Gradient-index optics

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

Gradient-index optics is the branch of materials science dealing with the production and characterization of gradually changing refractive index lenses. These lenses are able to eliminate the aberrations caused by traditional spherical lenses without requiring variation to the shape of the lens. Gradient index lenses may have a refraction gradient that is spherical, axial, or radial.

Gradient-Index Lenses in Nature

A variety of inhomogeneous lenses exists in nature that allow for different functions. The most obvious lens in nature is the lens present in the eye. Nature progressively varies the refractive index of the lens in order to optimise the optics required for survival. Eagles have the ability to maintain focus and high resolution at large distances. The eyes of an antelope have a broad field of view, which allows them to detect predators. In humans, the lens is able to highly resolve and reduce aberration for both short and long distances (Shirk et al, 2006). The quality of the lens does, however, deteriorate with age with noticeable effects usually occurring after the age of 40 in humans. Another phenomenon related to the varying refractive indexes of materials in nature is the Mirage. Since refractive index generally increases with density of the material, it is seen that cool air has a higher refractive index than hot air. Therefore, in a desert, light passes from the cool air to the warm air causing the path of the light ray reaching an observer to bend, thus giving them a misconception of the object’s spatial displacement, i.e. The object may appear closer to the observer than it actually is.

History

In 1854, J C Maxwell suggested a lens whose refractive index distribution would allow for every region of space to be sharply imaged. Known as the Maxwell Fisheye Lens, it involves a spherical index function and would be expected to be spherical in shape as well (Maxwell, 1854). This lens, however, is impractical to make and has little usefulness since, only points on the surface and within the lens are sharply imaged and extended objects suffer from extreme aberrations. In 1905, R W Wood used a dipping technique creating a gelatin cylinder with a refractive index gradient that varied symmetrically with the radial distance from the axis. Disk shaped slices of the cylinder were later shown to have plane faces with radial index distribution. He showed that even though the faces of the lens were flat, they acted like converging and diverging lens depending on whether the index was a decreasing or increasing relative to the radial distance (Wood, 1905). In 1964, was published a posthumous book of R. K. Luneburg where he discovered a lens that converge all rays of light onto a point which is located on the opposite surface of the lens (Luneburg, 1964). This also limits the applications of the lens, in that it is difficult to be used to focus visual light, however, it was thought to have had some usefulness in microwave applications.

Theory

When considering aberrations, it is necessary to interpret the projections of light in terms of Gaussian Principles. If a ray is emitted from point P, and passes near point P’, and Δx and Δy are the deviations from P’, then Δx and Δy are measures of the aberration for the rays perceived (Figure 1) (Marchand, 1976). Inhomogeneous GRIN lenses attempt to reduce the aberrations Δx and Δy to zero (See Figure 2).

Figure 1: Cartesian Interpretation of Aberrations. Rays of light leaving point P are refracted at the lens plane, and pass through point A. Point P’ is the point at which the light rays would ideally converge if an aberration free lens was used. Point A and point P’ differ by the values Δx and Δy.

Figure 2: How a Two Dimensional Gradient Index Lens Works. Gradient index lens converge all light rays onto a common point, by refracting light through a changing refractive index (many different refractive indices).

An inhomogeneous gradient-index lens possesses a refractive index whose change follows the function: n=f(x,y,z) of the coordinates of the region of interest in the medium. According to Fermat’s principle, the light path integral (L), taken along a ray of light joining any two points of a medium, is stationary relative to its value for any nearby curve joining the two points. The light path integral is given by the equation:

$\int _{S_{o}}^{S}n\,dS$

Where n is the refractive index and S is the arc length of the curve. If Cartesian coordinates are used, this equation is modified to incorporate the change in arc length for a spherical gradient, to each physical dimension:

$\int _{S_{o}}^{S}n(x,y,z){\sqrt {x'^{2}+y'^{2}+z'^{2}}}\,dS$

where prime corresponds to d/ds (Marchand, 1978). The light path integral is able to characterize the path of light through the lens in a qualitative manner, such that the lens may be easily reproduced in the future.

