The path that takes the longest time

Fermat’s principle is usually taught in four words: light takes the quickest path. It is a good rule, it gives Snell’s law and the law of reflection in a line of algebra each, and it is wrong in a way that cannot be patched by adding a caveat, because the counterexample can be built to order.

The three mirrors differ in nothing but curvature. The source, the detector, the ray and the reflection law are identical across the set. What changes is whether the neighbouring paths are longer or shorter than the one taken — and light has no way of knowing which, because the thing it responds to is not the length but the rate of change of the length.

The construction, and why it is not a trick #

Take a source $S$ and a detector $D$ and draw the ellipse with them as foci. Every point on that ellipse has the same total distance to the two of them; that is the definition of an ellipse, and it is the reason a whispering gallery works.

Now replace the elliptical mirror by any mirror tangent to it at the vertex. If the new mirror is flatter, it lies outside the ellipse everywhere except at the point of tangency — and a point outside the ellipse has a larger total distance to the two foci. So every neighbouring path is longer, and the central one is a minimum.

If the new mirror is more curved, it lies inside the ellipse, every neighbouring point has a smaller total distance, and the central path is a maximum. There is no third possibility and no room for argument: the sign is decided by whether the mirror’s curvature exceeds the ellipse’s, and the ellipse’s curvature at that vertex is a number one can write down.

That is the whole counterexample. It needs no exotic materials, no negative index, no metasurface. A shaving mirror with a source and a detector placed inside its centre of curvature is already in the maximum case for some geometries, and the light does not notice.

It is worth pausing on how sturdy the construction is, because a counterexample built from a special case can usually be dismissed as one. Nothing here depends on the mirrors being conics; that choice only makes the arithmetic exact. Any smooth mirror tangent to the ellipse at the reflection point falls into one of the three cases according to a single number — its curvature there — and curvature is a continuous parameter with no gap in it. So the maximum case is not a boundary or a coincidence: it occupies half of the available mirrors, in the same sense that half of all numbers are greater than a given one. If light really took the least-time route there would be a vast family of ordinary reflections it could not perform, and it performs all of them.

Nor does the case require the mirror to be strange to look at. The ellipse through a source and detector a metre apart, with the mirror half a metre away, has a radius of curvature at its vertex of a few tens of centimetres, which is an unremarkable concave mirror. The demonstration is a bench experiment rather than a thought experiment, and the reason it is never set up is that there is nothing to see: the light arrives exactly where the equal-angle construction says, and the fact that it took the slow road leaves no trace on the spot it lands.

What the principle is actually a shadow of #

The reason the sign does not matter is that Fermat’s principle is not a law. It is a corollary of the wave picture, and specifically of the fact that a sum of many contributions is dominated by wherever their phases stop changing.

The Cornu spiral’s chords are the amplitudes of near-field diffraction patterns, and the spiral is what a sum of wavelets looks like when their phases turn steadily. That is what the principle is actually a shorthand for: the stationary path is where neighbouring paths agree in phase and add, and every other region of the spiral curls up and contributes nothing. Fermat’s principle is the statement that only the tight part of the spiral survives.

The light arriving at the detector is a sum over every path from the source, each with a phase proportional to its optical length. Paths whose lengths differ by more than a wavelength have essentially random phases and cancel in pairs. Paths near a place where the length is not changing — where the first derivative vanishes — agree in phase over a whole band, and their contributions add.

Nothing in that argument mentions whether the second derivative is positive or negative. A band of paths whose lengths curve upward away from the centre and a band whose lengths curve downward both arrive in phase near the centre and both survive. The requirement is that the first derivative be zero, and “first derivative zero” is the definition of stationary, not of least.

Every mechanics problem distinguishes a minimum from a maximum — one is stable and one is not — because mechanics cares which way a system moves when disturbed. Optics does not: light does not settle into a path, so a maximum serves as well as a minimum, and the correct statement was always stationary rather than least. The word survived because the commonest cases are minima.

The two ordinary cases, done properly #

It is worth returning to the cases the principle is usually demonstrated on, because the demonstrations are correct and the correctness is a coincidence of geometry rather than a general fact.

Refraction through a graded medium is the case where the ray bends continuously, and the same principle governs it with no surface anywhere.

The bending is stationarity applied point by point in a medium whose index varies with height, with nothing to reflect from. That is where the principle earns its keep over Snell’s law: there is no boundary and no angle of incidence, and the ray still has a path, found by the same search.

That last sentence is the general rule and it is worth stating properly. Along a ray, the stationary path is a minimum until the first point at which neighbouring rays reconverge — the first conjugate point — and each time such a point is passed the character of the stationary value changes. A ray that has been through one focus is a saddle; through two, further still. The counting is exactly the same as the counting of nodes in a bound state, and it is called the Morse index in both places.

