Fermat's principle and the ray laws

Light in a medium of index n moves at c/n, so a stretch ds takes n ds/c. Add the pieces along any route and the travel time is T = (1/c) ∫ n ds. The integral is the optical path length, OPL = ∫ n ds: the vacuum distance that would take the same time. It is a length in metres, and it is what a wave actually counts, because the phase accumulated along the route is φ = k₀ OPL = 2π OPL/λ₀, with λ₀ the vacuum wavelength. A 1.00 mm plate of glass with n = 1.52 has OPL 1.52 mm; the extra 0.52 mm is 945 wavelengths of 550 nm light, about 5.9 × 10³ rad of phase that a 1.00 mm gap of air would not have supplied. Fermat's principle in its working form is a statement about this length: of all routes between two fixed points, light follows one for which OPL, and with it T and φ, is stationary under a small deformation of the route, δ(OPL) = 0.

The principle is not a preference of light; it is what a wave does. Every route from A to B contributes an amplitude proportional to e^(iφ), and what arrives at B is their sum. Deform a route slightly and, in general, φ changes at first order, so nearby routes arrive with phases spread over many cycles and cancel in pairs. Only around a route where dφ/dx = 0 does the first-order change vanish; there a whole band of neighbours arrives nearly in phase and reinforces. That band is the ray, and it has a width. Bend a straight path of length L through a point displaced sideways by h at its midpoint: each half grows from L/2 to √((L/2)² + h²) ≈ L/2 + h²/L, so the extra path is 2h²/L. The neighbours stay within half a wavelength, phase within π, out to 2h²/L = λ/2, that is h = ½√(λL). For L = 1.0 m and λ = 500 nm that is 0.35 mm either side: the ray is a tube about 0.7 mm across, and everything the ray laws say applies inside it.

Put a flat surface along the x axis, A a height a above it and B a depth b below, a horizontal distance d apart, and let the route cross at x. Then T(x) = [n₁√(a² + x²) + n₂√(b² + (d − x)²)]/c, and one differentiation gives c dT/dx = n₁ x/√(a² + x²) − n₂ (d − x)/√(b² + (d − x)²) = n₁ sin θ₁ − n₂ sin θ₂, the angles measured from the normal. Setting it to zero is Snell's law. Reflection is the same derivative with the second leg back on the first side, so n₂ = n₁ and sin θᵢ = sin θᵣ: the law of reflection. Away from the stationary point the derivative is a signed error whose sign says which way to move the crossing point, which is how a computer, or the figure below, finds the ray. Differentiate instead with respect to a displacement z of the crossing point sideways, out of the plane containing A, B and the normal: c dT/dz = n₁z/L₁ + n₂z/L₂, zero only at z = 0. That is the third ray law: the refracted and reflected rays lie in the plane of incidence.

Differentiate again for the flat surface: c d²T/dx² = n₁ cos²θ₁/L₁ + n₂ cos²θ₂/L₂, both terms positive, so at a plane interface or plane mirror the ray really is the quickest route and the time landscape is a single bowl. Curve the surface and the sign is no longer guaranteed. An ellipsoidal mirror with A and B at its foci reflects every wall point to B, because the two focal distances sum to the same 2a for every point of the wall: T is constant, d²T/dx² = 0, and the whole wall is stationary. A mirror that curves more tightly than that ellipsoid, touching it at one point P, puts its other points inside the ellipsoid, where the focal sum is smaller, so the path through P is longer than through any neighbour and the real ray is a local maximum of T. A mirror flatter than the ellipsoid gives a minimum. In all three cases the first derivative vanishes at P and the law of reflection holds there. The first derivative fixes the ray; the second only says what kind of stationary point it sits on, and nothing in the physics needs that to be a minimum.

Every imaging equation later in this section replaces sin θ by θ. The series sin θ = θ − θ³/6 + θ⁵/120 − … prices that move: the fractional error of writing sin θ ≈ θ is θ²/6, which is 0.13% at 5°, 0.51% at 10°, 2.0% at 20° and 4.6% at 30°, where the exact figure is already 4.7%. Propagate it through Snell's law. For 25° incidence on glass of index 1.52 the paraxial form n₁θ₁ = n₂θ₂ gives θ₂ = 25°/1.52 = 16.45°; the exact law gives sin θ₂ = 0.4226/1.52 = 0.2780, θ₂ = 16.14°. The paraxial answer is 0.30° high, 1.9%. Substituting the series on both sides shows the fractional error is (θ₁² − θ₂²)/6 = 1.85%: the two cubic terms partly cancel, and the denser side, with its smaller angle, contributes less. Keep the cubic terms and solve θ₂ − θ₂³/6 = (θ₁ − θ₁³/6)/1.52 by iteration and you get 16.139°, within 0.005° of exact. That third-order correction is the term the Seidel theory of aberrations keeps, and spherical aberration is what the paraxial equations lost when they dropped it.

The band of reinforcing paths met above is about √(λL) wide. A ray picture makes sense only if the aperture or obstacle that defines the beam is much wider than that band, so that the stationary path has its neighbourhood to itself. The ratio squared is the Fresnel number, N_F = a²/(λL), for an aperture of half-width a viewed at distance L. A coin of radius 1.0 cm at 1.0 m under 550 nm light has N_F = (10⁻²)²/(5.5 × 10⁻⁷ × 1.0) = 1.8 × 10²: a sharp geometrical shadow. A pinhole of radius 1.0 mm at the same distance has N_F = 1.8, and its patch of light is a diffraction pattern that no ray tracing predicts. In the same way the mirrors and lenses of this section must be smooth on the scale of λ and large compared with it. The formal statement is that geometrical optics is the λ → 0 limit of wave optics, the eikonal limit, in the way that classical mechanics is the ħ → 0 limit of quantum mechanics: sending λ → 0 sends N_F → ∞ and every diffraction angle λ/a to zero, leaving exactly the ray laws.