Power-law index profile

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

${\displaystyle n(r)={\begin{cases}n_{1}{\sqrt {1-2\Delta \left({r \over \alpha }\right)^{g}}}&r\leq \alpha \\n_{1}{\sqrt {1-2\Delta }}&r\geq \alpha \end{cases}}}$

where ${\displaystyle \Delta ={n_{1}^{2}-n_{2}^{2} \over 2n_{1}^{2}},}$

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

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