The grain that is in the light

Illuminate a wall with a laser and the wall acquires a texture: a fine, high-contrast granularity that swims about when the head is moved. Illuminate the same wall with a torch and there is nothing. The wall has not changed.

The texture is in the light. It is what a coherent wave becomes after it has been scattered by anything rough on the scale of a wavelength, and its properties turn out to be almost entirely independent of what did the scattering.

Why a rough surface does this #

A surface rough by more than a wavelength scatters light from each point with a phase determined by the local height. Heights vary over a wavelength or more, so the phases are effectively random and uniformly distributed.

At any point in the observing plane the field is the sum of contributions from every illuminated point of the surface, each with its own random phase. That sum is a random walk in the complex plane: many small steps in uniformly random directions.

A speckle field at a point is a random walk in the plane, with the steps being the contributions from each scatterer and the endpoint being the complex amplitude. That single identification supplies everything: the endpoint of a long random walk has a Gaussian distribution in each component, the intensity is the squared modulus, and the exponential intensity distribution follows without any optics being done at all.

A random walk of many steps has real and imaginary parts that are independent Gaussians of equal width, by the central limit theorem. The intensity is the sum of their squares. And the sum of two squared Gaussians is an exponentially distributed quantity — which is the entire statistical content of speckle, reached in three sentences and depending on nothing about the surface except that it was rough.

This is why two independent lamps never produce visible interference and a laser on a wall always does. The lamp’s phases are random and changing far faster than any detector can follow, so the pattern averages away within a femtosecond; the laser’s are random and static, so the pattern stands still.

The brightness most likely to be nothing #

The exponential distribution has a consequence that surprises everybody who has not seen it, and it is directly measurable.

The most probable intensity is zero. There is no peak near the mean, no typical brightness, and the darkest bin is the fullest. A coherently illuminated surface is more likely to be black at any given point than to be anything else.

The contrast — the standard deviation of the intensity divided by its mean — comes out at 1.01 in the computed pattern, against exactly one for an exponential distribution. That is the definition of a fully developed speckle, and it is a strong statement: the fluctuations are as large as the signal.

The exponential shape has a second consequence that matters for anybody trying to measure through speckle. A distribution with a peak has a well-defined typical value, so averaging a few samples gets close to it quickly; an exponential does not, so a photometric measurement made through a speckle pattern converges as slowly as it possibly could. In a coherent image, the signal-to-noise ratio of a single-pixel intensity measurement is exactly one whatever the light level — the fluctuation is as large as the signal, and turning the laser up does not help at all. That is a fundamentally different situation from shot noise, which does improve with brightness, and it is the reason coherent imaging is grainy in a way that photon-starved imaging is not.

It also explains why the texture looks the way it does. A distribution with a peak would give a mottled grey; an exponential gives isolated bright grains against a mostly dark field with genuine zeros in between. Those zeros are real. At each of them the complex amplitude has passed exactly through the origin, and around each is a point where the phase is undefined — an optical vortex, of which a typical speckle pattern contains one per grain.

The size that belongs to the instrument #

The claim that most needs testing is that the grain size has nothing to do with the surface.

They lie on a line of slope minus one. Doubling the aperture halves the grain, exactly, and the surface never enters.

The reason is that a speckle pattern is a diffraction pattern of the aperture, positioned randomly. Each grain is about the size of the smallest feature the aperture can produce, which is the resolution limit — $\lambda z/D$ for an aperture of diameter $D$ at a distance $z$. What the rough surface supplies is the randomness of the positions and phases; the scale is supplied by whatever the light passed through last.

That is a statement with two useful directions. It means speckle limits the resolution of every coherent imaging system at about one resolution element, so a coherent image is grainy at exactly the scale at which it is sharp. And it means measuring the grain size measures an aperture that may not be reachable otherwise — including the pupil of a living eye, which is what laser speckle refractometry does, and the moving surface of a retina, which is what laser Doppler flowmetry reads.

The swimming motion when the head moves is the same fact felt directly. The pattern on the retina is a diffraction pattern of the eye’s own pupil, so moving the eye moves the pattern; whether it appears to move with the head or against it depends on whether the eye is focused in front of the wall or behind it, which makes a laser speckle pattern a free refractometer.

The numbers, for a laser and a wall #

The scaling is only useful with a number attached to it, so here are three.

The diffraction limit of a circular aperture is the same length as a speckle grain seen through that aperture, and it is the same calculation. That is worth seeing plainly because it explains the one counter-intuitive fact about speckle: the grain size belongs to the observing system and not to the rough surface. Change the aperture and the grains change size; change the roughness and they do not.

A laser pointer on a wall two metres away, seen by eye. The relevant aperture is the pupil, about 3 millimetres. At 650 nanometres the grain size on the retina corresponds to an angular size of $\lambda/D \approx 2\times10^{-4}$ radians, which at two metres is about 0.4 millimetres on the wall. That is why the texture is just at the edge of being resolvable and why it looks finer if the pupil is dilated in a dark room.

