mech. Anal. 99, 147-164 (1987; Zbl 0634.35063)], where they prove that for \(X_ 1=X_ 2\) and \(Y_ 1=Y_ 2\) there exists in the class of symmetric lenses a unique analytic solution.