The spiral that says how much light arrives

Huygens’ construction is the oldest working picture of a wave: every point of a front is a source of a small spherical wavelet, and the front a moment later is the surface those wavelets touch. It produces reflection, refraction and the spreading of a wave past an obstacle, in the seventeenth century, with a compass and a ruler. What it cannot produce is a number. An envelope is a locus — a place — and a place carries no information about how strong the wave arriving there is.

The distinction between the arc and the chord is the whole of the subject. A long stretch of wavefront contributes a great deal of curve and, if the curve has wound round on itself, almost no chord: the contributions have cancelled. The spiral is a way of doing that bookkeeping without ever performing an integral by hand.

What the envelope leaves out #

Start with what the old construction does well.

It is right about where. It says nothing about how much, and worse, taken literally it says the wrong thing about direction: a spherical wavelet radiates backwards as much as forwards, so the envelope ought to include a second surface travelling back toward the source. No such wave exists. Fresnel repaired both defects in 1818, and the repair was a patch rather than a derivation. Blocking every other half-period zone turns the spiral’s cancellation into a lens, which is the sharpest demonstration that the cancellation is real. He treated the wavelets as amplitudes to be added with their phases — which is superposition — and inserted an obliquity factor that falls to zero backwards, a quarter-cycle phase advance, and a factor of $1/\lambda$, none of which he could justify. Kirchhoff derived all three from the wave equation sixty-four years later.

What adding with phases buys is visible with only two sources: sum their fields rather than enveloping their surfaces and lines appear along which the contributions cancel and lines along which they reinforce. Every diffraction pattern is that calculation with the two sources replaced by a continuum of them across an aperture — which is exactly the continuum Huygens proposed, used in the way he did not use it.

One step of the sum the spiral performs continuously is two waves of unequal amplitude added with a phase between them. The reason the total comes out as a chord rather than as an arc is already visible there: the resultant of two contributions is shorter than their two lengths added, whenever they are not exactly in step. Stack a continuum of such steps, each turned a little further than the last, and the running total curls.

The phase that goes as the square of the distance #

Take a straight edge, illuminated from far away, and a point on a screen. The contribution from a piece of the wavefront a distance $s$ off the direct line has to travel further to reach that point, by an extra path length that is, for small angles,

$$\Delta \approx \frac{s^2}{2L},$$

with $L$ the harmonic mean of the source and screen distances. The extra phase is $2\pi\Delta/\lambda = \pi s^2/\lambda L$. Defining

$$v = s\sqrt{\frac{2}{\lambda L}}$$

makes that phase exactly $\pi v^2/2$, and the running sum of the contributions from $0$ to $v$ becomes

$$C(v) = \int_0^v \cos\frac{\pi t^2}{2}\,\mathrm{d}t, \qquad S(v) = \int_0^v \sin\frac{\pi t^2}{2}\,\mathrm{d}t.$$

Plot $S$ against $C$ and the result is the spiral in the hero figure. It has three features and each of them is a physical statement.

Arc length is $v$. Moving along the curve by an amount is moving along the wavefront by the corresponding distance. This is what makes the geometry read as a construction rather than a graph.

The two eyes at $(\pm\tfrac12, \pm\tfrac12)$ are where the far parts of the front pile up. Contributions from far out on the wavefront arrive with wildly varying phases and go round and round in ever tighter circles, converging to a point rather than adding up. So a wavefront with nothing in its way contributes the chord from one eye to the other, of length $\sqrt2$.

Amplitude is the chord. The vector sum of a set of contributions is the vector from the first to the last, and on this curve that is the straight line between the two points.

A quarter, not a half #

Put a straight edge exactly at the direct line, so that half the wavefront is blocked, and ask what arrives at the point on the geometrical shadow boundary. The surviving half of the front runs from $v = 0$ to $v = \infty$, which on the curve is from the centre to one eye. That chord is $\sqrt{1/2}$, exactly half the full one.

Half the amplitude is a quarter of the intensity. This is not an approximation and not a coincidence: it is the statement that the field is what adds and the intensity is what is measured, and everything counter-intuitive about diffraction comes from that one square.

It is worth noticing how much has been settled by a construction with no calculus visible in it. The quarter follows from the observation that the surviving half of the wavefront runs from the curve’s centre to one of its eyes, and that observation is available to anyone holding a drawing of the curve. No integral has to be evaluated, no series summed, and no special function looked up. That is the sense in which Fresnel’s repair is a construction rather than a formula: the geometry does the work, and the work it does is the work an integral would otherwise have to do.

Reading the rest of that curve off the spiral is a matter of moving the starting point. A point outside the shadow sees more than half the front: the chord runs from an eye to somewhere partway round the spiral’s first turn, and as the observation point moves outward the far end of the chord walks round that turn, getting longer and shorter as it goes. That is the fringes. The longest such chord runs to $v = 1.217$, has length 1.656 against the open front’s 1.414, and gives an intensity of exactly 1.370 — the brightest fringe, brighter than no obstacle.

