The mirror that sends a wave back to where it began

Every front is a source set out Huygens’ construction: each point of a wavefront acts as a source of secondary wavelets, and the next front is their envelope. The spiral that says how much light arrives turned it into a calculation, the wave that comes from the rim found diffraction concentrated at an aperture’s edge, the backward wave Huygens had to remove found the obliquity factor that kills the wavelets’ backward half, and the fan of plane waves inside every beam rewrote a beam as a spectrum of directions.

Kirchhoff’s theorem, behind all of those, says something stronger than Huygens did: the field everywhere inside a closed surface is fixed by the field and its slope on that surface. The theorem is usually read as a way of calculating. Read as an instruction, it becomes a machine. Record a wave over a surface as it passes; re-emit, from every point of the surface, exactly the recorded wave in reverse order; and the field inside must be the original wave running backwards, converging on the source that made it. This essay follows that machine — the time-reversal mirror — through the limit that stops it making a perfect point, the smaller limit a partial mirror imposes, and the surprise that disorder between source and mirror makes it better.

Running a wave backwards #

The wave equation for sound in still air, or for light in a transparent medium, has a property the heat equation lacks: replacing the time $t$ by $-t$ turns any solution into another solution. A wave that spreads out from a source, run backwards, is a wave that converges onto it. The equation that only runs forwards found diffusion forbidding exactly this; waves allow it, as long as nothing absorbs them.

A converging wave is almost never seen in nature, because nothing arranges the necessary conditions: every point of a large sphere would have to emit, at precisely the right moments, the reverse of what it received. The solution that is thrown away found the same converging, “advanced” solutions of Maxwell’s equations discarded on the grounds that no physical process sets them up. A time-reversal mirror is a physical process that sets them up. Each element of the mirror is a transducer that records the wave arriving at it, stores the recording, and plays it back reversed in time. In the frequency domain, reversing time is the same as taking the complex conjugate of each frequency’s amplitude, and for monochromatic light the same device is called a phase-conjugate mirror.

There is one more ingredient, which makes the mirror work in media nobody has mapped. Swap the ends and nothing changes found reciprocity: in a medium without magnetic fields or flow, a wave going from A to B and a wave going from B to A are affected in exactly the same way. The wave that went from the source to each mirror element was shaped by everything along its path; the reversed wave going back from that element to the source is shaped by the same path in exactly the reverse way, and undoes it.

The spot a closed mirror makes #

The figure computes the simplest case, a mirror that completely surrounds a point source in empty space. The recorded wave, played back, converges from every direction and makes a spot about 0.6 wavelengths across at half amplitude. It is not a point. The returning wave cannot carry any detail finer than about half a wavelength, because such detail lived only in the evanescent near field of the source — the part of its field that decays within a wavelength and never travelled to the mirror at all. How far apart two things have to be found the same half-wavelength limit on any image formed by travelling waves.

There is a second reason, more particular to time reversal. In the original experiment the source emitted and nothing absorbed. In the reversed one the wave converges onto the source’s position, but there is nothing there to absorb it, so it passes straight through and diverges again. The converging and diverging waves overlap and interfere, making a standing-wave pattern whose central spot is the one in the figure. Mathematically the reversed field is the imaginary part of the Green’s function — the difference of the outgoing and incoming waves — whose central lobe is a Bessel function. Experiments that add an “acoustic sink”, a transducer at the focus that emits the reverse of the original source at the right moment, cancel the diverging wave and have made spots well below half a wavelength, because the sink restores the missing near field. Without it, half a wavelength is the floor.

A mirror on one side only #

Real mirrors do not surround their source. A row of transducers on one side sees the source only through a cone, and can send the wave back only through the same cone.

The figure moves the mirror to one side: 32 elements spanning ten wavelengths, forty wavelengths from the source. It sees the source through a cone fourteen degrees across, and it focuses exactly as a lens of that size and distance would: a spot 4.8 wavelengths wide, about the wavelength times the distance divided by the aperture. The reversed wave carries only the directions the mirror received, and the range of directions sets the size of the spot, as the fan of plane waves inside every beam found for any beam. Time reversal has no special power here: in empty space a partial mirror is just a lens with an unusual way of being aimed.

