Abstract
We show the existence of a lens, when its lower face is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves a given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper face of the lens satisfies a pde of Monge-Ampère type.
Highlights
We solve the following inverse problem in geometric optics concerning the design of a lens: rays are emitted from a planar source with unit direction e(x) and energy density I(x) for every x ∈
The surface σC,w is parametrized by f (x, C, w) = (φ(x), u(φ(x))) + d(x, C, w)m(x), where (φ(x), u(φ(x))) is the point of incidence of the ray with direction e(x) on the graph of u, m(x) is the unit direction of the refracted ray at (φ(x), u(φ(x))), and d(x, C, w) designates the length of the trajectory of the ray with direction m(x) inside the lens. d(x, C, w) is given in (2.3) below, and the constant C is chosen so that d is positive; see Figure 2(b)
It is proved in [13] that if a lens sandwiched between the lower surface u and the upper surface f, refracts all rays with direction e(x) into the direction w, and f is a regular surface at each point, the upper surface is parametrized by f (x, C, w) = (φ(x), u(φ(x))) + d(x, C, w)m(x) with d(x, C, w) given by (2.3)
Summary
We solve the following inverse problem in geometric optics concerning the design of a lens: rays are emitted from a planar source with unit direction e(x) and energy density I(x) for every x ∈. Of the parameters involved, whereas to avoid singularities one needs to control the curvature of the surface u and that of the potential h, recall e = Dh. In Section 4, we construct refracting surfaces σ so that the lens sandwiched between u and σ refracts incident rays with direction e(x), x ∈ , into a far field target ∗; see Figure 1. In this case, u is assumed to be concave, h convex, and u, , ∗ and e are so that σC,w satisfies the conditions in Theorem 2.5 for each w ∈ ∗. + (1 + κ1κ2)|u(φ(y)) − u(φ(x))|, which concludes the proof of the proposition
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