A point inside the shadow sees less than half: the chord runs from the eye to a point already spiralling in toward the other eye, and it shortens monotonically. There are no fringes inside a straight edge’s shadow, ever, and the reason is that the far end of the chord is winding up rather than going round.

The same aperture, four distances #

The spiral makes the near field and the far field one calculation. A slit is two edges, so its amplitude is the chord between two points on the curve, both of which move as the observation point moves — and how far apart they are on the curve is fixed by the slit’s width in units of $\sqrt{\lambda L/2}$, which is the Fresnel number.

At a large Fresnel number the two points on the curve are far apart in arc length, both deep in the spiral’s turns, and the chord between them wobbles as the pattern is traversed — those are the ripples across the opening. At a small Fresnel number both points are near the centre of the curve, where it is nearly straight, and the chord is nearly the arc: contributions add almost in phase, and the pattern is the broad single lobe everyone recognises.

The far-field limit is worth having beside it, because it is the case everybody meets first. There the pattern is a Fourier transform of the aperture and the width of the central lobe is inversely proportional to the width of the slit — a regime the spiral’s centre describes, and a limit of this construction rather than a different phenomenon. The crossover between the two happens where the aperture’s own width becomes comparable with $\sqrt{\lambda z}$.

The Fresnel number is the ratio of two lengths that both belong to the situation rather than to the aperture: the aperture’s half-width, and the scale $\sqrt{\lambda L}$ that the wavelength and the distance make between them. When the first is larger, the aperture contains many zones and the near field is a shadow with structure at its edges; when the second is larger, the aperture is smaller than one zone and everything that gets through arrives in phase. A slit half a millimetre wide is a near-field object at arm’s length and a far-field object across a room, in green light, and neither statement is about the slit.

The word “diffraction” is therefore doing two jobs. In the far field it names a Fourier transform; in the near field it names a chord of a spiral; and the same aperture does one, then the other, as a screen is walked away from it.

Zones, and the spot that should not be there #

Fresnel’s own way of organising the integral was to divide the wavefront into half-period zones — annular strips whose contributions arrive alternately in phase and out of phase, each strip separated from the next by half a wavelength of extra path. On the spiral, one zone is one half-turn.

Counting them settles the qualitative behaviour of almost any aperture immediately. An opening that admits exactly one zone is the brightest arrangement possible on axis — twice the amplitude of the unobstructed wave, so four times the intensity, because there is nothing left to cancel it. An opening admitting two zones is nearly dark, because the second zone cancels the first. An opening admitting a hundred is the geometrical case, where all but the outermost half-zone has cancelled in pairs and the surviving contribution is what the ray picture calls “the light that got through”.

The zone picture makes one prediction that ended an argument. A circular obstacle blocks a set of complete zones and leaves the rest, and on the axis behind it every surviving zone is at the same distance, so their contributions all add in phase: there should be a bright spot at the exact centre of a circular shadow. Poisson produced this in 1818 as a reduction to absurdity of Fresnel’s theory. Arago went and looked, and it was there. That story belongs to the diffraction ladder, and it belongs here as the moment the amplitudes stopped being optional.

Six decibels at the horizon #

The quarter that this essay’s first refutation is about has an afterlife in a subject that never mentions Fresnel: it is the number every radio link is designed around.

A microwave path between two towers is blocked, partly, by whatever the terrain does in between. If a hilltop rises exactly to the straight line joining the antennas, the situation is the straight edge of the figures above with the observation point exactly on the shadow boundary — half the wavefront removed, a quarter of the power arriving. In the units the trade uses, that is a loss of six decibels, and it is quoted as the knife-edge diffraction loss at grazing incidence in every propagation handbook.

Six decibels is a great deal to lose from a link budget, and the loss grows quickly as the obstruction rises further. Which raises the design question: how far below the line must the terrain be for the obstruction to cost nothing?

The zone counting answers it. What matters is whether the obstacle intrudes into the first half-period zone — the region around the direct path within which contributions arrive within half a wavelength of the direct one — and its radius at a point along the path is

$$r = \sqrt{\frac{\lambda d_1 d_2}{d_1 + d_2}},$$

with $d_1$ and $d_2$ the distances to the two ends. For a six-gigahertz link across ten kilometres that is eleven metres at the midpoint, which is a great deal more than the metre or two of line-of-sight clearance a surveyor’s intuition suggests.

The rule of thumb is to keep sixty per cent of that radius clear, at which point the loss is essentially zero — and, entertainingly, an obstacle sitting a little further out than that produces a small gain, because the observation point is on the first bright fringe. Radio engineers do not design for the fringe, because the terrain and the atmosphere both move; but it is there, and it is the 1.37 of the edge figure.

The other Fresnel lens #

A confusion worth settling: the Fresnel lens on a lighthouse and the Fresnel zones of this essay are the same man and unrelated physics.