What the degenerate case is for #

The middle mirror — the one on which every path is exactly equal — looks at first like a curiosity between two interesting cases. It is the most useful of the three, and every focusing instrument ever built is an attempt to make it.

A lens forms an image because every route from the object point to the image point takes the same time — which is a degenerate stationary point rather than an isolated one. That is the case the principle handles least comfortably and the one that matters most: the second derivative vanishes in every direction, so the usual classification does not apply, and what a lens does is arrange precisely that degeneracy.

That is the design rule, and it is the same rule for a glass lens, a parabolic radio dish, a Fresnel zone plate and a gravitational lens. Make the optical path from source to image equal along every route the instrument admits, and the contributions arrive in phase and add in amplitude rather than in intensity — which is the difference between an image and a blur. The instrument’s whole job is to manufacture a degeneracy that nature does not usually supply.

Read the other way round, the same rule says what an instrument cannot do. Equal path length can be arranged for one pair of conjugate points, or for one wavelength, or over one aperture — not for all of them at once. Every aberration is a residual inequality of optical path, and the catalogue of aberrations is a catalogue of the ways the equality fails as the object moves, the colour changes or the aperture opens. A sphere failing to focus is the simplest entry in it.

Where the discarded second derivative reappears #

If the second derivative does not matter, why compute it? Because it is not discarded — it is deferred. The stationary-phase argument says the leading contribution comes from the stationary point, and the correction to it is governed by how fast the phase curves away.

A spherical mirror has no focus: rays at different heights cross the axis at different places, and the envelope of the crossings is a caustic. Every point on that caustic is where the second derivative — the term Fermat’s principle discards — has finally become the whole answer. The stationary condition locates the path and says nothing about how tightly the neighbouring paths cluster around it, and a caustic is where they stop clustering.

Where the curvature is large, the band of paths that agree in phase is narrow and the ray is a good summary. Where the curvature is small, the band is wide, and the light arriving at a detector is a genuinely extended bundle. Where the curvature vanishes — at a caustic — the band is unbounded and the ray picture produces infinity, which is the honest signal that rays have stopped being enough.

The width of the useful band is worth a number, because it is not small. For a path of length $L$ at wavelength $\lambda$ the paths that stay within half a wavelength of the stationary one span a transverse distance of order $\sqrt{\lambda L}$ — a millimetre for green light over two metres, and two centimetres over a kilometre. That is the first Fresnel zone, and it is why a microwave link needs clearance from obstacles that are nowhere near the straight line between its antennas.

The width of a diffraction pattern is set by the aperture and the wavelength together, and it is the same $\sqrt{\lambda L}$ scale that decides how wide the bundle of contributing paths is. That is the honest boundary of the whole principle: it locates a ray as the centre of a bundle whose width it cannot report, and everything about resolution lives in the width.

The same word, in mechanics #

The correction this essay makes to Fermat has an exact twin one subject over, and it is worth following because the mechanical version is where the error does the most damage.

Hamilton’s principle is universally called the principle of least action, and the derivation of the equations of motion from it uses only that the first variation vanishes. Everything said above therefore applies unchanged: the true trajectory is a stationary point of the action, and whether it is a minimum is a separate question with a geometry-dependent answer.

The cleanest demonstration is a free particle constrained to a sphere. Its trajectories are great circles, and between two points on a sphere there are two great-circle arcs — the short way and the long way round. Both are perfectly good solutions of the equations of motion; a bead sliding on a frictionless sphere will travel either. The short arc is a minimum of the length. The long arc is not a maximum, because a path can always be made longer by wiggling it, and it is not a minimum either: it is a saddle, on which some deformations shorten the path and others lengthen it.

What separates the two arcs is that the long one passes through the antipode of its starting point, and the antipode is a conjugate point — a place where a whole family of neighbouring trajectories, launched in different directions, reconverges. That is the same conjugate point the mirror section above named, doing the same job: the character of the stationary value changes each time one is crossed.

The rule is general and Jacobi established it in the 1840s. The action of a true trajectory is a minimum up to its first conjugate point and a saddle thereafter, and the number of conjugate points crossed is the number of independent directions in which the action decreases. A harmonic oscillator is the easiest case to check: the action between two endpoints is a minimum if the transit takes less than half a period, and a saddle if it takes more, because half a period is exactly when every trajectory launched from one point reconverges at another.

That is not an obscure regime. Half a period of an oscillator, or the far side of a focus in an optical system, or the long way round a sphere are all ordinary configurations, and in every one of them the quantity the principle is named after is not least. The name has survived for two centuries because nothing in any derivation depends on it, which is exactly the situation in which a wrong name is hardest to dislodge.

The dispute the principle settled, two centuries late #

Fermat proposed his principle in 1662, and the reason he proposed it was to attack a rival derivation of the same law.