A thicket that sharpens the focus #

Now fill the space between the source and the mirror with scatterers.

The figure puts a slab of two hundred scatterers, fifteen wavelengths thick, across the path. A wave from the source reaches the mirror having bounced among them — some of it directly, most after one or several scatterings — and arrives as a jumble with no recognisable wavefront. An ordinary lens pointed at the source through such a slab would form no image at all. The time-reversal mirror records the jumble, reverses it, and sends it back through the same slab; reciprocity guarantees that every scattering is undone in reverse order, and the wave converges on the source. The focus is not worse than in empty space. It is better: 1.5 wavelengths wide against 4.8.

The reason is that the scatterers have enlarged the mirror. Waves from the source that would have missed the mirror entirely — heading off at wide angles — were redirected into it by scattering, so the mirror has recorded waves that left the source over a much wider range of directions than its own aperture subtends. Played back through the slab, those waves retrace their routes and arrive at the source from those same wide angles. The slab acts as a lens much wider than the mirror, and a wider lens makes a smaller spot. Mathias Fink’s group in Paris demonstrated this with ultrasound in 1995, focusing through a forest of two thousand steel rods to a spot six times narrower than the same mirror achieved in water alone. It is a rare case in physics where adding disorder improves an instrument’s resolution, and it works only because the instrument does not need to know what the disorder is.

A long jumble in, a short pulse back #

The same undoing happens in time.

A short pulse emitted by the source arrives at a mirror element as a long signal: first the waves that took the most direct routes, then a long tail — the coda — of waves that wandered longer among the scatterers, each path arriving at its own time. The figure shows one element’s recording: nearly fifty periods of ringing from a pulse a few periods long. When every element plays its recording backwards, the latest arrivals are emitted first and the earliest last, and every path is retraced so that all its contributions arrive back at the source at the same instant. The fifty-period jumble recompresses into a pulse a few periods long.

That recompression is the principle of several technologies. Underwater acoustic communication, where the ocean’s reflections from the surface and floor smear every transmitted signal, uses time reversal to recompress a message at the receiver. Wireless systems that reverse the channel’s measured response before transmitting — precoding, in the engineering language — focus radio energy on a single receiver in a cluttered room. And one of the most striking demonstrations used a single transducer in a closed reverberating cavity: the coda of a single pulse, bouncing for a long time off the cavity’s walls, carried enough information that playing it back from one point refocused a sharp pulse at the source, the cavity’s reverberation acting as a mirror of many elements.

Why the mirror is not a trick with the arrow of time #

Running a wave backwards sounds as though it should violate the second law of thermodynamics, and it is worth being exact about why it does not. The scattered wave arriving at the mirror is not disordered in the thermodynamic sense: it is a single, perfectly definite wave, complicated but fully determined by the source and the medium, and carrying no entropy of its own. What makes it look disordered is only that its structure is spread over many elements and a long time. The mirror records that structure and re-emits it, supplying the energy from its own amplifiers; it does not collect heat from the medium or turn randomness into order.

The distinction matters in practice. A wave that has been partly absorbed — its energy turned into the random motion of molecules in the medium — has lost exactly the part that cannot be reversed, because the mirror never recorded where that energy went. A medium that has moved since the recording has scrambled the paths in a way the recording does not describe. Time reversal works for as much of the wave as remained a wave, in a medium that stayed still, and it fails in proportion to what did not. That is also why the arrival that keeps arriving — the tail a pulse leaves behind in two dimensions — is not an obstacle: it is part of the wave, recorded and reversed like the rest.

A corner cube for waves of any shape #

An ordinary mirror reverses only one component of a wave’s direction; the corner that sends light home reverses all three, returning a ray along the line it came from whatever its angle, but only for a ray — a plane wave — and only if the wave arrives through a clear medium. A time-reversal mirror is the generalisation to waves of arbitrary shape. It returns not one direction but the entire pattern of directions, amplitudes and delays that the source produced, so that the wave retraces every path, curved, scattered and reflected. A corner cube sends a laser beam back to the telescope that fired it at the Moon; a time-reversal mirror would send back the whole distorted wavefront a star’s light acquires in the atmosphere, undistorted by the return trip, which is the principle behind the adaptive optics that correct telescope images — with the atmosphere measured and the correction applied by a deformable mirror, rather than recorded and replayed.

