The grating that photographs itself

Henry Fox Talbot reported it in 1836 in a paper about looking at gratings through a lens, and it reads as an observation nobody had expected to make: moving the eyepiece back and forth, the grating reappeared over and over, and between the reappearances the bright lines had changed places with the dark ones.

Why it happens #

Take an object with period $d$, decompose it into its spatial frequencies, and propagate.

Each Fourier component of the transmitted field is a plane wave leaving at its own angle, $\sin\theta_m = m\lambda/d$. Travelling a distance $z$ along the axis, a wave at that angle acquires a phase that differs from the axial one by

$$\Delta\phi_m = -\frac{\pi \lambda z m^2}{d^2}$$

in the paraxial approximation, since $\cos\theta_m \approx 1 - \tfrac12(m\lambda/d)^2$.

Now put $z = 2d^2/\lambda$. Every phase becomes $-2\pi m^2$, which is a whole number of turns for every $m$ whatever it is. So every component comes back to the phase it started with, the sum is the sum it started as, and the field is the object.

The orders a grating produces are the components being propagated, and the important feature is the $m^2$. Each order acquires a phase on the way that goes as the square of its index — so at a distance where every one of those phases is a whole multiple of $2\pi$, all the components arrive back in step and the original pattern reassembles itself. The revival is a coincidence of squares, and it is exact rather than approximate.

It is worth pausing on how little that argument used. Nothing about the shape of the object entered: not the duty cycle, not whether the grating blocks or retards, not how many harmonics it has. All that was needed is that the object be periodic, so that its spectrum is a set of orders labelled by integers. Any periodic object whatever revives at $2d^2/\lambda$, and two objects of the same period revive at the same distance however different they look.

That $m^2$ is what makes the effect exact rather than approximate. A revival that needed every one of a hundred orders to be nearly in phase would be a poor image; a revival in which every phase is precisely a multiple of $2\pi$ is a perfect one.

The planes in between #

The same expression predicts what happens at simple fractions of the Talbot distance, and the predictions are strange enough to be worth checking.

At $z = z_T/2$ the phase is $-\pi m^2$, which is $0$ for even $m$ and $-\pi$ for odd $m$. Multiplying the odd orders by $-1$ and leaving the even ones alone is precisely a shift of half a period. So the plane halfway to the revival holds the object translated sideways, and the translation costs nothing and involves nothing moving.

At $z = z_T/4$ the phase is $-\pi m^2/2$, which is $0$ for even $m$ and $-\pi/2$ for odd. That turns the odd part of the object by a right angle relative to the even part, and what appears depends on what the even part is.

None of it is interference between a pair of beams, which is the model most accounts of diffraction start from. When two waves meet they simply add, and what is being added here is a whole discrete spectrum at once, each member carrying a phase that depends on the square of its order.

Forty-five years to an explanation #

Talbot’s 1836 note is a page long and contains no theory. He was using a lens and observed that “it was very curious to observe that though the grating was greatly out of focus, yet the appearance of the bands was by no means confused” — and then that the colours changed as the eyepiece moved, and that at some positions the bright and dark bands exchanged places.

Rayleigh explained it in 1881, and it is worth noticing what he had that Talbot did not. Not better apparatus: the observation is easy. What he had was the habit of decomposing a periodic object into its Fourier components and asking what happens to each — an approach that arrived with Fourier’s work on heat in 1822, spread into optics through the 1860s and 1870s, and became the standard way to think about diffraction only in that period.

Rayleigh’s paper gives the distance as $2d^2/\lambda$ and remarks that a grating so illuminated could be copied without a lens, which is precisely the lithographic application taken up a century later.

The forty-five-year gap is characteristic of the near field. Fraunhofer diffraction — the pattern far away — was worked out first, because far away is where the arithmetic is easiest and where an astronomer’s instruments live. The intermediate region, where the quadratic phase matters and the far-field approximation does not apply, was harder to compute and had no obvious use, and it was mostly left alone until there were reasons to build things a millimetre apart.

