= Helical rays in a radially graded refractive index
{title2=$\omega^2=-\mu'(R)/[R(\mu(R)+R\mu'(R))]$}
For a positive <refractive index> $\mu(r)$ in <cylindrical coordinates>, a <light ray> with constant radius $R>0$ and varying axial coordinate has the form $\theta(z)=\theta_0+\omega z$ when it exists. <Fermat principle> makes $L=\mu(r)\sqrt{1+r'^2+r^2\theta'^2}$ stationary. Conservation of the momentum conjugate to the cyclic angle makes $\theta'$ constant along a constant-radius path. The radial <Euler-Lagrange equation> is then
$$
\mu'(R)(1+R^2\omega^2)+\mu(R)R\omega^2=0,
$$
giving the displayed formula. A nonzero real <helix> requires $0<-R\mu'(R)/\mu(R)<1$. If $\mu'(R)=0$, $\omega=0$ gives a straight axial <light ray>; a generic radial profile need not support a constant-radius <helix>. Its <helical pitch> is $2\pi/|\omega|$. This ray model is useful for a <graded-index optical fiber> when the <refractive index> varies smoothly and <geometrical optics> applies; it does not by itself impose wave-mode quantization or boundary confinement.
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