Luneburg lens

A Luneburg lens (original German Lüneburg-Linse) is a spherically symmetric gradient-index lens. A typical Luneburg lens's refractive index n decreases radially from the center to the outer surface. They can be made for use with electromagnetic radiation from visible light to radio waves.

For certain index profiles, the lens will form perfect geometrical images of two given concentric spheres onto each other. There are an infinite number of refractive-index profiles that can produce this effect. The simplest such solution was proposed by Rudolf Luneburg in 1944. Luneburg's solution for the refractive index creates two conjugate foci outside the lens. The solution takes a simple and explicit form if one focal point lies at infinity, and the other on the opposite surface of the lens. J. Brown and A. S. Gutman subsequently proposed solutions which generate one internal focal point and one external focal point. These solutions are not unique; the set of solutions are defined by a set of definite integrals which must be evaluated numerically.

Each point on the surface of an ideal Luneburg lens is the focal point for parallel radiation incident on the opposite side. Ideally, the dielectric constant ${\displaystyle \epsilon _{r}}$ of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index ${\displaystyle n}$ falls from ${\displaystyle {\sqrt {2}}}$ to 1), according to

where ${\displaystyle R}$ is the radius of the lens. Because the refractive index at the surface is the same as that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of ellipses.

Maxwell's fish-eye lens is also an example of the generalized Luneburg lens. The fish-eye, which was first fully described by Maxwell in 1854 (and therefore pre-dates Luneburg's solution), has a refractive index varying according to

where ${\displaystyle n_{0}}$ is the index of refraction at the center of the lens and ${\displaystyle R}$ is the radius of the lens's spherical surface. The index of refraction at the lens's surface is ${\displaystyle n_{0}/2}$. The lens images each point on the spherical surface to the opposite point on the surface. Within the lens, the paths of the rays are arcs of circles.

The properties of this lens are described in one of a number of set problems or puzzles in the 1853 Cambridge and Dublin Mathematical Journal. The challenge is to find the refractive index as a function of radius, given that a ray describes a circular path, and further to prove the focusing properties of the lens. The solution is given in the 1854 edition of the same journal. The problems and solutions were originally published anonymously, but the solution of this problem (and one other) were included in Niven's The Scientific Papers of James Clerk Maxwell, which was published 11 years after Maxwell's death.

In practice, Luneburg lenses are normally layered structures of discrete concentric shells, each of a different refractive index. These shells form a stepped refractive index profile that differs slightly from Luneburg's solution. This kind of lens is usually employed for microwave frequencies, especially to construct efficient microwave antennas and radar calibration standards. Cylindrical analogues of the Luneburg lens are also used for collimating light from laser diodes.