Every front is a source

Huygens’ rule, stated in 1690, is so simple that it reads as a restatement rather than a law: every point on a wavefront may be treated as a source of a spherical wavelet, and the front an instant later is the surface touching all of them. It is a geometric recipe with no differential equations in it, it can be executed with a compass, and it accounts for four separate phenomena that were otherwise four separate rules.

That first result is the one usually skipped, and it is the one that makes the rest of the construction believable. If a wavelet from each point really spreads in all directions, why does a beam of light not spray sideways immediately? The answer is in the picture: it does, and its neighbours’ wavelets cancel it everywhere except along the envelope. The straightness of a beam is a cancellation, not an absence of spreading.

Where the ray went #

The word missing from the paragraph above is ray. Huygens’ construction has no rays in it at all — it propagates surfaces, and a ray is defined afterwards, as the normal to the family of surfaces. That ordering is worth keeping, because it makes clear which of the two is the derived object.

The same bookkeeping does a different job when the source is moving: circles emitted at successive instants, bunched ahead of it and stretched behind. Every construction on this page is that one picture used differently — the fronts are drawn from where and when each was emitted, and whatever geometry results is the answer. A ray drawn on such a figure would be a line from the source to a listener, and it would carry none of the information the spacing carries.

Refraction, as a delay #

The construction’s best-known success is a derivation of Snell’s law that needs nothing but a ratio of speeds.

The mechanism is a delay and nothing else. One end of the front reaches the boundary before the other; in the interval, the first end has advanced into the slow medium by less than the second end has advanced through the fast one; the front is left tilted. Anyone who has pushed a shopping trolley with one stiff wheel onto a carpet has performed the experiment.

Two things follow immediately from that reading and are hard to see from Snell’s law as an algebraic rule. The bend depends only on the ratio of the speeds, so refractive index is dimensionless and a boundary between two materials with the same wave speed is invisible. And the frequency cannot change at a boundary — the far side is being driven by the near side, arrival for arrival — so it is the wavelength that changes, and it changes by exactly the factor the wavelets did.

The same event in the ray language is what most of optics uses, and it contains strictly less. A ray gives the direction and is silent about the wavelength, about the phase, and about the fact that a front exists at all. Both descriptions are correct and one of them is poorer, and the difference shows up the moment two paths have to be compared — which is every interference calculation ever done.

Where the front runs out #

The interesting case is the one Huygens’ contemporaries used against him: what happens when a front is cut short.

The relationship in that figure is the origin of the whole ray approximation. Make the aperture enormous compared with the wavelength and the spreading angle goes to zero, the beam stays a beam, and geometrical optics is exact. Make it comparable and the beam is gone. For sound, wavelengths are metres and every doorway is a diffracting aperture, which is why a conversation is audible round a corner; for light they are half a micron and almost nothing in ordinary experience is small enough, which is why it took until 1665 for anyone to notice that light diffracts at all.

The same arithmetic settles a question about hearing that has nothing obviously to do with waves. Low notes have wavelengths of metres and spread through a doorway into the whole room; high notes have wavelengths of centimetres and come out of it as a beam. That is why the bass from a neighbouring room arrives and the words do not, why a tweeter has to be aimed and a woofer does not, and why the position of a subwoofer in a room matters much less than anyone selling one admits.

What the intensity actually does is the measure of what the 1690 version was missing. Huygens’ construction gives the directions of the zeros correctly and says nothing whatever about how bright anything is between them, because it propagates surfaces and never adds amplitudes. The envelope is an outline of where the disturbance has got to; the pattern is what the contributions arriving at a point produce when they are summed with their phases, and that sum is a nineteenth-century idea rather than a seventeenth-century one.

Reflection, and the one case Huygens got wrong on purpose #

The reflection derivation is the same delay argument with the wavelets kept in the first medium. A front arriving obliquely reaches one end of the mirror before the other; the wavelets from the early points have grown larger by the time the last point is struck; the envelope is a front tilted the other way, and the angle out equals the angle in.

What makes that worth a paragraph is a case Huygens applied it to that nobody had explained: Iceland spar, a crystal of calcite which splits an incident beam into two, one obeying the ordinary law and one not. His treatment gives the second beam its own construction with ellipsoidal wavelets instead of spherical ones — because the speed in the crystal depends on direction — and it works. The ordinary and extraordinary rays come out at the observed angles, from a rule about the shape of a wavelet.

That is a remarkable thing to have got right in 1690, and it is also where the construction’s limits are clearest. Huygens could describe double refraction and could not explain why passing the beam through a second crystal made the intensities depend on the relative orientation, which is polarisation and which needs the wave to be transverse. He assumed it was longitudinal, like sound, because nothing in the geometry says otherwise. The construction propagates a surface and does not care what is oscillating on it, which is its great strength and the reason it could not settle that question.

