Power-law index profile
For optical fibers, a power-law index profile is an index of refraction profile characterized by n ( r ) = { n 1 1 − 2 Δ ( r α ) g r ≤ α n 1 1 − 2 Δ r ≥ α {\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 Δ = n 1 2 − n 2 2 2 n 1 2 , {\displaystyle \Delta ={n_{1}^{2}-n_{2}^{2} \over 2n_{1}^{2}},} and n ( r ) {\displaystyle n(r)} is the nominal refractive index as a function of distance from the fiber axis, n 1 {\displaystyle n_{1}} is the nominal refractive index on axis, n 2 {\displaystyle n_{2}} is the refractive index of the cladding, which is taken to be homogeneous ( n ( r ) = n 2 f o r r ≥ α {\displaystyle n(r)=n_{2}\mathrm {\ for\ } r\geq \alpha } ), α {\displaystyle \alpha } is the core radius, and g {\displaystyle g} is a parameter that defines the shape of the profile. α {\displaystyle \alpha } is often used in place of g {\displaystyle g} .