The photograph that keeps the spectrum

When two waves meet, they simply add, and everything that superposition has produced since — beats, fringes, pulses from locked phases, lattices of light — comes from adding two or more waves that started in different places. The simplest pair of waves to add, though, is a wave and its own reflection. They have the same frequency and the same wavelength by construction, they keep a fixed phase relation without any effort, and they overlap automatically in the space in front of any mirror. Their sum is a standing wave, which is old news on a guitar string. In light it took until 1890 for anyone to see one, and when they did, it answered a question about light that had been open for sixty years, and then within a year produced the strangest colour photographs ever made.

A wave that does not travel #

A wave travelling toward a mirror can be written as $\cos(kz + \omega t)$, with $z$ the distance from the mirror. A metal reflects light with its electric field inverted, so the reflected wave is $-\cos(kz - \omega t)$. Their sum, by the identity for the difference of two cosines, is

$$E = 2\sin(kz)\,\sin(\omega t).$$

The position and the time have separated. Every point in front of the mirror oscillates in step with every other, at an amplitude $2|\sin kz|$ that depends on where the point is but not on when. Nothing moves along $z$. Where $\sin kz = 0$ — at the mirror and every half-wavelength from it — the field is zero at every instant. These are nodes. Between them, at the antinodes, it swings to twice the amplitude of either wave alone.

The magnetic field of the same light does the opposite. In a travelling wave the electric and magnetic fields rise and fall together; in the reflected wave one of them is inverted and the other is not, so where the electric fields of the two waves cancel the magnetic fields add. The magnetic envelope is $2|\cos kz|$, largest exactly at the mirror. The two fields have swapped places: the standing wave has electric nodes and magnetic antinodes at the same points, a quarter-wavelength from electric antinodes and magnetic nodes. The node that is not standing still showed that a perfect reflector is what makes those nodes true zeros, and a good metal mirror is close enough to perfect for this purpose.

For green light, half a wavelength is 275 nanometres in air and 183 in a photographic emulsion, whose refractive index of about 1.5 shortens the wavelength in proportion. The pattern is therefore a stack of sheets, parallel to the mirror and a few hundred nanometres apart: invisible to any microscope of 1890, and finer than the grains of any ordinary photographic plate.

Which field darkens silver #

The question Otto Wiener set out to answer in 1890 is one that sounds strange now. Light was known to be a transverse wave, vibrating across its direction of travel; polarisation showed that. Augustin Fresnel had held that the vibration of polarised light was perpendicular to its plane of polarisation, Franz Neumann that it lay in it, and both theories fitted the optics of crystals equally well. When Maxwell’s theory arrived, the question changed form without being settled: light had two fields at right angles, and the old theories had in effect each chosen one. Which one does matter respond to?

A standing wave answers that directly, because the two fields have their nodes in different places. Wiener coated a glass plate with a photographic film of collodion about a thirtieth of a wavelength thick — thin enough to sample one level of the standing wave rather than average over many — and laid it on a silvered mirror, not flat, but tilted at an angle of about a minute of arc. The film rose through the standing wave on a long slant, cutting each sheet of the pattern along a line, so that sheets a few hundred nanometres apart in height became bands a millimetre apart along the film.

Developed, the film showed a series of dark bands, which was itself the first direct photograph of a standing light wave. The decisive part was the very beginning. If the magnetic field exposed the silver, the film would be darkest where it touched the mirror. If the electric field did, the film would be clear there, with its first dark band half a band-spacing along. Wiener’s film was clear at the contact. Whatever acts on the photographic emulsion is the electric field. Repeating the experiment with light striking the mirror at 45 degrees, where light polarised one way still makes a standing wave and light polarised the other way, its incoming and reflected fields now at right angles, makes none, he identified the electric field with Fresnel’s vibration and settled the old argument at the same time. Paul Drude and Walther Nernst repeated it in 1892 with a fluorescent film instead of a photographic one and found the same; later experiments with photoelectric layers did too. Matter feels the electric field of light because its electrons are charges, and the magnetic force on an electron moving at ordinary speeds is smaller by the ratio of its speed to the speed of light.

A colour written as a depth #

In 1891 Gabriel Lippmann turned the same pattern into a photograph. He made a plate with an emulsion of exceptionally fine grain, so fine as to be nearly transparent, and placed it with the emulsion side against a pool of mercury, which served as the mirror. Light from the scene, focused by an ordinary camera lens, passed through the glass and the emulsion, reflected from the mercury, and came back through the emulsion again. Every point of the image therefore contained a standing wave of the light falling on it, running up through the thickness of the emulsion, with sheets of maximum exposure half a wavelength apart.

