Graded-index rays and fiber guiding

For an isotropic medium, stationary optical path gives the vector ray equation $d(n\mathbf t)/ds=\nabla n$, where $\mathbf t$ is a unit tangent and s is arclength. A transverse index gradient changes direction as the ray propagates. In a nearly axial system with small slope, replace arclength by z and take n nearly equal to its on-axis value in the derivative coefficient. The transverse equation then relates ray curvature to the transverse index gradient.

Choose the local profile $n(y)\simeq n_0(1-g^2y^2/2)$, where g has units of inverse length. Index is highest at the axis and decreases away from it. Its gradient is approximately $-n_0g^2y$, giving a restoring curvature. This is a local paraxial model requiring $|gy|\ll1$ and small ray slopes. Extending the quadratic expression to arbitrarily large radius would eventually give an unphysical negative index, so the stated profile is never a global material specification.

The paraxial solution depends on both launch height and launch slope. A ray launched parallel to the axis at a positive height reaches the axis after a quarter of the spatial oscillation period, with negative slope. At half a period it returns to zero slope on the opposite side. The full pitch is $2\pi/g$ along z, not a time period. A quarter-pitch rod can therefore turn a parallel ray distribution into a focusing distribution at its exit.

In reduced coordinates with $u=n_0\theta$ to this order, the GRIN segment is a rotation-like matrix with cosine diagonal terms and sine coupling terms. Its determinant is one, preserving the first-order invariant. It differs from the damping-free oscillator lab because the laboratory oscillator starts at unit displacement with zero velocity and uses time; here the independent variable is propagation distance and physical launch conditions vary. The analogy is useful only after those variables and limitations are declared.

For a straight step-index fiber, a higher-index core is surrounded by lower-index cladding. A meridional ray must strike the core-cladding wall beyond the core-side critical angle to remain guided in the ideal ray picture. At a flat input face, Snell’s law links the external launch angle to the internal axial angle. Combining the two conditions gives $n_{\mathrm{ext}}\sin\beta_{\max}=\sqrt{n_{\mathrm{core}}^2-n_{\mathrm{clad}}^2}$, with beta measured from the fiber axis outside the entrance.

The external index must be retained: immersion changes the accepted external angle even though the core-cladding index contrast is unchanged. This elementary expression concerns meridional rays and an ideal straight fiber, not arbitrary skew-ray trajectories, bend losses, or coupling efficiency from a structured field. When the core size is only a few wavelengths, discrete electromagnetic modes replace a dense ray bundle. A graded profile can reduce modal travel-time differences, but a sinusoidal ray drawing alone does not establish a fiber’s dispersion or bandwidth.