Engineering:Power-law index profile
For optical fibers, a power-law index profile is an index of refraction profile characterized by
- [math]\displaystyle{ 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]\displaystyle{ \Delta = {n_1^2 - n_2^2 \over 2 n_1^2}, }[/math]
and [math]\displaystyle{ n(r) }[/math] is the nominal refractive index as a function of distance from the fiber axis, [math]\displaystyle{ n_1 }[/math] is the nominal refractive index on axis, [math]\displaystyle{ n_2 }[/math] is the refractive index of the cladding, which is taken to be homogeneous ([math]\displaystyle{ n(r)=n_2 \mathrm{\ for\ } r \ge \alpha }[/math]), [math]\displaystyle{ \alpha }[/math] is the core radius, and [math]\displaystyle{ g }[/math] is a parameter that defines the shape of the profile. [math]\displaystyle{ \alpha }[/math] is often used in place of [math]\displaystyle{ g }[/math]. Hence, this is sometimes called an alpha profile.
For this class of profiles, multimode distortion is smallest when [math]\displaystyle{ 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]\displaystyle{ g }[/math], the profile becomes a step-index profile.
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