Where it lives #

The Talbot distance contains the pitch squared, and that square confines the effect to a narrow range of objects.

So the effect is a property of gratings between a few micrometres and a few hundred, which is exactly the range of every ruled grating, every lithographic mask and every wire mesh. That is not a coincidence — both ranges are set by what can be made and what a bench can hold — but it does explain why the phenomenon is met so often by people not looking for it.

What the near field does at a straight edge makes the contrast: a single edge produces fringes that never repeat, because a single edge has a continuum of spatial frequencies rather than a discrete set. Periodicity is what turns a spreading pattern into a recurring one — the discrete orders can come back into step, and a continuum cannot.

The pattern between the planes #

Everything above concerns particular depths. What happens at the depths in between is the more interesting object, and it has a name.

Plot the intensity as a function of position across the grating and distance behind it, and the result is a two-dimensional pattern called the Talbot carpet. It is periodic in both directions — period $d$ across, period $z_T$ along — and inside one cell it is not smooth at all. At every rational fraction $p/q$ of the Talbot distance the pattern is a superposition of $q$ copies of the object, displaced and weighted; at irrational fractions it is none of those. The result is a structure with detail at every scale, built out of nothing but a square wave and a quadratic phase.

That the fractional planes give $q$ copies is worth stating because it is what the effect is used for when a copy is not wanted. A Talbot array illuminator is a grating placed a rational fraction of a Talbot distance in front of a target, arranged so that the light which arrived spread over the whole period is concentrated into $q$ narrow lines — with no absorption anywhere and therefore no loss. It converts a uniform beam into a comb of bright spots at essentially unit efficiency, which no aperture mask can do.

The fractions also explain the doubled frequency at a quarter of the way. Two copies displaced by half a period is a pattern with twice as many features, and $q = 2$ is the first fraction after the trivial one.

What it is used for #

Because the self-image is exact and the intermediate planes are not, the effect is a sensitive way of measuring anything that disturbs a wavefront.

Wavefront sensing. Put a grating in a beam and photograph two planes behind it. A wavefront that is not flat shears the self-image, and the shear measures the wavefront’s slope. That is a Talbot interferometer, and it needs no reference beam.

Lithography without contact. A mask held one Talbot distance from a wafer prints its own pattern with no optics between them, and without touching. The technique has obvious appeal and an obvious weakness — the depth of focus is a fraction of the Talbot distance, so the gap has to be held to a micrometre.

X-ray phase imaging. Soft tissue absorbs X-rays almost not at all and refracts them slightly. A grating interferometer converts that refraction into a displacement of a Talbot self-image, which is measurable — so a Talbot pattern is what makes phase-contrast X-ray imaging possible on a laboratory source.

Everything has a wavelength, so a beam of atoms passing through a material grating shows the same revivals at a distance set by the de Broglie wavelength. That is worth having because it removes any suspicion that the effect belongs to light: it is a property of wave propagation behind a periodic screen, and the atoms behave identically at a wavelength ten thousand times shorter.

What it costs #

The illumination has to be coherent across several periods. A source of finite size illuminates the grating with a spread of directions, and each direction produces its own displaced self-image; the sum washes out unless the source’s transverse coherence length exceeds a few periods. With sunlight through a fine mesh the revival is visible and blurred, which is what Talbot saw.

How far a source’s light can be relied on to be in step across a transverse distance sets how many periods of the grating contribute, and therefore how sharp the revival is. A perfectly coherent source revives the pattern exactly; a partially coherent one revives a blurred version, and the blur grows with distance. That is the cost of the effect, and it is why it is a laboratory phenomenon rather than an everyday one.

And the light has to be one colour and one source. Fringe visibility falls as the delay between contributions approaches the length over which a wave can be relied on to remember its own phase, which for a broad spectral line is a few micrometres.

