23.239.7.192: /* Relationship to other complexity classes */

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Relationship to other complexity classes

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	~~There~~ is ~~nothing~~ to ~~write about myself~~ at all.<br>~~Great~~ to be a ~~member~~ of ~~this site~~.<br>~~I really hope I am useful at all~~<br><br>~~My blog post dermology anti-wrinkle cream~~ (~~[http:~~//~~fanbuzzy~~.~~com/~~index.~~php?do~~=/~~profile~~-~~7482~~/~~info~~/ ~~fanbuzzy~~.~~com~~])		[[Image:Grin-lens.png\|frame\|right\|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''' ('''GRIN''') '''optics''' is the branch of [[optics]] covering optical effects produced by a gradual variation of the [[refractive index]] of a material. Such variations can be used to produce [[lens (optics)\|lens]]es with flat surfaces, or lenses that do not have the [[Optical aberration\|aberrations]] typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that is spherical, axial, or radial.

			== In nature ==
			The [[Lens (anatomy)\|lens]] of the [[eye]] is the most obvious example of gradient-index optics in nature. In the [[human eye]], the [[refractive index]] of the lens varies from approximately 1.406 in the central layers down to 1.386 in less dense layers of the lens (Hecht 1987, p. 178). This allows the eye to image with good resolution and low aberration at both short and long distances (Shirk ''et al.'', 2006).

			Another example of gradient index optics in nature is the common [[mirage]] of a pool of water appearing on a road on a hot day. The pool is actually an image of the sky, apparently located on the road since light rays are being [[refracted]] (bent) from their normal straight path. This is due to the variation of refractive index between the hot, less dense air at the surface of the road, and the denser cool air above it. The variation in temperature (and thus density) of the air causes a gradient in its refractive index, causing it to increase with height (Tsiboulia, 2003). This index gradient causes refraction of light rays (at a shallow angle to the road) from the sky, bending them into the eye of the viewer, with their apparent location being the road's surface.

			The Earth's atmosphere acts as a GRIN lens, allowing observers to see the sun for a few minutes after it is actually below the horizon, and observers can also view stars that are below the horizon (Tsiboulia, 2003). This effect also allows for observation of electromagnetic signals from satellites after they have descended below the horizon, as in [[radio occultation]] measurements.

			== Applications ==
			The ability of GRIN lenses to have flat surfaces simplifies the mounting of the lens, which makes them useful where many very small lenses need to be mounted together, such as in [[photocopier]]s and [[image scanner\|scanner]]s. The flat surface also allows a GRIN lens to be easily fused to an [[optical fiber]], to produce [[collimated light\|collimated]] output.

			In imaging applications, GRIN lenses are mainly used to reduce aberrations. The design of such lenses involves detailed calculations of aberrations as well as efficient manufacture of the lenses. A number of different materials have been used for GRIN lenses including optical glasses, plastics, [[germanium]], [[zinc selenide]], and [[sodium chloride]].

			Certain optical fibres ([[graded-index fiber\|graded-index fibres]]) are made with a radially-varying refractive index profile; this design strongly reduces the [[dispersion (optics)\|modal dispersion]] of a [[multi-mode optical fiber]]. The radial variation in refractive index allows for a sinusoidal height distribution of [[ray (optics)\|rays]] within the fibre, preventing the rays from leaving the [[Core (optical fiber)\|core]]. This differs from traditional optical fibres, which rely on [[total internal reflection]], in that all modes of the GRIN fibres propagate at the same speed, allowing for a higher temporal bandwidth for the fibre (Moore, 1980).

			== Manufacture ==
			GRIN lenses are made by several techniques:
			* [[Neutron]] irradiation (Sinai, 1971) – [[Boron]]-rich glass is bombarded with neutrons 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) – An organic [[monomer]] is partially polymerized using [[ultraviolet light]] at varying intensities to give a refractive gradient.
			* [[Ion exchange]] (Hensler, 1975) – Glass is immersed into a liquid melt with [[lithium]] ions. As a result of [[diffusion]], [[sodium]] ions in the glass are partially exchanged with lithium ones, with a larger amount of exchange occurring at the edge. Thus the sample obtains a gradient material structure and a corresponding gradient of the refractive index.
			* Ion Stuffing (Mohr, 1979) – [[Phase (matter)\|Phase]] separation of a specific glass causes pores to form, which can later be filled using a variety of [[Salt (chemistry)\|salt]]s or concentration of salts to give a varying gradient.

			== History ==
			In 1854, [[James Clerk Maxwell\|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 fish-eye\|''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, [[Robert W. Wood\|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 [[Rudolf Luneburg\|R. K. Luneburg]] where he described a [[Luneburg lens\|lens]] that focuses incident parallel rays of light onto a point 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 has some usefulness in [[microwave]] applications.

			== Theory ==
			An inhomogeneous gradient-index lens possesses a refractive index whose change follows the function
			<math>n=f(x,y,z)</math> 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 (optics)\|ray of light]] joining any two points of a [[Optical medium\|medium]], is [[Stationary process\|stationary]] relative to its value for any nearby curve joining the two points. The light path integral is given by the equation
			:<math alt="L = (integral from So to S) n ds">L=\int_{S_o}^{S}n\,ds</math>, where ''n'' is the refractive index and ''S'' is the arc length of the curve. If [[Cartesian coordinate]]s are used, this equation is modified to incorporate the change in arc length for a spherical gradient, to each physical dimension:
			:<math alt="L = (integral from So to S) n(x,y,z)(x'^2 + y'^2 + z'^2)^(1/2) ds">L=\int_{S_o}^{S}n(x,y,z)\sqrt{x'^{2}+y'^{2}+z'^{2}}\, ds</math>
			where prime corresponds to d/d''s'' (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 the method of production used. For example, GRIN lenses made from a radial gradient index material, such as [[SELFOC Microlens]] (Flores-Arias ''et al.'', 2006), have a refractive index that varies according to:
			:<math alt="n_r = n_o (1- (Ar^2/2))">n_{r}=n_{o}\left ( 1-\frac{A r^2}{2} \right )</math>, where ''n''<sub>''r''</sub> the refractive index at a distance, ''r'', from the [[optical axis]]; ''n''<sub>o</sub> is the design index on the optical axis, and ''A'' is a positive constant.

			==References==
			{{reflist}}
			*Flores-Arias M T, Bao C, Castelo A, Perez M V, Gomez-Reino C, (2006). ''Optics Communications'' 266, 490-494
			*Hecht, Eugene (1987). ''Optics'', 2nd ed., Addison Wesley, ISBN 0-201-11609-X.
			*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
			*Tsiboulia, A B (2003). "Gradient Index (GRIN) Lenses". In Ronald G. Driggers. ''Encyclopedia of Optical Engineering'', Volume 1. New York, NY: Marcel Dekker. 675-683. ISBN 9780824742508.
			*Wood, R W (1905). ''Physical Optics'', p. 71. Macmillan, New York.

			{{Glass science}}

			[[Category:Optics]]
			[[Category:Fiber optics]]
			[[Category:Glass engineering and science]]

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23.239.7.192: /* Relationship to other complexity classes */

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q906766 (Report Errors)

en>Mastergreg82 at 08:59, 6 July 2012