Can one make a laser out of cardboard?

TL;DR

This paper investigates how Lambertian reflective tubes in 2D and 3D affect light collimation and brightness, revealing that 3D tubes produce an infinitely bright spot at the center as the source recedes.

Contribution

It provides a mathematical analysis of light behavior in Lambertian tubes, highlighting the difference in collimation and brightness growth between two and three dimensions.

Findings

In 2D, the exiting rays are approximately collimated.

In 3D, the brightness at the center becomes infinite as the source moves away.

The study characterizes the optical effects of Lambertian reflections in semi-infinite tubes.

Abstract

We consider two dimensional and three dimensional semi-infinite tubes made of "Lambertian" material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three dimensional tube will see "infinitely bright" spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away.