Contrast bought with bandwidth #

A single frequency refocused through a scattering slab gives a spot surrounded by speckle — the grainy background that the grain that is in the light found in any wave scattered by disorder. The spot is the one place where every scattered contribution arrives in phase; everywhere else they arrive with random phases and add to a random amount. Using many frequencies helps, because the speckle pattern is different at each frequency while the spot is always at the source. Summed over frequencies, the spot grows coherently and the speckle only on average, and the contrast rises, as the figure shows: from 23 times the background with one frequency to 52 with twenty-five. Frequencies close enough together make nearly the same speckle and add little; the useful ones are those separated by more than the inverse of the coda’s duration, which is why a longer, more tangled coda means more independent looks through the disorder and a cleaner focus.

What the mirror is good for #

The most widely used application is medical. A kidney stone, struck by an ultrasound pulse, reflects some of it; a time-reversal mirror records the echo and sends back its reversal, amplified, which converges on the stone through the tissue in between — tissue whose varying sound speed would defocus an ordinary focused beam — and breaking the stone takes less energy delivered elsewhere. Iterating the process, sending back the reversed echo of the reversed echo, converges on the most strongly reflecting target in the field, a procedure that selects targets automatically. The same idea is used to focus ultrasound through the skull, whose thickness and density vary and distort a beam; recordings from a source placed or imaged at the target let the reversed wave correct the distortion. In non-destructive testing, reversed echoes locate flaws in metal parts, and in seismology, reversing recorded seismograms numerically and propagating them back into a model of the Earth locates the sources of earthquakes and tremors.

Where the reversal stops #

The mirror relies on three conditions, and each can fail. The medium must not change between recording and playback; moving scatterers, flowing fluids and changing temperatures break the retracing, which is why focusing through a living body must be fast. The medium must not absorb much: absorption is not time-reversible, because a reversed absorbing medium would have to amplify, and absorption both weakens the returning wave and blurs the focus in proportion to the energy lost. And the medium must be reciprocal: in the presence of a magnetic field or a flow, the wave from B to A is not the reverse of the wave from A to B, and swap the ends and nothing changes found the devices that exploit exactly that asymmetry. The model in the figures is two-dimensional, uses point scatterers and a simplified form of the wave’s spreading from each point, and treats the transducers as ideal; real experiments are three-dimensional and sample the field at discrete points with finite bandwidth, all of which the model’s numbers should be read against.

What the pictures cannot show #

The figures show profiles and single traces, not the wave itself moving. What an experiment shows, filmed with a scanning hydrophone, is striking: a jumble of wavelets spreading across the slab, then, after the playback, the same jumble gathering itself up and running backwards into a single converging front that collapses onto the source. The pictures also cannot show that the focus is found without any knowledge of the medium. Nobody computed the slab’s scattering; the mirror measured its effect and undid it, and the same mirror would work through a different slab, a skull or a harbour without being changed.

Still open: how much can be recovered from how little #

A time-reversal mirror records only part of a wave: a finite aperture, a finite band, a finite time. How close to the original source the refocused wave can come, given what was recorded, is a question about information: how many independent pieces of the scattered field the mirror captures, and how the medium’s disorder multiplies them. In strongly scattering media, and in reverberant cavities, remarkably few transducers suffice, because the medium mixes the field so thoroughly that each transducer samples much of it. Whether time reversal can be pushed to image through thick biological tissue with light — where the field changes in milliseconds and the recording must be done optically, by digital phase conjugation — and how deep it can reach before the tissue’s motion and absorption defeat it, are open questions in optical imaging.

The habit worth carrying away is to ask of any wave equation whether it can be run backwards, and what would be needed to do it. A lossless, reciprocal medium lets a recorded wave be sent back along every path it took, so a time-reversal mirror refocuses on its source without knowing anything about the medium — and scatterers that would ruin a lens widen the mirror, shrinking its focus from 4.8 wavelengths to 1.5 in the model here. Half a wavelength stays the floor, because the part of the source’s field that was finer than that never travelled far enough to be recorded.