Fermat’s principle and optical path

Assume a transparent, isotropic, stationary medium with a real positive phase index $n(\mathbf r)$ at the chosen vacuum wavelength $\lambda_0$. Optical path length is the integral of index times geometric arclength. It has units of metres, although it is not the distance a ruler measures. Multiplying it by the vacuum wavenumber gives propagation phase. A uniform glass segment of length two centimetres and index one and a half therefore contributes three centimetres of optical path.

In a nondispersive medium the speed is $c/n$, so optical path divided by c is travel time. For a dispersive material, however, phase index and group index differ. A narrow pulse’s delay requires the group index, whereas monochromatic ray geometry is usually formulated with phase index at a fixed frequency. Calling every optical-path calculation a pulse arrival-time calculation silently introduces an assumption about dispersion. We retain the monochromatic formulation throughout this derivation.

Place a horizontal interface between media with indices $n_1$ and $n_2$. A source is a perpendicular distance a above it, and the receiver is a perpendicular distance b below it with horizontal displacement X. Let x be the crossing coordinate measured from the source’s vertical projection. Each segment is straight because its index is uniform. The remaining freedom is a single real number, so the path integral becomes an ordinary differentiable function.

Differentiating each square root produces a horizontal displacement divided by that segment’s length. Those ratios are the sines of the respective angles to the interface normal. Setting the derivative to zero balances the index-weighted sines. This route also explains which angle convention belongs in Snell’s law; measuring angles from the interface would replace sines with cosines. Endpoint positions stay fixed while the crossing point moves, which is essential to interpreting the derivative.

For this planar two-segment geometry, positive indices and nonzero perpendicular distances make the second derivative positive. Its stationary crossing is therefore the unique minimum in that restricted family. The same conclusion does not follow for an arbitrary curved reflector or an optical system with multiple images. Stationary paths can have different second variations, and several physical rays can connect the same endpoints. Fermat’s principle is a local variational condition, not an algorithm that compares every possible route.

The wave interpretation helps diagnose the misconception. Nearby nonstationary contributions rapidly acquire different phases and tend to cancel, whereas the phase changes only to second order near a stationary path. This is an asymptotic explanation of rays when wavelengths are small compared with system scales; it does not eliminate diffraction. Near a caustic, several rays crowd together and a naive ray amplitude can diverge even though the wave field remains finite. Stationarity survives as a geometric guide while the amplitude approximation needs repair.