The refractive index gradient of GRIN lenses can be mathematically modelled according to its method of production. For example, GRIN lenses made from a radial gradient index material, such as SELFOC® (Flores-Arias et al, 2006), express a refractive index that varies according to:

$n_{r}=n_{o}\left(1-{\frac {Ar^{2}}{2}}\right)$

Where, n_r the refractive index at a distance, r, from the optical axis; is the design index on the optical axis and A is a positive constant.

Design and Manufacture of GRIN Lenses

The GRIN lens is designed in two separate phases. In the first phase, the lens is evaluated by analytical techniques. The second phase involves the design of the lens. When considering the manufacturing of GRIN lenses, there are two important features of any technique: the depth of the gradient; and the magnitude of the change in index-of-refraction, Δn, throughout the lens (Moore, 1980). Production techniques involve:

• Neutron Irradiation (Sinai, 1971) – Boron-rich glass is bombarded with neutrons in order to cause a change in the boron concentration, and thus the refractive index of the lens.
• Chemical Vapour Deposition (Keck et al., 1975) – Involving the deposition of different glass with varying refractive indexes, onto a surface to produce a cumulative refractive change.
• Partial Polymerisation (Moore, 1973) – Organic monomer is partially polymerized using UV light at varying intensities in order to give a refractive gradient.
• Ion Exchange (Hensler, 1975) – Ions, such as lithium replace sodium ions already present in a glass substrate. When this technique is modulated spatially, a gradient index lens is formed.
• Ion Stuffing (Mohr, 1979) – Phase separation of a specific glass causes pores to form, which can later be filled using a variety of salts or concentration of salts to give a varying gradient.

Applications

The main uses of GRIN lenses involve applications in telecommunications and optical imaging. In telecommunications, a long fibre many kilometres in length but only a 20-100µm diameter have refractive gradients engineered to vary radially from the centre, with refractive index decreasing as distance from the centre increases. This allows for a sinusoidal height distribution of the ray within the fibre, preventing the ray from touching the walls. This differs from the traditional optical fibres which rely on total internal reflection, in that all modes of the GRIN fibres propagate at the same velocity, thus allowing for a higher temporal bandwidth for the fibre (Moore, 1980). Imaging using GRIN lenses is used to mainly reduce the aberrations and increase focus. This involves detailed calculations of aberrations as well as the efficient manufacture of the lenses. Recently, a number of different methods of gradient production have been implemented, such as, optical glasses; plastics; germanium; zinc selenide; and sodium chloride.

References

Flores-Arias M T, Bao C, Castelo A, Perez M V, Gomez-Reino C, (2006). “Optics Communicaitons”266, 490-494

Hensler J R, "Method of Producing a Refractive Index Gradient in Glass," U.S. Patent 3,873,408 (25 Mar. 1975).

Keck D B and Olshansky R, "Optical Waveguide Having Optimal Index Gradient," U.S. Patent 3,904,268 (9 Sept. 1975).

Luneberg R K, (1964). “Mathematical Theory of Optics.” Univ. of California Press, Berkeley.

Marchand E W (1976). J. Opt. Soc. Amer. 66, 1326.

Marchand E W (1978). “Gradient Index Optics”. New York Academic Press.

Maxwell, J.C. (1854). Cambridge and Dublin Math. J. 8, 188

Mohr R K, Wilder J A, Macedo P B, and Gupta P K, in Digest of Topical Meeting on Gradient- index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WAL.

Moore, D.T. (1980). Applied Optics. 19, 1035-1038

Moore R S, "Plastic Optical Element Having Refractive Index Gradient," U.S. Patent 3,718,383 (Feb. 1973).

Shirk J.S, Sandrock M, Scribner D, Fleet E, Stroman R, Baer E, Hilter A. (2006) NRL Review pp 53-61

Sinai P, (1970). Applied Optics. 10, 99-104

Wood, R.W. (1905). “Physical Optics,” p. 71. Macmillan, New York.