Power-law index profile

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For optical fibers, a power-law index profile is an index of refraction profile characterized by

$n(r) = \begin{cases} n_1 \sqrt{1-2\Delta\left({r \over \alpha}\right)^g} & r \le \alpha\\ n_1 \sqrt{1-2\Delta} & r \ge \alpha \end{cases}$

where $\Delta = {n_1^2 - n_2^2 \over 2 n_1^2},$

and $n (r)$ is the nominal refractive index as a function of distance from the fiber axis, $n 1$ is the nominal refractive index on axis, $n 2$ is the refractive index of the cladding, which is taken to be homogeneous ( $n(r)=n_2 \mathrm{\ for\ } r \ge \alpha$ ), $α$ is the core radius, and $g$ is a parameter that defines the shape of the profile. $α$ is often used in place of $g$ . Hence, this is sometimes called an alpha profile.

For this class of profiles, multimode distortion is smallest when $g$ takes a particular value depending on the material used. For most materials, this optimum value is approximately 2. In the limit of infinite $g$ , the profile becomes a step-index profile.