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Fermat's Principle and Snell's Law

Source lecture(s): PHY622 Ch. 2

Intuition

Light takes the path of stationary optical path length. Not "shortest" — stationary, which includes maxima and saddle points, and the distinction matters for mirages and gravitational lensing. From this one statement the whole of geometrical optics follows, including the law of refraction that Snell found empirically.

\[\delta\int n\,ds = 0\]

Deriving Snell's law

Light goes from \(A\) in a medium of index \(n_1\) to \(B\) in \(n_2\), crossing a plane interface. Let the crossing point be at horizontal position \(x\). The travel time is

\[T(x) = \frac{n_1\sqrt{a^2 + x^2}}{c} + \frac{n_2\sqrt{b^2 + (d-x)^2}}{c}\]

Setting \(dT/dx = 0\):

\[\frac{n_1x}{\sqrt{a^2+x^2}} = \frac{n_2(d-x)}{\sqrt{b^2+(d-x)^2}}\]

and recognising each fraction as a sine of the angle from the normal:

\[\boxed{\;n_1\sin\theta_1 = n_2\sin\theta_2\;}\]

Snell's law is a stationarity condition. This is the simplest possible illustration of the variational method — a single variable rather than a whole function — and it is worth doing by hand once, because everything later is the same idea with an infinite-dimensional space of paths.

The lifeguard

The standard intuition, and it is a good one. A lifeguard on sand must reach a swimmer in water. Running is fast, swimming is slow. The quickest route is not the straight line: it bends, spending more distance on the fast medium. Set (running speed)/(swimming speed) \(= n_2/n_1\) and the optimal path obeys exactly Snell's law.

Stationary, not minimal

Fermat originally said "least time", and it is usually wrong to say so:

  • Concave mirror. A ray reflecting off the inside of an ellipse from one focus to the other takes a path that is a maximum among nearby paths, not a minimum.
  • Mirages and gravitational lensing. Multiple images occur precisely because several distinct paths are each stationary. If stationarity meant minimality there could be only one image.

The correct statement is \(\delta T = 0\). The same correction applies to Hamilton's principle, which is stationary action, not least action, despite the traditional name.

What it connects to

  • Hamilton's principle. Same structure with time as parameter, and the optical–mechanical analogy runs deep: \(n\,ds\) ↔ \(p\,dq\). Hamilton built his mechanics by noticing this, and de Broglie and Schrödinger later exploited it — wave mechanics is to classical mechanics as wave optics is to ray optics.
  • Geodesics. In a medium of varying \(n\), light follows a geodesic of the optical metric. In general relativity light follows a geodesic of spacetime; gravitational lensing can be computed with an effective refractive index.
  • Wave optics. Fermat's principle is the stationary-phase limit of the path integral over all routes — nearby paths interfere constructively only near a stationary one. Feynman's formulation makes "why does light take that path?" answerable: it takes all of them, and the others cancel.

Common mistakes

  • Saying "least time". Stationary. Maxima occur, and multiple stationary paths are what makes multiple images possible.
  • Applying it where the wavelength is not small. It is the geometrical-optics limit; diffraction is exactly where it fails.
  • Forgetting \(n\) can vary continuously. The interface derivation is a special case; in a gradient-index medium the Euler–Lagrange equation gives a curved ray, which is why mirages bend and why fibre optics works.

Knowledge graph position

Prerequisites: calculus of variations. Leads to: Hamilton's principle, geometrical optics, the optical–mechanical analogy and hence wave mechanics.

Quiz

Q1 (conceptual). Why is "least time" the wrong statement of Fermat's principle?

Answer

The correct condition is stationary time, \(\delta T = 0\). Reflection inside a concave mirror gives a path that is a local maximum; and multiple images in mirages and gravitational lensing exist precisely because several distinct paths are separately stationary, which minimality would forbid.

Q2 (computational). Light passes from air (\(n=1\)) into glass (\(n=1.5\)) at 30° from the normal. Find the refracted angle.

Answer

\(\sin\theta_2 = (1)(\sin30°)/1.5 = 0.5/1.5 = 0.333\), so \(\theta_2 = 19.5°\) — bent toward the normal, as it must be on entering a slower medium.

Q3 (MCQ). The deep analogy between Fermat's principle and Hamilton's principle is that:

  • (a) both minimise distance
  • (b) both are stationarity conditions, with \(n\,ds\) playing the role of \(p\,dq\)
  • (c) both apply only to straight lines
  • (d) both require constant velocity
Answer

(b). Hamilton noticed this correspondence and built his mechanics around it; it is the historical route by which de Broglie and Schrödinger arrived at wave mechanics.