The same wall through a camera at f/8. The aperture is now the lens’s, and the grain in the image plane is $\lambda \times f/D = 650\,\text{nm} \times 8 \approx 5$ micrometres — comparable with a pixel, which is why speckle in a photograph of laser-lit scene is a per-pixel effect rather than a visible texture.

A star through a four-metre telescope. The grain in a short exposure is $\lambda/D \approx 1.4\times10^{-7}$ radians, which is 0.03 arcseconds, against the arcsecond or so of atmospheric blur. The pattern therefore contains structure a thousand times finer than the blurred image appears to have, and recovering it is what speckle interferometry does.

The general rule to carry away is that a speckle grain is one resolution element of whatever aperture is looking. That is not an approximate statement — it is the same diffraction calculation with a random phase distribution behind it instead of a point source — and it is why the grain size is the most useful single measurement that can be made on a speckle pattern.

Getting rid of it, and what that costs #

Speckle is a nuisance in every coherent imaging application, and the arithmetic of removing it is unforgiving.

Adding $N$ independent patterns reduces the contrast as $1/\sqrt{N}$, which is ordinary averaging and holds no surprises. The difficulty is entirely in the word independent.

There are only a few places to get independence. Polarisation gives a factor of two and no more, because there are two of them. A spread of wavelengths gives as many independent patterns as there are coherence lengths across the surface roughness — so a source with a shorter coherence length gives more, which is precisely what coherence length measures. Motion gives as many as fit inside the exposure, at the cost of blurring anything that is also moving.

Each of those spends something the system wanted. Polarisation is often carrying information; spectral spread costs resolution in a spectrometer and chromatic sharpness in a projector; time costs frame rate. That is why a laser projector is harder to engineer than a lamp-based one, why medical ultrasound images still look grainy after forty years of effort, and why synthetic-aperture radar images are processed in “looks” that trade resolution for contrast at exactly this square-root rate.

Speckle used on purpose #

The same statistics that make speckle a nuisance make it an unusually sensitive measuring instrument, because a pattern with contrast one changes visibly for a very small change in anything.

A speckle pattern is a great many interference patterns at once, and correlating one against another is what speckle metrology does. Displace the surface and the pattern shifts; deform it and the pattern decorrelates — so a quantity that looks like noise becomes a measurement of displacement at a fraction of a wavelength, using no reference beam and no alignment.

Speckle interferometry compares two patterns from the same object before and after it is deformed. Because each grain shifts by an amount set by the local displacement, subtracting the two images gives fringes at contours of constant displacement, and the sensitivity is a fraction of a wavelength over a whole surface at once — with no reference surface, no polished optics, and no contact.

Laser speckle contrast imaging reads the contrast rather than the pattern. A moving scatterer makes the pattern change during the exposure, which lowers the contrast in that region; mapping the contrast therefore maps the flow. It is used to image blood perfusion through skin in real time, using nothing but a laser, a camera and the arithmetic in the previous section run backwards.

Stellar speckle interferometry was the first technique to beat atmospheric seeing. A short exposure of a star through a turbulent atmosphere is a speckle pattern whose grain size is set by the telescope’s aperture rather than by the atmosphere’s; correlating many such exposures recovers detail at the telescope’s diffraction limit, which long exposures had been throwing away since telescopes were built.

The same statistics elsewhere #

Nothing in the derivation used light. What was used was a sum of many contributions with random phases, and that occurs widely.

Interference built up from individual arrivals is the same superposition with two sources instead of many, and the difference is only in the number of phases being added. What makes speckle random is that the phases are random; what makes a two-slit pattern regular is that they are not. The statistics change and the physics does not.

Radio fading in a multipath environment is speckle in time and space: the received amplitude is a random walk over paths, so the power is exponentially distributed and deep nulls are common. That is Rayleigh fading, and the whole design of a mobile radio link is a response to its statistics.

Ultrasound speckle is the granularity of every medical ultrasound image, and it is not tissue texture: it is the same random walk over sub-resolution scatterers.

A rough sea seen in sunlight does the same thing incoherently, which is why the glitter path is a field of bright points rather than a smooth reflection; the statistics differ because the sun is not coherent, but the geometry — a random surface mapping directions to positions — is the same fold that makes a caustic.

Sea clutter in a radar return has the same origin and nearly the same distribution, with departures that carry information about the sea state.

In each of those the contrast being one, rather than something smaller, is the thing that has to be designed around, and the square-root law for averaging it down is the same law.

Why it took until lasers to notice #

Speckle is a consequence of coherence and nothing else, so in principle it could have been seen at any point in the history of optics. In practice it was described in passing by Newton, seen and named by Exner in 1877 with sunlight through a fine aperture, and then very nearly forgotten for eighty years.

A thermal source has a coherence area far smaller than a laser’s, so getting a visible speckle pattern from one requires an aperture small enough to throw most of the light away. That is why the phenomenon waited for lasers: it was always there in principle, and until 1960 nobody had a source bright enough to see it with the aperture stopped down that far.

