The backward wave Huygens had to remove

Huygens’ construction is one of the most productive pictures in physics. Every point of a wavefront is a source of a spherical wavelet; the wavefront a moment later is the envelope of those wavelets. From that alone come the law of reflection, Snell’s law, the shape of a wave passing an obstacle and most of what happens at a boundary.

Taken literally it also predicts something that never happens. A spherical wavelet spreads in every direction, backwards included. Sum the wavelets over a whole plane and there ought to be a wave travelling back the way the original came, as strong as the one going on.

The usual response is to take the envelope on the forward side and say nothing about the other one. That is an instruction rather than an explanation, and what makes the construction respectable is that there is a real answer.

The sum is done rather than argued about. Every candidate reproduces the incident wave on the forward side, which is the part of the construction that has always worked and which any correct rule has to preserve. Behind the plane the three disagree completely: an isotropic wavelet gives a full backward wave, a cosine rule gives one of the same size with the sign reversed, and $(1+\cos\theta)/2$ gives four parts in a thousand — the residue of a finite aperture rather than a physical wave.

Where the factor comes from #

The factor is not a repair invented to remove an embarrassment. It falls out of solving the wave equation.

Kirchhoff’s derivation starts from the equation rather than from a picture: given the field and its normal derivative on a closed surface, Green’s theorem gives the field at any point inside. Applied to a plane with a wave crossing it, the result has exactly the form of Huygens’ construction — a sum of secondary sources over the surface — with three things attached that Huygens’ geometry has no room for.

The first is the obliquity factor $(1+\cos\theta)/2$: full strength forwards, half at right angles, nothing backwards. The second is a phase advance of a quarter period, so each secondary wavelet leads the primary wave rather than following it. The third is a factor of one over the wavelength, which is a dimensional necessity — a sum over an area has to be divided by an area to give back an amplitude, and the only length available is the wavelength.

None of the three can be guessed from the geometric statement, and all three are needed for the construction to reproduce the wave it started from. That is the sense in which Huygens’ principle is correct: it is the shape of the answer, and the coefficients are supplied by the equation.

The quarter-period advance deserves a second look, because it is the one that sounds impossible. It says the secondary sources are ahead of the wave that produced them, which cannot be a causal statement. It is not one: the wavelets are a device for re-expressing the field on one surface as the field somewhere else, not a physical sequence of emissions, and their phase is whatever makes the identity hold. The construction is an integral theorem wearing the clothes of a mechanism.

Adding up the wavefront #

Fresnel’s way of doing the sum is worth having because its result is so strange. Divide the wavefront into annular zones, each reaching the observation point half a wavelength further than the last. Every zone has the same area — the annulus gets thinner exactly as fast as it gets bigger — so every zone would contribute equally were it not for the obliquity factor, which makes each slightly weaker than the last.

The contributions therefore alternate in sign and shrink slowly, and the running sum spirals in. Where it converges is the surprise: to half the first zone’s contribution. The whole illuminated wavefront, out to infinity, delivers what half of its innermost zone would deliver on its own.

Two consequences follow, and both are experiments.

Block everything except the first zone — a hole a millimetre or so across at a metre — and the intensity at the point is four times what it was with nothing in the way, because the amplitude has doubled. An obstruction makes it brighter.

Block every second zone instead. The remaining contributions are all in phase and add rather than cancelling, and the intensity is enormous — a lens made of nothing but a pattern of rings, which is the zone plate and which focuses by arithmetic rather than by refraction. It has a focal length that depends on the wavelength in the opposite direction from a lens’s, which is one of the few places the two kinds of optics disagree about a sign.

Putting the two spirals beside each other says what the obliquity factor is doing that is more than removing a backward wave. Without it, the alternating series has no reason to converge at all: the sum after ten zones and the sum after eleven differ by a whole zone’s contribution, and there is no limit to approach.