A finite grating has edges. The exact revival is a property of an infinite periodic object. A grating with $N$ periods produces a self-image over the middle of the field and a mess within a few periods of each edge, and the good region shrinks as $z$ grows.

Monochromatic light has been assumed. The Talbot distance depends on the wavelength, so white light gives a superposition of revivals at different depths. Interestingly the coloured planes that result are themselves useful, and the effect is sometimes deliberately exploited to make a plane whose colour reports its distance.

The grating has been an amplitude grating. A phase grating — one that retards the light rather than blocking it — has a quite different set of orders, no zeroth order at all if the retardation is half a wave, and its quarter-Talbot plane is where the doubled-frequency image appears most strikingly. Everything about the distances is the same, because the distances depend on the periodicity alone; everything about which plane shows what is different.

And the paraxial approximation is doing real work. The expansion of $\cos\theta_m$ to two terms is what makes the phase quadratic in $m$, and quadratic in $m$ is what makes every phase a multiple of $2\pi$ together. Keeping the next term breaks the exactness, and for a pitch of a few wavelengths it breaks it badly.

The same carpet in time #

The argument used two things and no more: a discrete spectrum, and a propagation that multiplies each component by a phase proportional to the square of its index. Neither is optical, and the exact analogue in time is worth following because it is used industrially.

Take a periodic train of light pulses at a repetition rate $f_r$ and send it down a dispersive optical fibre. Its spectrum is a comb — a set of lines spaced by $f_r$, labelled by an integer $m$. Dispersion means the group delay depends on frequency, so travelling a distance gives each line a phase whose leading term is quadratic in its offset from the centre, which is quadratic in $m$.

That is the same expression. The transverse coordinate has become time, spatial frequency has become optical frequency, and diffraction has become group-velocity dispersion — and the underlying equation is the same paraxial one in both cases. Everything in this essay carries over unchanged.

So there is a length of fibre at which every comb line’s phase is a whole number of turns together, and the pulse train emerges exactly as it went in, having been smeared beyond recognition in between. That is the temporal Talbot effect, and it was worked out in 1981.

The fractional planes carry over too, and this is where the use lies. At a rational fraction of the Talbot length the train becomes $q$ interleaved copies of itself — which is to say its repetition rate has been multiplied by $q$, with no gating, no modulation and no loss whatever. A ten-gigahertz train becomes a forty-gigahertz train by passing through a length of fibre. Nothing was added and no light was thrown away; the energy was redistributed in time by a quadratic phase, exactly as the array illuminator redistributes it in space.

The correspondence runs the other way as well, and is worth stating as a general translation rule. Diffraction in space and dispersion in time are the same operator, so every effect in one has a partner in the other: a lens is a quadratic phase in space and a chirp is a quadratic phase in time; a focus is a compressed pulse; the far field of an aperture is the spectrum of a pulse. Recognising the pairing turns half of Fourier optics into pulse shaping and back again.

And in a box #

The deepest version of the same argument is quantum mechanical, and it is deep because nothing had to be arranged for it.

A particle in a one-dimensional box has energies proportional to $n^2$. Any state at all is a superposition of those, and time evolution multiplies each term by $e^{-iE_nt/\hbar}$ — which is a phase proportional to $n^2$ times the elapsed time.

That is the Talbot condition with $t$ in place of $z$. There is a time at which every one of those phases is a whole number of turns together, and at that time the wavefunction returns to exactly what it was, however complicated it was and however thoroughly it had smeared out in between. It is called the revival time, it is $4mL^2/\pi\hbar$ for a box of width $L$, and it is exact.

Plotting the probability density against position and time produces a figure with the same structure as the Talbot carpet, and it is called a quantum carpet for that reason. At rational fractions of the revival time the wavefunction is a superposition of a small number of displaced copies of its initial shape — the same fractional structure, from the same integers.

