Luneburg 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.

[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.

of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index

The fish-eye, which was first fully described by Maxwell in 1854[5] (and therefore pre-dates Luneburg's solution), has a refractive index varying according to where

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.

[7] 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.

[5] 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,[8] 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.

Cylindrical analogues of the Luneburg lens are also used for collimating light from laser diodes.

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.

A custom-made Luneburg lens antenna can be designed to meet specific requirements, including different sizes, feeds, and operating frequencies.

The phase centre of the feed horn must coincide with the point of focus, but since the phase centre is invariably somewhat inside the mouth of the horn, it cannot be brought right up against the surface of the lens.

In contrast, if multiple feeds are used with a parabolic reflector, all must be within a small angle of the optical axis to avoid suffering coma (a form of de-focussing).

Apart from offset systems, dish antennas suffer from the feed and its supporting structure partially obscuring the main element (aperture blockage); in common with other refracting systems, the Luneburg lens antenna avoids this problem.

This uses just one hemisphere of a Luneburg lens, with the cut surface of the sphere resting on a reflecting metal ground plane.

The arrangement halves the weight of the lens, and the ground plane provides a convenient means of support.

However, the feed does partially obscure the lens when the angle of incidence on the reflector is less than about 45°.

It helped a mass of attendees stay connected and share the most exciting moments without causing major network problems.

The antenna is perfect for events like festivals or parades when people are roaming the grounds and not seated in one location.

In the plane, the circular symmetry of the system makes it convenient to use polar coordinates

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.

In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for

A circle, shaded sky blue at the center, fading to white at the edge. A bundle of parallel red lines enters from the upper right and converges to a point at the opposite edge of the circle. Another bundle does the same from the upper left.
Cross-section of the standard Luneburg lens, with blue shading proportional to the refractive index
Numerical simulation of a Luneburg lens illuminated by a point source at varying positions.
A Luneburg lens converts a point source into a collimated beam when the source is placed at its edge.
A circle, shaded sky blue at the center, fading to white at the edge. A bundle of red curves emanate from a point on the circumference and re-converge at a point at the opposite edge of the circle. Another bundle does the same from the upper left.
Cross-section of Maxwell's fish-eye lens, with blue shading representing increasing refractive index
Type 984 3D radar on HMS Victorious , 1961, using a Luneburg lens