A lighthouse lens is a refracting lens with most of its glass removed. Take a thick convex lens, cut it into concentric annular rings, and slide each ring back so that the whole thing becomes a flat plate with a stepped surface. Each ring still refracts light through the same angle it did before, because refraction depends on the surface’s slope and not on the glass behind it, so the assembly focuses like the original while weighing a fraction as much. Nothing about wavelengths, phases or interference enters, and the device works identically for every colour and for sound.

A zone plate is the diffractive object, and it does belong here. Block the alternate half-period zones of a wavefront — leaving open the ones whose contributions arrive in phase — and everything that survives adds up at the axial point, which is a focus made by removing light rather than by bending it. Its focal length depends on the wavelength, and it has an infinite series of further foci at odd fractions of the main one, both of which a glass lens does not.

The distinction matters practically because zone plates are the only focusing elements available where glass is opaque. X-rays cannot be refracted usefully, so an X-ray microscope focuses with a zone plate whose finest ring sets its resolution; the same construction with reflective rather than absorbing rings is how a soft X-ray telescope is built.

Two devices, both called Fresnel’s, one a mass-saving trick for a refracting lens and one an application of the arithmetic in this essay. The name they share is a biography rather than a physics.

Where the wavelength enters, and where it does not #

One property of the edge pattern is worth naming because it is what the two figures at different wavelengths demonstrate, and it is easy to miss.

Written in the variable $v$, the pattern has no parameters in it at all. There is one curve, one first fringe at $v = 1.217$ with an intensity of 1.370, one first minimum at $v = 2.345$, and one quarter at the boundary — and none of those numbers knows the wavelength, the distance or the size of anything.

All of the physical content is in the conversion, $s = v\sqrt{\lambda L/2}$. That single length $\sqrt{\lambda L}$ is the only scale in the problem, and every feature of the pattern sits at a fixed multiple of it. Change the wavelength by a factor of fifty and everything moves outward by a factor of seven; change the screen distance by a factor of a hundred and everything moves outward by ten.

Which explains why diffraction is invisible in ordinary life and unavoidable in radio. At green light and a metre, $\sqrt{\lambda L}$ is 0.7 millimetres, so the fringes at a shadow’s edge are sub-millimetre and low in contrast against any real illumination. At a wavelength of five centimetres and a path of ten kilometres it is over twenty metres, so the same pattern is spread across the landscape and the “shadow” of a hill is a hundred metres of gradual fading rather than an edge.

The two regimes are one figure with the axis relabelled, which is the strongest form of the claim that diffraction is not a small correction to geometrical optics but the same phenomenon at a different ratio.

Where the model stops #

Everything here is scalar. The field has been treated as a single number, which throws away polarisation. That is an excellent approximation when the aperture is much larger than a wavelength and a bad one when it is comparable — a slit narrower than a wavelength diffracts quite differently for light polarised along it and across it, and the scalar theory has no way to say so.

The obstacle is assumed to be a mathematical shadow. A real edge is made of something: it has a thickness, an edge radius, and a material that has its own reflectivity and its own phase shift. The theory here replaces it with the statement that the field is exactly zero over one half-plane and exactly unobstructed over the other, which is Kirchhoff’s boundary condition and is known to be inconsistent — the field it produces does not satisfy the assumed boundary values when evaluated back at the screen. It works because the inconsistency is confined to within a wavelength of the edge.

The paraxial approximation is in the exponent. The path difference was expanded to second order in $s/L$, which is what makes the phase a clean quadratic and the integrals Fresnel’s. At large angles the next term matters and the curve is not this curve.

What the pictures cannot show #

The spiral is a picture of a sum, and the sum is over a continuum. What is drawn as a smooth curve is the limit of adding infinitely many infinitesimal contributions, and no drawing can show the individual wavelets that the construction is nominally about — they have been integrated away before the curve exists. The construction’s own objects are therefore invisible in the figure that makes it quantitative, which is a common enough fate for a good picture.

The figures also draw intensity, and intensity is a time average. Nothing here shows the field oscillating; the fringes are what a detector reports after averaging over many cycles, and a fast enough instrument would see something quite different and much less useful.

Where the ladder goes next #

This ladder began with every point of a front being a source and has now given the wavelets amplitudes and phases, which is what turns a construction into a calculation. The rungs above it: the Fresnel–Kirchhoff integral in full, with the obliquity factor derived rather than inserted; the zone plate, which is a lens made by blocking alternate zones and has a focal length that depends on wavelength in the opposite sense to a glass one; the transition to the eikonal limit, where the stationary-phase point of this integral becomes Fermat’s path; and the angular-spectrum method, which does the whole thing in the Fourier domain and makes the near field and the far field the same propagator evaluated at two distances.

The habit worth carrying away is the distinction the hero figure was drawn for. The arc is what was added; the chord is what arrived. Wherever contributions carry phases, the total is not the sum of their sizes, and a picture that shows only where things came from cannot say how much of them is left.