The spacing of the sheets depends on the wavelength. Red light writes sheets farther apart than green, and green farther than blue, so the colour of the light at each point of the image is recorded as the period of a structure in the emulsion’s depth. Light of several wavelengths writes the sum of their patterns, because exposure is proportional to intensity and the intensities of different wavelengths add without interfering — the rule what adding does to the energy found for any two sources that cannot keep a fixed phase with each other. Averaged over the minutes of an exposure, a red wave and a green wave beat through every phase relation many times over, and their cross term averages to nothing. The figure shows what that sum looks like. A single spectral line writes a perfect stack of sheets, as deep as the emulsion goes. A band of wavelengths writes sheets that agree near the mirror, where every wavelength has its node, and drift apart with height, as their periods differ; their sum fades into a uniform grey within a distance that is shorter the broader the band, roughly $\lambda^2/(2n\,\Delta\lambda)$. White light writes a few strong sheets near the mirror and nothing above.

This is the same arithmetic as the phases that turn a glow into pulses, turned on its side. There, many frequencies added with equal phases at one instant made a pulse in time, and the width of the pulse was set by the spread of frequencies. Here, many wavelengths all with a node at one place — the mirror — make a burst of fringes in space next to it, and the depth of the burst is set by the spread of wavelengths. The mirror is the place where every component is in step.

Reading it back #

Developed, the exposed sheets become layers of metallic silver, or in later processes layers of slightly different refractive index, spaced exactly as the light spaced them. Lippmann then removed the mercury and looked at the plate by reflected white light. Each developed layer reflects a little of whatever falls on it. For most wavelengths those small reflections come back with assorted phases and cancel. For the wavelength that wrote the layers, each reflection travels exactly one wavelength farther, there and back, than the one before, and they all return in step and add.

Each layer acting alone is a weak reflector, a few per cent at most, and the plate’s colour is entirely an effect of their adding. The layer that makes a reflection vanish used the same addition in reverse: a single coating a quarter-wavelength thick arranges for two reflections to return half a wavelength apart and cancel, so that a lens transmits what it would otherwise have reflected. The Lippmann plate arranges for dozens of reflections to return a whole wavelength apart and reinforce. The difference between a coating that removes a reflection and a stack that makes a colour is only the spacing and the number of the layers.

The plate reflects the colour that made it, and nothing else, because the developed record is a periodic structure, and the gap a repeat opens is exactly a band of wavelengths that such a structure reflects. The condition is the Bragg condition, twice the layer spacing times the index equal to the wavelength, and the plate satisfies it by construction, because the light set the spacing in the first place. The figure computes the reflection directly, adding the weak reflection from each slice of the developed record with its phase, and finds the peak at the recorded wavelength to within a nanometre.

The width of the peak depends on how many layers take part. A thin emulsion holds only a few layers, their reflections agree over a broad range of wavelengths, and the reproduced colour is pale and unsaturated. A thick emulsion holds many, and the agreement is sharp. With eleven layers the peak is 46 nanometres wide; with twenty-seven, 18; with eighty-two, 6. This is the same result as what a thousand slits buy: the resolving power of a grating, the wavelength divided by the smallest difference it can separate, is the number of lines that take part. A Lippmann plate is a grating with its lines stacked in depth instead of laid side by side, and it obeys the same rule. Broad-band light, which wrote only a few layers near the mirror, replays a broad colour; a pure spectral line, which wrote layers all the way up, replays a pure one.

Two yellows #

Lippmann’s process has a property that no other colour photography shares. Every other method — the three-colour separations of the 1860s, the Autochrome plates of 1907, colour film, a digital camera’s sensor — records colour as three numbers at each point, measured through three broad filters, red, green and blue. That is all that is needed to fool the eye, which itself measures colour with three kinds of cone and nothing else; the cones are also packed only as finely as the eye’s own diffraction limit can use, as how far apart two things have to be found, so the eye discards detail in colour and in space by the same economy. It means, though, that the recording keeps only what the eye keeps. Two lights with quite different spectra that happen to stimulate the three cones in the same proportions look identical to the eye and are recorded identically by the film.

The pure yellow of a sodium lamp and an equal mixture of green and red light are such a pair, near enough. A three-colour film would record them as the same yellow and reproduce them by mixing the same dyes. A Lippmann plate records each as what it is. Exposed to the sodium light, it writes one stack of layers and replays one peak at 589 nanometres. Exposed to the mixture, it writes two superposed stacks of different periods and replays two peaks, at 544 and 631 nanometres. Each is lower than the single peak, because each line wrote half as much exposure and the reflected power goes as the square of the layers’ contrast. To the eye looking at the two plates, the colours are again nearly the same. To a spectrometer pointed at them, they are as different as the two lights were.

That makes a Lippmann photograph not a picture of a scene’s colours but a record of its spectra, one at every point. The difference is the one that the phases that turn a glow into pulses drew from the opposite direction, where two lights with the same spectrum could be entirely different in time: a measurement keeps some of what was there and discards the rest, and which part it keeps decides what can be asked of it afterward.

