The eikonal equation and the ray limit

Begin with the scalar Helmholtz model in a transparent isotropic medium of real index n at fixed angular frequency. Write the complex field as $\psi=A\exp(ik_0S)$, with real leading amplitude A, vacuum wavenumber $k_0=2\pi/\lambda_0$, and eikonal S measured in metres. The dimensionless phase is $k_0S$. This scalar starting point already omits vector polarization effects and requires care near material boundaries; it is not a complete Maxwell derivation for every spatially varying medium.

Substitution into the Helmholtz equation produces terms proportional to the square, first power, and zeroth power of $k_0$. The leading square term vanishes when the gradient magnitude of S equals n. Slow amplitude variation means that the neglected amplitude-curvature term is small compared with the leading oscillatory terms, typically when the wavelength in the medium is much shorter than the relevant variation length. A short vacuum wavelength alone is insufficient if an aperture edge or a sharp feature introduces a competing small scale.

Surfaces of constant S are wavefronts. Their normal is its gradient, so the unit ray tangent in an isotropic medium is $\mathbf t=\nabla S/n$. Along a ray, the rate of change of S with arclength is n. Integrating shows that the difference in eikonal values equals optical path along that ray, provided a consistent local phase branch is chosen. The local wavevector is $k_0\nabla S$, with magnitude $k_0n$ and wavelength $\lambda_0/n$.

Taking a derivative of the eikonal relation along its gradient recovers $d(n\mathbf t)/ds=\nabla n$, joining phase geometry to the GRIN ray equation. Initial or boundary wavefront data are still required; specifying only n does not identify which family of rays a source launches. The eikonal may need several branches where distinct ray families reach the same point. A single-valued smooth phase field cannot describe all possible interference from those branches without additional construction.

The next-order imaginary terms give $2\nabla A\cdot\nabla S+A\nabla^2S=0$, equivalent to the displayed divergence law. It conserves scalar flux $A^2\nabla S$ through a ray tube. In a uniform medium with cross sections perpendicular to the rays, amplitude therefore scales as the inverse square root of tube area. This is why a spreading beam grows dimmer without absorption. In a varying index, the factor n enters that flux, so the same area-only rule cannot be transferred unchanged.

At a caustic the geometric tube cross section can vanish, making this leading amplitude diverge. The actual wave field remains finite after an appropriate diffraction or uniform asymptotic treatment. Turning points, interference fringes, narrow apertures, wavelength-scale guiding, and abrupt interfaces also demand supplementary wave analysis or boundary matching. These failures do not mean every ray direction is useless; they mean the approximation is nonuniform in a particular region. Compare scales and observables before deciding whether a ray, scalar wave, or vector electromagnetic model is adequate.