With it, the sum converges, and it converges to a value that does not depend on where the wavefront was cut off. That is the property that makes the construction usable — an answer that changed with the size of an arbitrary aperture would not be an answer — and it comes from the same factor that removes the backward wave.

Where the spiral has to be helped #

The convergence of the zone sum needs a comment, because the honest version of it is less tidy than the textbook version.

The obliquity factor falls from one to a half as the angle goes from zero to a right angle, and then continues to zero — but the part of the plane at more than a right angle from the observation point does not exist for a plane wavefront. So on an unbounded plane the factor only falls to a half, the alternating series does not converge, and the partial sums oscillate for ever between two values whose mean is the right answer.

What makes it converge physically is that no wavefront is unbounded. A real beam has an edge, a real source has a finite extent, and the contributions from far out on the plane are not there. The figures include a gentle taper for exactly this reason, and the taper is not a trick: it is the mildest possible statement of a fact about every real experiment.

That is worth knowing because it shows where the classical treatment is doing something quietly. The famous result — the whole wavefront gives half the first zone — is a statement about an Abel sum, and its physical content is that the answer does not depend on where the wavefront is cut off, provided the cut-off is many zones out. Different windows give the same answer, which is what makes the result meaningful.

The obstacle that makes it brighter #

The first zone’s consequence deserves a paragraph of its own, because it is the most testable statement in the essay and the least believable.

Put an opaque disc in the beam, on the axis, covering the first several zones. Behind it, on the axis, there is a bright spot — as bright as if the disc were not there. The reason is the spiral: the disc removes the first several turns of it, and the remaining turns still spiral in to a point at very nearly the same distance from the origin, because the spiral’s centre is what the sum converges to and the early terms only decide where it starts.

Poisson produced this as a reductio against Fresnel’s theory in 1818: if light is a wave then there is a bright spot in the middle of a circular shadow, which is absurd. Arago looked, and it is there. It is one of the cleanest instances of a prediction that was offered as a refutation and settled the matter the other way.

The same argument says the spot is fragile in a specific way: it needs the disc to be circular and the illumination to be on the axis, because both are what makes the remaining zones concentric. A disc with a rough edge, or a source off the axis, dilutes it — and how quickly it dilutes is a measurement of the edge.

What the construction cannot be asked #

The division between what the geometric construction gets right and what it cannot address is clean, and it is worth stating because it explains the principle’s odd status.

What it gets right is every direction: the law of reflection, Snell’s law, the angular positions of diffraction minima, the shape of a front after an obstacle. Those depend only on where the wavelets arrive in phase, which is geometry, and no coefficient enters.

What it cannot give is any amplitude. How bright a diffraction pattern is, how the intensity divides between orders, what happens right at the edge of a shadow — all of those need the factors Kirchhoff supplies, and Huygens’ statement contains none of them.

The historical shape of that is worth recording. Huygens published the construction in 1690 and used it entirely for directions. Fresnel added the interference of the wavelets in 1818, which is what turns a construction about envelopes into a theory of diffraction, and got the amplitudes approximately right by an argument that was largely intuition — including the obliquity factor and the quarter-period phase, both of which he put in because the answers came out wrong without them. Kirchhoff derived them in 1882, from the wave equation, and showed that Fresnel’s intuitions were the leading terms of a theorem.

What it gets right, drawn #

It is worth ending the criticism with the case that vindicates the picture, because the division is so clean.

The refracted angle in that figure is not fitted or corrected. It is the direction in which the wavelets from every point of the incident front arrive in phase, and it comes out as $\sin\theta_1/\sin\theta_2 = v_1/v_2$ with no factor of any kind. Every one of the missing ingredients — the obliquity, the quarter period, the one over the wavelength — is a multiplicative constant across the whole front, and a constant multiplying every wavelet equally cannot move the place where they agree in phase.

