MATH Seminar

Title: A solution to a problem of determining the sides of a lens
Seminar: Analysis and Differential Geometry
Speaker: Hasan Palta of Emory University
Contact: Vladimir Oliker, oliker@mathcs.emory.edu
Date: 2011-09-20 at 4:00PM
Venue: MSC W301
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Abstract: Suppose that a beam of light with the positive vertical direction $\bf{k}$ is crossing a domain $\Omega$ in the horizontal plane $z=0$ with some intensity $I\in L^1(\bar{\Omega})$ and is refracted at both sides of a lens in such a way that the final direction is also $\bf{k}$ and that the beam illuminates a set $T_d$ in the plane $z=d$ with intensity $L\in L^1(\bar{T}_d)$. Let $n_1$ and $n_2$ be the refractive indices of the ambient environment and of the lens, respectively. Such a construction generates a mapping $P:\Omega\to T$ where $T$ is the orthogonal projection of the domain $T_d$ onto $z=0$. We consider the inverse problem of recovering the two sides $z\in C(\bar{\Omega})$ and $w\in C(\bar{T})$ of the lens for given domains $\Omega$ and $T_d$ and the corresponding intensities $I$ and $L$. In analytic formulation, this problem requires a solution to a nonlinear partial differential equation of Monge-Amp\`{e}re type. In this talk, we present a different approach to this problem, describe an algorithm giving approximate solutions using general properties of geometric optics and give some examples.