A Luneburg lens (original German Lüneburg lens) 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.^[1] 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.^[2][3] These solutions are not unique; the set of solutions are defined by a set of definite integrals which must be evaluated numerically.^[4]

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.

Radar reflector

Luneburg reflectors (the marked protrusion) on an F-35

A radar reflector can be made from a Luneburg lens by metallizing parts of its surface. Radiation from a distant radar transmitter is focussed onto the underside of the metallization on the opposite side of the lens; here it is reflected, and focussed back onto the radar station. A difficulty with this scheme is that metallized regions block the entry or exit of radiation on that part of the lens, but the non-metallized regions result in a blind-spot on the opposite side.

Removable Luneburg lens type radar reflectors are sometimes attached to military aircraft in order to make stealth aircraft visible during training operations, or to conceal their true radar signature. Unlike other types of radar reflectors, their shape doesn't affect the handling of the aircraft.^[9]

For any spherically symmetric lens, each ray lies entirely in a plane passing through the centre of the lens. The initial direction of the ray defines a line which together with the centre-point of the lens identifies a plane bisecting the lens. Being a plane of symmetry of the lens, the gradient of the refractive index has no component perpendicular to this plane to cause the ray to deviate either to one side of it or the other. In the plane, the circular symmetry of the system makes it convenient to use polar coordinates ${\displaystyle (r,\theta )}$ to describe the ray's trajectory.

Given any two points on a ray (such as the point of entry and exit from the lens), Fermat's principle asserts that the path that the ray takes between them is that which it can traverse in the least possible time. Given that the speed of light at any point in the lens is inversely proportional to the refractive index, and by Pythagoras, the time of transit between two points ${\displaystyle (r_{1},\theta _{1})}$ and ${\displaystyle (r_{2},\theta _{2})}$ is

${\displaystyle T=\int _{(r_{1},\theta _{1})}^{(r_{2},\theta _{2})}{\frac {n(r)}{c}}{\sqrt {(r\,d\theta )^{2}+dr^{2}}}={\frac {1}{c}}\int _{\theta _{1}}^{\theta _{2}}n(r){\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta ,}$

where ${\displaystyle c}$ is the speed of light in vacuum. Minimizing this ${\displaystyle T}$ yields a second-order differential equation determining the dependence of ${\displaystyle r}$ on ${\displaystyle \theta }$ along the path of the ray. This type of minimization problem has been extensively studied in Lagrangian mechanics, and a ready-made solution exists in the form of the Beltrami identity, which immediately supplies the first integral of this second-order equation. Substituting ${\displaystyle L(r,r')=n(r){\sqrt {r'^{2}+r^{2}}}}$ (where ${\displaystyle r'}$ represents ${\displaystyle {\tfrac {dr}{d\theta }}}$), into this identity gives

${\displaystyle n(r){\sqrt {r'^{2}+r^{2}}}-n(r){\frac {r'^{2}}{\sqrt {r'^{2}+r^{2}}}}=h,}$

where ${\displaystyle h}$ is a constant of integration. This first-order differential equation is separable, that is it can be re-arranged so that ${\displaystyle r}$ only appears on one side, and ${\displaystyle \theta }$ only on the other:^[1]

${\displaystyle d\theta ={\frac {h}{r{\sqrt {{\big (}n(r){\big )}^{2}r^{2}-h^{2}}}}}\,dr.}$

The parameter ${\displaystyle h}$ is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for ${\displaystyle \theta }$ as a function or ${\displaystyle r}$. In general it provides the relative rates of change of ${\displaystyle \theta }$ and ${\displaystyle r}$, which may be integrated numerically to follow the path of the ray through the lens.

• Ball lens
• BLITS (Ball Lens In The Space) satellite
• Gravitational lenses also have a radially decreasing refractive index.