The reason is that a thermal source has a tiny coherence area and a short coherence length, so producing a fully developed pattern from one means stopping the beam down until there is nearly nothing left. Exner managed it and the effect stayed a curiosity, because nobody could do anything with a pattern that faint.

The laser changed the economics rather than the physics. A milliwatt of laser light has a coherence area larger than any aperture likely to be pointed at it, so a fully developed pattern is available at full brightness. Within a couple of years of the first lasers the effect had been rediscovered, and it went from curiosity to nuisance to instrument inside a decade — which is a common trajectory for phenomena that had been invisible only for want of a source.

There is a moral in that worth carrying to any argument about whether an effect is “important”. Nothing about speckle changed in 1960. What changed was the availability of a source that made it easy to produce, and half of what is now known about coherence was learned by making a nuisance measurable.

The pattern is not noise, and it can be inverted #

Everything above treats speckle as randomness to be averaged down. It is not random in the sense that matters: a static rough surface performs a fixed linear operation on the incoming light, and a fixed linear operation can be measured and undone.

That observation, acted on in 2007, produced one of the more startling demonstrations in optics. Put a layer of white paint on a slide — opaque, strongly scattering, the sort of thing light is supposed not to get through in any useful way — and illuminate it with a beam whose wavefront can be adjusted pixel by pixel. Watch one point behind the slide and adjust the pixels one at a time, keeping any change that makes that point brighter.

After a few thousand adjustments the light behind the paint is not a speckle field. It is a focus, brighter than the surrounding speckle by a factor of roughly the number of pixels controlled — a thousand in the first demonstration, and orders of magnitude more since.

Nothing has been done to the paint. The scattering was always deterministic; what was missing was knowledge of which incident phases arrive at the chosen point in step, and the optimisation finds them. Measuring the whole map at once — the transmission matrix relating every input mode to every output mode — turns the scattering medium into an optical element that can be used deliberately, and images have been sent through layers of tissue, through eggshell, and along single multimode fibres whose mode mixing had made them useless for imaging.

The reframing is the point. A speckle pattern is not disorder; it is a complicated but perfectly definite transformation, and calling it noise is a statement about what has been measured rather than about the light.

The pattern nobody can copy #

The same determinism has a use that depends on the pattern being unrepeatable by anyone else.

Take a small token of some strongly scattering material — a resin block with particles stirred through it — and record the speckle pattern it produces for a beam arriving at a particular angle. The pattern depends on the position of every scatterer in the block, so it is an extraordinarily detailed fingerprint of an object whose microstructure nobody controlled during manufacture and nobody can measure well enough to reproduce.

Reading the token is easy and forging it is not. Duplicating the pattern would mean placing several million scatterers to within a fraction of a wavelength, which is beyond any fabrication process, and the token gives a different pattern at every illumination angle — so a copy that matched one reading would fail the next.

Devices built this way are used to authenticate documents and components, and to generate cryptographic keys that exist nowhere except in the physical object: the key is not stored, it is measured, and an attacker who takes the token apart destroys the thing they were trying to read.

It is an unusual security argument, in that its guarantee comes from a manufacturing limit rather than from a computation being hard. And it rests on the statistics at the top of this page — a pattern of contrast one, with grains as fine as the aperture allows, carries an enormous number of independent numbers, and that is what makes it worth reading.

What the pictures cannot show #

The pattern here is fully developed. That requires many scatterers per resolution element and a surface rough by more than a wavelength. A weakly rough surface gives a lower contrast and a distribution that is not exponential, and the transition between the two is its own subject.

Everything is monochromatic and one polarisation. Real speckle from a real laser is partially polarised and partially averaged over the source’s linewidth, and the measured contrast is correspondingly below one.

The transform is discrete. The pattern is band-limited by construction, which is also true of real speckle but with a different cutoff; the histogram’s agreement with the exponential is to eleven per cent in the well-populated bins, and the residual is sampling rather than physics.

The aperture is uniformly illuminated and circular. A real pupil is apodised, obstructed or aberrated, and the autocorrelation of the pattern reports its actual shape rather than the ideal one — which is the basis of measuring an aperture from its speckle and is a feature rather than a limitation.

And nothing here is dynamic. The interesting applications all use the pattern’s change with time, which needs a model of what is moving, and that is a different calculation from the static statistics drawn here.

The ladder from here #

Later rungs on this anchor: the joint statistics of intensity and phase, and the optical vortices at the zeros; partially developed speckle and what a low contrast measures; the speckle of a summed field, where the object’s own structure begins to show through; and speckle in the image plane rather than the far field, where the grain size is set by the imaging system’s point spread function and the pattern carries the object’s shape.

The neighbouring ladders are how far a wave can remember, which is the coherence length that decides how many independent patterns a source can supply, why two lamps never interfere, which is the same randomness moving too fast to see, and how far apart two things have to be, which is the resolution limit the grain size turns out to equal.