Luneberg lens
Encyclopedia
A Luneburg lens is a spherically symmetric, variable-index
refracting
structure which will form perfect geometrical image
s of two given concentric spheres onto each other. The Luneburg lens is a spherical lens
(usually a ball lens) generally having a gradient
of decreasing refractive index
n radially out from its center. The focusing properties of the Luneburg lens can be achieved through an infinite number of refractive-index solutions. The simplest such solution was proposed by Rudolf Luneburg in 1944. Luneburg's solution for the refractive-index creates two conjugate foci
outside of 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.
Like a Luneburg lens, a gravitational lens
also has a radially decreasing refractive index.
of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index
falls from 
to 1), according to
where
is the radius of the lens. Because the refractive index at the surface is the same as a that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of ellipse
s.
The practical implementation is normally a layered structure 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.
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.
.
It focuses each point on the spherical surface of radius R to the opposite point on the same surface. Within the lens, the paths of the rays are arcs of circles.
The hemispherical Luneburg lens design is smaller and more reliable than traditional satellite dish
antennas.
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
refracting
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
structure which will form perfect geometrical image
Image
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...
s of two given concentric spheres onto each other. The Luneburg lens is a spherical lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
(usually a ball lens) generally having a gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of decreasing refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
n radially out from its center. The focusing properties of the Luneburg lens can be achieved through an infinite number of refractive-index solutions. The simplest such solution was proposed by Rudolf Luneburg in 1944. Luneburg's solution for the refractive-index creates two conjugate foci
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
outside of the lens. The solution takes a simple and explicit form if one focal point
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
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
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
which must be evaluated numerically.
Like a Luneburg lens, a gravitational lens
Gravitational lens
A gravitational lens refers to a distribution of matter between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer...
also has a radially decreasing refractive index.
Luneburg's solution
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 constantDielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive indexRefractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
falls from to 1), according towhere
is the radius of the lens. Because the refractive index at the surface is the same as a that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of ellipseEllipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s.
The practical implementation is normally a layered structure 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
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
, especially to construct efficient microwave antennas and radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
calibration standards.
Luneburg lens radar reflector
A radar reflectorRetroreflector
A retroreflector is a device or surface that reflects light back to its source with a minimum scattering of light. An electromagnetic wave front is reflected back along a vector that is parallel to but opposite in direction from the wave's source. The device or surface's angle of incidence is...
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.
Maxwell's fish-eye
Maxwell's fish-eye 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.It focuses each point on the spherical surface of radius R to the opposite point on the same surface. Within the lens, the paths of the rays are arcs of circles.
Publication and attribution
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 eleven years after Maxwell's death.Luneburg lens satellite antenna
A Luneburg lens made of polyethylene and polystyrene has been designed for train-mounted satellite dishes.The hemispherical Luneburg lens design is smaller and more reliable than traditional satellite dish
Satellite dish
A satellite dish is a dish-shaped type of parabolic antenna designed to receive microwaves from communications satellites, which transmit data transmissions or broadcasts, such as satellite television.-Principle of operation:...
antennas.