That is the general statement of what the construction is good for. Directions come from phases and amplitudes come from coefficients, and Huygens’ geometry contains the phases exactly and the coefficients not at all. Reflection, refraction, the angular positions of every diffraction feature and the shape of every wavefront are the first kind. Brightness is the second.

It also explains why the principle survived so long without its factors: for two hundred years the questions asked of it were all of the first kind, and the first experiment to ask a question of the second kind was Fresnel’s.

Where the model stops #

Kirchhoff’s theory is not exact and is known not to be. It requires the field and its normal derivative to be specified on the surface, and for a screen with a hole in it the natural assumption — the incident field in the hole, zero on the screen — is mathematically inconsistent, since a field vanishing with its derivative on any patch vanishes everywhere. The theory works anyway, to a good approximation, whenever the hole is large compared with the wavelength; the exact treatments due to Sommerfeld and to Rayleigh remove the inconsistency and disagree with Kirchhoff mainly near the edges.

The obliquity factor as written is for a scalar wave. Light is not scalar, and a full treatment carries the polarisation through; the result is that the effective obliquity depends on the polarisation relative to the aperture, which matters for apertures comparable with the wavelength and not otherwise.

The zone construction assumes the observation point is on the axis. Off-axis the zones are not circles and the sum is harder, which is why the elegant half-the-first-zone result is quoted for the axis and the general case is done numerically.

And the taper is a stand-in for a real aperture. Nothing in the essay computes what a particular edge does, and near an edge the difference between the treatments is exactly where they part company — which is the subject the Cornu spiral belongs to and where the corrections to Kirchhoff are measurable.

Why the principle is worth keeping #

Given that the geometric statement is incomplete and the integral theorem is available, it is fair to ask why anybody still teaches the construction.

The answer is that the integral theorem is unusable as a way of thinking. It is an exact statement that requires a numerical evaluation for every question, and it gives no picture of why an answer is what it is. The construction gives the picture and gets every angle right, which is most of what anybody wants to know most of the time.

The relationship is a common one and worth naming. A picture that is exact about one class of question and silent about another is far more useful than a formula that is approximately right about both, provided the boundary between the classes is known. Huygens’ construction is safe for directions and mute about brightness, and knowing that is worth more than knowing Kirchhoff’s integral.

Where it becomes dangerous is where the boundary is forgotten — which is what happens when a ray argument is pushed past the point where a wave picture is needed, and it is the same failure one level up. The construction does not warn anybody that it has stopped applying, because it produces an answer either way.

What the pictures cannot show #

The polar plot of the obliquity factor draws an amplitude against angle and cannot show that the factor is a property of the derivation rather than of any physical wavelet. There is no little source at each point of the wavefront radiating with that pattern; there is an integral, whose integrand happens to have that angular dependence, and the difference matters as soon as anybody asks what the wavelets are made of.

The zone figure draws the sum on the axis and hides the fact that the zones are enormous. For light at a metre, the first zone is about half a millimetre across and the tenth is about a millimetre and a half — so an obstacle of a few millimetres covers a handful of zones and the interesting effects are all at that scale. For sound the zones are metres across, which is why a wall does much less to a low note than the ray picture suggests and why a barrier’s usefulness depends on frequency as much as on its height.

Where the ladder goes next #

The Huygens ladder began with every front being a source, went through the spiral that says how much light arrives and the wave that comes from the rim. This rung asks what the construction has to be given to be correct. The rungs after it: the angular spectrum, which re-expresses the same content as a sum over plane waves and makes the propagation a multiplication rather than an integral; the exact solutions for a half-plane, where Kirchhoff’s inconsistency is removed and the difference is measurable; and the reciprocity of the whole construction, in which source and observation point can be exchanged.

The habit worth carrying away is to ask what a successful picture predicts that nobody looks for. Huygens’ construction was right about everything anybody tested for two hundred years and wrong about something nobody thought to check — and the repair, when it came, arrived from the equation rather than from the picture, bringing two more factors the picture could not have suggested.