Editing Gradient-index optics (section)

== 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 = \int_{S_0}^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 = \int_{S_0}^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.''<ref>{{Cite book|last=Marchand|first=Erich W.|title=Gradient index optics|date=1978|publisher=Academic Press|isbn=978-0124707504|location=New York|oclc=4497777}}</ref> 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]],<ref>{{Cite journal|last=Flores-Arias|first=M.T.|last2=Bao|first2=C.|last3=Castelo|first3=A.|last4=Perez|first4=M.V.|last5=Gomez-Reino|first5=C.|date=2006-10-15|title=Crossover interconnects in gradient-index planar optics|journal=Optics Communications|language=en|volume=266|issue=2|pages=490–494|doi=10.1016/j.optcom.2006.05.049|issn=0030-4018|bibcode=2006OptCo.266..490F}}</ref> 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> is 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.