Mathematical retroreflectors

Abstract

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object notched angle is a new one; a proof of its retroreflectivity is given.Mathematics subject classifications: 37D50, 49Q10

Comparative Study on Efficiency of Mirror Retroreflectors

“…It was proved in [1,5] that there exist Tube retroreflectors with retro-reflectivity ratio arbitrarily close to 1. More precisely, suppose that the sides of the retroreflector are perfectly reflecting, k = 1; then we have r(B(n, d, ε)) − −− → certain family of retroreflectors with the size of small segments going to zero, ε → 0, and their number going to infinity, n = n(ε) → ∞, and with fixed distance d between the segments.…”

“…Here we provide analytical derivation of the retro-reflectivity ratio r(α, k) in the limit when β → 0, δ → 0. It can be made rigorous with using methods from [5]. However, here we limit ourselves by numerical verification of our heuristics.…”

“…It is even not known if such retroreflectors really exist. What we know, however, is that there exist asymptotically perfect families of retroreflectors [1,5,7]. For arbitrarily small ε > 0, one can choose a retroreflector in such a family so that the portion of light rays reflected from it in wrong directions is smaller than ε.…”

“…Thus, the retroreflector Notched angle depends on three parameters: α, β, and δ. It is proved in [5] that for a certain family of retroreflectors B(α, β, δ) with δ = δ(α) − −− → α→0 0, β = β(α), β/α − −− → α→0 0 the retro-reflectivity ratio r(B(α, β(α), δ(α))) goes to 1 as α → 0.…”

“…Thus, the retroreflector Tube depends on three parameters: n, d, and ε. It is proved in [1,5] that for a certain family of retroreflectors B(n, d, ε) with ε → 0, n = n(ε) → ∞, and with d fixed, the retro-reflectivity ratio r(B(n(ε), d, ε)) goes to 1 as ε → 0. Notched angle is obtained by replacing the lateral sides of the rectangle with a broken line whose segments are parallel and perpendicular to the inlet (see Fig.…”