The angle at which the construction has no answer #

Run the refraction construction the other way — from glass into air, so the wavelets in the second medium are larger than those in the first — and something instructive happens at a particular angle. The construction stops having a solution.

The envelope is the line tangent to all the wavelets, drawn from the points where the front met the boundary. With the second medium faster, each wavelet has grown by more than the front advanced along the surface in the same interval, and past some incidence there is no straight line touching them: the first wavelet has already outrun the point where the last one starts. Geometrically there is nothing to draw.

The angle at which the last possible tangent lies flat along the surface is the critical angle, and it is where the construction’s own arithmetic gives $\sin\theta_2 = 1$: 41.8° for glass into air, 48.8° for water into air. Beyond it the construction produces no transmitted front at all, and the reflection has to take everything.

This is worth dwelling on because of how the construction fails. It does not give a wrong answer, or a small answer, or an answer needing correction. It reports that the object it was asked to build does not exist — which is exactly the right report, since a propagating wave in the second medium genuinely does not exist beyond that angle.

What the construction cannot then say is what does exist. There is a field beyond the boundary: it clings to the surface, falls off exponentially with distance into the second medium over a fraction of a wavelength, carries no energy away, and is entirely real. It is measurable by putting a second piece of glass close enough to intercept it, at which point light crosses a gap it is forbidden to propagate in — frustrated total internal reflection, and the direct optical analogue of a particle tunnelling through a barrier.

An evanescent field is not a wavefront and has no wavelets, so no envelope construction can produce it. It is a solution of the wave equation with an imaginary wavenumber in one direction, and it belongs to the algebra rather than to the compass. That is the sharpest available statement of what Huygens’ recipe left out: it propagates surfaces, and there are solutions that are not surfaces.

Where the envelope folds #

The construction assumes that the envelope of the wavelets is a well-behaved surface with a definite normal at every point. For a plane front in a uniform medium it is. For almost any front that has been through a curved boundary, it is not — and the places where it is not are among the most visible optical phenomena there are.

Propagate a front that has been refracted through a spherical drop and the envelope eventually folds over on itself, developing a cusp and then two branches. Along the fold, neighbouring parts of the front arrive in the same place travelling in the same direction. The rays pile up, and geometrical optics — which computes intensity by asking how much a bundle of rays has spread — returns infinity.

That fold is a caustic, and the bright curve at the bottom of a coffee cup on a sunny morning is one seen directly. So is the flickering net of light on the floor of a swimming pool, which is the caustic of the disturbed surface above. So is the rainbow: light entering a raindrop, reflecting once inside and leaving has a deflection that varies with where on the drop it entered, and that deflection has a minimum at about 138°, which puts a caustic at 42° from the antisolar point. Rays crowd at that angle and there are none beyond it, which is why a rainbow has a bright edge with a dark sky outside it.

The infinity is the construction’s, not nature’s. What actually happens at a caustic is decided by adding amplitudes with phases across the fold rather than counting rays, which is the repair the whole of this essay is about, and Airy did it for the rainbow in 1838. The answer is finite, it is brightest slightly inside the geometric angle, and it oscillates — which is what the faint supernumerary bows just inside a strong rainbow are, and what no ray-counting argument can produce.

That gives a clean statement of when Huygens’ recipe is enough. It is exact for the geometry, always; it is useful for the intensity everywhere the envelope is smooth; and it is worse than useless precisely where the envelope folds, which is where the interesting things are.

The repair, and how long it took #

The original construction has two defects, and both are consequences of the same omission.

It has no amplitudes. An envelope is a locus, and a locus cannot say whether the wave arriving there is strong or weak — so the construction predicts where a shadow’s edge is fuzzy and not what the fringes in it look like. And, taken literally, a spherical wavelet radiates backwards as well as forwards, so the envelope should include a second surface travelling back toward the source. No such wave exists.

Fresnel repaired both in 1818, by treating the wavelets as things whose amplitudes add with their phases — which is superposition — and by inserting an obliquity factor that falls to zero in the backward direction. The factor was chosen because it had to be; there was no argument for it beyond the absence of the backward wave. He also had to add a phase advance of a quarter of a cycle and a factor of $1/\lambda$ to make the intensities come out right, and both of those were similarly unmotivated.

Kirchhoff supplied the justification in 1882 by deriving the whole apparatus from the wave equation directly. The obliquity factor turns out to be $(1+\cos\theta)/2$, exactly zero backwards, and the quarter-cycle phase and the $1/\lambda$ appear on their own. What had been a geometric recipe with three empirical patches became a theorem — which is a common enough sequence, and worth noticing because the recipe was used successfully for a hundred and ninety years while being, strictly, unjustified.

