Power-law index profile
For optical fibers, a power-law index profile is an index of refraction profile characterized by
- <math> 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}</math>
where <math>\Delta = {n_1^2 - n_2^2 \over 2 n_1^2},</math>
and <math> n(r)</math> is the nominal refractive index as a function of distance from the fiber axis, <math>n_1</math> is the nominal refractive index on axis, <math>n_2</math> is the refractive index of the cladding, which is taken to be homogeneous (<math>n(r)=n_2 \mathrm{\ for\ } r \ge \alpha</math>), <math>\alpha</math> is the core radius, and <math>g</math> is a parameter that defines the shape of the profile. <math>\alpha</math> is often used in place of <math>g</math>. Hence, this is sometimes called an alpha profile.
For this class of profiles, multimode distortion is smallest when <math>g</math> takes a particular value depending on the material used. For most materials, this optimum value is approximately 2. In the limit of infinite <math>g</math>, the profile becomes a step-index profile.