Power-law index profile

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 ($$n(r)=n_2 \mathrm{\ for\ } r \ge \alpha$$), $$\alpha$$ is the core radius, and $$g$$ is a parameter that defines the shape of the profile. $$\alpha$$ 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.