Why it did not win #

Lippmann received the Nobel Prize in Physics for the process in 1908, and it was abandoned as a practical method within a few years of the award. The reasons were all consequences of the physics. The emulsion’s grains had to be much smaller than the layer spacing, a few tens of nanometres, and so fine an emulsion is very insensitive; exposures ran to minutes in bright sunlight and much longer indoors. The mercury mirror was awkward, and the image was unique: like a daguerreotype, a Lippmann plate cannot be printed or copied, because the colour lives in a structure inside one piece of gelatine.

Viewing it was awkward too. The colours appear only by reflection, in a narrow range of angles, and they change with angle, because light crossing the layers obliquely sees a shorter path between them and the Bragg condition selects a shorter wavelength; tilted, a Lippmann plate shifts toward the blue. The reflection from the front surface of the glass washes the colours out, so the plates were usually viewed through a shallow glass prism cemented to the front. And the gelatine swells and shrinks with processing and humidity, changing the layer spacing and with it every colour in the picture. The Autochrome, a plate covered with dyed starch grains acting as millions of tiny colour filters, was slower than ordinary film but enormously faster than a Lippmann plate, could be viewed by transmission, and won.

The hologram it became #

The idea did not die. In 1958 Yuri Denisyuk, in Leningrad, read about Lippmann’s method and asked what would happen if the mirror were replaced by an object. Light passing through a thick emulsion and scattered back from an object behind it forms a standing wave in the emulsion just as the light reflected from mercury does, but now the pattern encodes not only the wavelength of the light but the direction and phase of the light scattered from each point of the object. Developed, the layers reflect white light into the same directions with the same phases. The result is a reflection hologram, viewable in ordinary white light, which Denisyuk published in 1962 and which is still the form of most holograms seen in museums.

The connection runs through every use of superposition so far. Superposition makes a pattern whenever waves with a fixed phase relation overlap; a recording medium thick enough to hold the pattern in three dimensions keeps not just intensity but the information that produced it; and the recorded pattern, lit again, diffracts light back into the waves that wrote it. Lippmann’s colour photograph is a hologram of a mirror. Wiener’s film was the first measurement of the pattern, Lippmann’s plate the first use of it, and Denisyuk’s hologram the general case.

Where the picture stops #

The figures treat light as a scalar wave meeting the mirror at normal incidence, and the mirror as a perfect reflector that inverts the field. A real metal mirror has a reflection phase slightly different from 180 degrees, which moves the first node a few nanometres into or out of the metal and shifts every layer by the same amount; mercury is a good mirror but not a perfect one. The exposure is taken as proportional to intensity and the developed refractive-index change as proportional to exposure, when real emulsions saturate and respond non-linearly, which writes harmonics of the layer spacing and gives a Lippmann plate weak reflections at half the recorded wavelengths and false colours in the replay. The replay is computed to first order in the reflection from each layer, which is accurate for faint layers; a strongly developed plate reflects so much that the light reaching deep layers is depleted, and its reflection peak saturates and broadens. The emulsion is taken as transparent and grainless, when real emulsions absorb, scatter and shrink.

The domain of the recording is light coherent over the depth of the emulsion, a few micrometres, which is why white light records only near the mirror and why a Lippmann plate works with sunlight at all: coherence over a few micrometres is all that a standing wave against a mirror requires. A hologram of an extended object needs coherence over the depth of the scene, which is why Denisyuk’s work had to wait for the laser to become practical.

Still open: what the old plates actually recorded #

A few hundred Lippmann plates from the 1890s survive, and in principle each holds the spectra of a scene from more than a century ago. Recent work measuring the light the old plates reflect has found that it is a distorted copy of what was there: the peaks are shifted by shrinkage of the gelatine, broadened by the finite depth, accompanied by harmonics from the emulsion’s non-linearity, and shaped by the phase of the mirror. Recovering the original spectrum from the measured one is an inverse problem of the same kind as the image that is a diffraction pattern twice posed for a lens, where what reaches the detector is the object’s spectrum filtered and the question is what the filter has thrown away, and how much of the original can be recovered, and with what confidence, is not settled. Meanwhile the principle has returned in nanofabrication, where layered structures written to order — by lithography or by light in new photosensitive materials — produce colours that do not fade, because they are made by structure and not by dye.

The habit worth carrying away is to ask what a pattern made by superposition would record if something could hold it. Light added to its own reflection from a mirror stands still, with electric nodes at the mirror and every half-wavelength in the medium — 183 nm for green light in gelatine — and an emulsion thick enough to keep the sheets stores the spectrum of the light as their spacing. Lit again, the layers return the wavelengths that wrote them, with a purity set by how many layers there are, exactly as a grating’s resolution is set by its lines.