Theory
An inhomogeneous gradient-index lens possesses a refractive index whose change follows the function 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
- ,
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:
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 the method of production used. For example, GRIN lenses made from a radial gradient index material, such as SELFOC (Flores-Arias et al., 2006), have a refractive index that varies according to:
- ,
where nr the refractive index at a distance, r, from the optical axis; no is the design index on the optical axis, and A is a positive constant.
Read more about this topic: Gradient-index Optics
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