The bright spot that was meant to end the argument #

The best-documented episode in the construction’s history is a prediction made in order to destroy it.

In 1818 the French Academy set a prize competition on diffraction, expecting to reward a defence of the particle theory of light. Fresnel submitted the wave treatment described above. Poisson, on the judging committee and firmly of the other party, worked out a consequence: applied to a circular obstacle, Fresnel’s integral requires that the contributions from every point of the obstacle’s rim arrive at the centre of the shadow exactly in step, since all of them are the same distance from it. So there should be a bright spot in the middle of the shadow of a ball bearing — an obvious absurdity, and grounds for rejecting the whole theory.

Arago went and looked. The spot is there. It has been called Poisson’s spot ever since, after the man who used it as a reductio, and Arago’s spot after the man who checked. It had in fact been observed by Delisle a century earlier and by Maraldi, and forgotten, which is what happens to an observation with no theory to belong to.

The episode is worth the space because of what the prediction depended on. It is not a qualitative claim that waves bend round things; it is a quantitative consequence of adding amplitudes with phases over a continuum of secondary sources — the exact thing Huygens’ 1690 version could not do. A theory that predicts something ridiculous, and turns out to be right, is doing more work than one that predicts what everyone expected.

What it is really a statement about #

The lasting value of the construction is not that it is a convenient way to derive Snell’s law. It is that it identifies what a wave equation implies: that the future of a field can be computed from its values on a surface, with each element of the surface contributing independently.

That is a strong claim about locality and it is why the same recipe works for sound, for water waves, for light and for the wavefunction of a particle. It is the reason an antenna array can be designed by giving each element a phase and adding the wavelets; the reason ultrasound imaging can steer a beam electronically with no moving parts; and the reason a seismologist can propagate a wavefront through a model of the Earth’s interior by marching a surface forward rather than by solving the whole field at once. In each case what is being exploited is that the surface carries everything needed, so the calculation never has to look behind it.

There is a caveat that the same sentence contains. The claim holds exactly for a wave equation with a second time derivative and no first one — a lossless, non-dispersive medium. Add dissipation, or make the speed depend on frequency, and a pulse’s shape changes as it goes, so the front at one instant no longer determines the front at the next without knowing the history. A dispersive medium is exactly the case where the wavelets from different frequencies have different radii, and the envelope of a superposition of unlike wavelets is not a wavefront in any useful sense.

Where the wavelet’s radius comes from is worth stating plainly, because the construction never computes it. The speed of a wave on a string is set by the tension and the mass per unit length; the speed of a wave in any other medium is set by the corresponding pair of properties. Huygens’ construction takes that number as given and asks what geometry follows from it. Everything on this page is downstream of a quantity the medium supplies and the construction is silent about — which is a strength rather than a gap, since it is why the same construction serves light, sound and water without alteration.

It is also why it fails in the cases where the claim fails — in a medium whose properties change within a wavelength, near a sharp edge where the field on the surface is not what the unobstructed field would have been, and in any situation where the wave is not a scalar. The vector nature of light means that a rigorous treatment of a slit has to keep track of polarisation, and the answers for the two polarisations differ measurably for slits under a wavelength wide.

What the picture cannot show #

The wavelets in these figures are circles, which is correct in the plane of the page and wrong in three dimensions in a specific way: a spherical wavelet’s amplitude falls as $1/r$ and a circular one’s as $1/\sqrt r$, so a construction drawn in two dimensions is quantitatively a picture of a different problem. The geometry — the envelope, the angles, the zeros — is unaffected, and every amplitude in a plane drawing is wrong.

The figures also draw a small number of wavelets, and the construction requires a continuum. Nine circles have a visible envelope only because the eye supplies the curve; with nine actual sources the pattern between them would be a mess of interference, which is what a phased-array antenna or a diffraction grating is. The transition from a discrete set of sources to a continuous front is not decorative — it is the difference between a grating with orders and an aperture with a single spreading beam.

And the domain of validity is narrower than the recipe’s simplicity suggests: a homogeneous medium, a scalar wave, a front large compared with the wavelength, and an interest in directions rather than intensities. Outside it, the construction is still the right picture and the calculation has to be Kirchhoff’s.

The ladder from here #

Later rungs on this anchor: the Fresnel–Kirchhoff integral written out, with the obliquity factor derived rather than inserted; Fresnel zones, which turn the integral into an argument about annuli and explain the bright spot at the centre of a circular shadow; the zone plate, which is a lens built by blocking alternate zones; the eikonal limit, in which the construction becomes Fermat’s principle; and Huygens’ own use of it, which was to explain the double refraction of Iceland spar by giving the wavelets an elliptical shape.

The neighbouring ladders are what a travelling wave is, which is what is being propagated, refraction, which is the construction’s first success, and diffraction, which is its second and the one that needed the repair.