Descartes had obtained the sine law of refraction in 1637 by an analogy with a tennis ball crossing a thin cloth: the component of motion parallel to the surface is unchanged, and the perpendicular component is altered by the surface. To make the ball bend toward the normal on entering water, as light does, the analogy requires the ball to speed up — so Descartes’ derivation asserts that light travels faster in the denser medium.

Fermat found that unacceptable, and derived the same sine law from the requirement that the travel time be stationary, with light travelling slower in the denser medium. Both derivations give $n_1\sin\theta_1 = n_2\sin\theta_2$. They disagree completely about what the index means: for Descartes it is the ratio of speeds one way up, for Fermat the other.

The dispute was not settled by argument, and it was not settled quickly. Newton’s corpuscular theory sided with Descartes, because a corpuscle attracted into the denser medium must accelerate; Huygens’ wave theory sided with Fermat, because a wavefront delayed at one end must tilt. For nearly two centuries the two camps had the same formula, the same experimental support, and opposite claims about a quantity nobody could measure.

Foucault measured it in 1850. Using a rotating mirror to compare the travel time of light through a tube of water against an equal path in air, he found the water path slower, by very nearly the factor 1.33 that refraction implies. Fizeau reached the same conclusion within weeks by a different method.

The measurement was billed at the time as the crucial experiment between the wave and particle theories of light, and in a narrow sense it was: it killed the specifically Newtonian corpuscle, which required the opposite sign. It did not settle what light is, as the following century demonstrated at length. What it did settle is which of two derivations of one formula was tracking the physics — and the answer was the one built on a stationary quantity rather than on a mechanical analogy.

Where the model stops #

Stationary phase needs the phase to be smooth, and an edge is not. At a sharp obstruction the sum over paths has no stationary point near the geometric boundary and the whole estimate has to be replaced, which is the Fresnel construction rather than a ray.

The principle says nothing about intensity. It locates the paths and is silent about how much light travels along each, which is why an optical design cannot be done from Fermat alone. What sets the intensity is how the neighbouring paths converge or diverge, which is the second derivative again — so the quantity the principle does not need is exactly the quantity a photometric calculation is made of.

The principle presumes the endpoints are given. Fermat’s construction asks which route joins two named points, and an experiment usually specifies a source and a direction rather than a source and a destination. For most geometries the difference is bookkeeping; near a caustic it is not, because several distinct stationary routes join the same pair of points and the sum over them is what produces the interference fringes seen inside a rainbow’s bright arc.

And “time” is a shorthand. The stationary quantity is the optical path length, which is the geometric length weighted by refractive index, and it equals a time only where the phase velocity is what the index says. In a dispersive medium the ray follows the phase index and the energy follows the group index, and the two paths are not the same path.

What the pictures cannot show #

The hero figure draws three path-length curves and cannot draw the reason light responds to their derivatives. The sum over paths is not a thing that happens in space; it is a way of organising an integral, and any attempt to draw it as a spray of arrows crossing the room asserts something about photons that the calculation does not contain.

Nor can any of these figures show a maximum being taken. The light does not deliberate and reject the shorter routes; the shorter routes are all there, all contributing, and cancelling one another because their lengths change quickly with position. The drawing shows the lengths and cannot show the cancellation, which is the mechanism.

Where this ladder ends #

This is the last rung of the Fermat anchor, and the anchor is closed with it. The ladder ran from the stationary path located by search rather than by Snell, through a medium with no surface in it at all, to the word the principle has to be stated with and the construction that forces it.

What remains to be said about light and paths is not on this ladder. Where the ray summary breaks down is diffraction’s; how the amplitudes add rather than merely which paths survive is Huygens’; what a curved surface does to an image is imaging’s; and the variational principle read as mathematics — the Euler–Lagrange equation, the second variation, the Morse index — is a subject rather than a rung and belongs to the collection that owns mathematical pictures. The anchor is closed because the next essay against it would be one of those wearing the wrong name.

A second anchor closes with it, at one rung, and by enumeration rather than by depth. refraction has exactly one essay against it — what happens to a ray at a boundary — and every further thing there is to say about a ray crossing a surface is already somewhere else on this site. That the index depends on colour is dispersion’s. That past a certain angle nothing crosses is total internal reflection’s. How much of the light turns back, and how that depends on polarisation, is polarisation’s. Bending with no surface at all is Fermat’s, which has just closed. And what a ray does at a surface for the purpose of making a picture belongs to the collection that owns the geometry of drawing rather than to this one. An anchor can therefore be finished at its first rung, and being finished is a statement about scope rather than about how much has been written.

The habit worth carrying away is a small correction with a wide reach. A principle stated as an extremum is almost always a principle about a derivative. Least time, least action, least energy: in every case the derivation needs only the first variation to vanish, the word “least” is a description of the common case, and somewhere there is a configuration in which the quantity is a maximum and nothing whatever goes wrong.