Two comparisons sharpen what is doing the work. A harmonic oscillator’s energies are proportional to $n$ rather than $n^2$, so its phases come back into step after a single period and a wavepacket in it never spreads at all — which is why a coherent state of an oscillator behaves like a classical particle. A generic anharmonic potential has energies that are neither, so its phases never all return together and its revivals are partial and approximate. Exactness requires the spectrum to be a quadratic function of an integer, and the box and the grating both have one for the same reason: a wave confined to a period.

Revivals of this kind have been watched. Rydberg atoms, whose highly excited levels have very nearly the $n^2$ spacing of a box, show a wavepacket that spreads around its orbit, disappears, and reassembles on schedule; so do atoms held in an optical lattice. What Talbot noticed by racking an eyepiece back and forth in 1836 is the same arithmetic that governs how long a quantum system takes to forget its initial state and how long after that it remembers it again.

Where the model stops #

The revival is not an image in the sense a lens makes one. It reproduces one particular object at one particular distance, and it reproduces nothing else. Move the object and the pattern does not follow; change the pitch and the distance changes as its square. Nothing is being focused, and there is no conjugate relationship of the kind the lens equation describes.

What a lens does to the same spectrum is lay it out on a plane and transform it back, so a whole class of objects is imaged at once. Free propagation does something weaker: it reassembles one particular object at one particular distance, and only because that object was periodic. A lens images anything and a Talbot plane images only what was already repeating.

The intensity revives at more places than the field does. Because a detector sees $|A|^2$ and not $A$, there are additional planes at which the intensity pattern repeats although the field has not — the half-Talbot plane is one — and the distinction matters as soon as anything downstream is sensitive to phase.

The atom case is worth one more sentence, because it looks as though it needs a wave shared between particles and does not: the pattern appears one arrival at a time, each atom passing through the grating alone and landing where the revived intensity says.

And nothing here is a quantum effect, though the atom version reads like one. The argument is about a wave and a periodic object, and it works identically for light, for sound in a periodic waveguide, and for the surface of water behind a row of posts.

A localised packet spreads and does not come back, which is the contrast worth holding beside the revival. Both are the same propagation acting on a superposition; the difference is entirely in whether the components’ phases are commensurate. A continuum of them never re-aligns, and a discrete set does — periodically, forever, in principle.

What the pictures cannot show #

The waterfall figure stacks nine planes and draws each one’s intensity as a curve, which makes the pattern legible and makes it look like nine separate experiments. What is actually there is a continuous two-dimensional structure — the pattern is defined at every depth, not at nine of them — and its full form is an intricate fractal-like figure whose fine detail is not resolvable by any drawing of finite width.

Nor can any of these figures show the field. Everything drawn is $|A|^2$, and the whole mechanism lives in the phase of $A$: what happens between the planes is that each order’s phase advances at its own rate, which is invisible in every one of the profiles above. The half-period shift at $z_T/2$ is the one place the phase leaves a visible trace, and it does so only by moving the intensity sideways.

Where this ladder goes next #

Four rungs stand on diffraction. The first found where rays stop being enough and a shadow acquires a bright centre; the second turned diffraction into a resolution limit; the third counted what a thousand slits buy. This one asks what happens behind a grating rather than far away from it, and finds that free space reassembles the object exactly, at a distance containing the pitch squared.

The habit worth carrying away is about the difference a discrete spectrum makes. When a propagation multiplies each component by a phase, a discrete set of components can return to step and a continuous set cannot. The same distinction is why a plucked string sounds a note and a struck drum a thud, why an ideal crystal has sharp reflections and a glass does not, and why a wave packet spreads for ever while a periodic wave does not spread at all.

What is left on this ladder is the effect of breaking the periodicity. A grating with one line missing produces a self-image with a localised disturbance that propagates in a well-defined way, and following that disturbance is how a defect in a periodic structure is measured without touching it.