The depth a broad spectrum can see

Why two lamps never interfere found that interference needs light whose phase is correlated from one moment to the next, and how far a wave can remember found how far: the coherence length, inversely proportional to the width of the light’s spectrum. The fringe that measures a star used spatial coherence to measure the size of a star, the fringe and the spectrum are one measurement found that the fringes of a moving-mirror interferometer are the Fourier transform of the light’s spectrum, and the faint light a bright one makes measurable found that mixing a weak signal with a strong reference lifts it above the noise.

Put those together and point the interferometer at something that reflects light from many depths at once. The coherence length becomes a ruler: fringes appear only where the reference path matches the path to a reflector, to within the coherence length, and the broader the spectrum, the finer the ruler. That is optical coherence tomography, invented in 1991 and now performed tens of millions of times a year, mostly on the back of the human eye. This essay follows why a broad spectrum, the enemy of ordinary interference, is what makes it work; how the depth is read either by moving a mirror or by reading colours; and why the shape of a spectrum matters as much as its width.

Fringes that exist only when the paths match #

A Michelson interferometer splits light into two arms and recombines it. Moving one arm’s mirror changes the path difference, and the detector sees fringes — bright and dark in turn every half wavelength of mirror travel. With a laser the fringes continue for metres of travel. With a broader source they do not.

The figure compares two sources centred at 840 nanometres. Light one nanometre wide makes fringes that hardly fade over forty micrometres of mirror travel; its coherence length is about three hundred micrometres. Light fifty nanometres wide makes fringes only within a few micrometres of equal paths, inside an envelope 6.2 micrometres across at half height. Away from equal paths each colour still makes its own fringes, but with slightly different spacing, and summed over the whole band they average to nothing. Only where the paths are equal are all the colours in step at once.

For ordinary interferometry that is a nuisance. For locating something it is ideal. A laser interferometer can measure a displacement to a tiny fraction of a wavelength, but it cannot say where the two paths are equal: every fringe looks like every other, and the count is lost the moment the beam is interrupted. A broadband interferometer produces one burst of fringes, at one place, and says unambiguously: here the paths match.

Why not time the echo #

Ultrasound imaging measures depth the direct way: send a click, time the echo, and multiply by the speed of sound. Light could be used the same way in principle, and in practice cannot. A reflector ten micrometres deeper than another returns its echo later by the time light takes to travel twenty micrometres, about seventy femtoseconds. No detector responds that fast; the fastest photodiodes resolve tens of picoseconds, a thousand times too slow.

The interferometer sidesteps the problem by never measuring a time at all. It compares the echo with a copy of the outgoing light that has travelled a known path, and asks only whether the two are still correlated. The correlation of broadband light falls off in seventy femtoseconds of delay, so the comparison resolves delays that short — the timing is done by the light’s own coherence, and the detector only has to register a steady brightness. The coherence length is a clock built into the light, and a broad spectrum makes it tick fast.

White-light fringes, and colours that only thin films show #

The principle is much older than the instrument. Albert Michelson, aligning his interferometers, used white light to find the position of exactly equal paths: with a lamp, a handful of coloured fringes appear only there, and a coloured central fringe marks zero path difference unmistakably, where monochromatic light would give endless identical fringes. Engineers used the same trick from the 1980s to find faults in optical fibres, sending broadband light down the fibre and locating reflections by white-light interferometry to a few micrometres, and coherence tomography, first demonstrated on a retina and an artery in 1991, is that fibre technique pointed at tissue and scanned sideways.

The same limit explains a familiar observation. A soap film or a film of oil on water shows colours only where it is thin — a few wavelengths of light at most. The layer that makes a reflection vanish found why a thin layer’s two reflections interfere; the reason a thick one shows no colour is that daylight’s coherence length is a couple of micrometres, and reflections from the two faces of a thicker film are no longer correlated. A window pane has two reflecting faces too, and no colours, for exactly that reason. Coherence tomography turns the disappearance of colour in thick films into a measurement: it finds, for every reflection in a sample, the one reference distance at which the colours come back.

A ruler whose divisions are set by colour #

The width of that burst — the axial resolution — follows from the Fourier relation between spectrum and fringes. For a spectrum with a smooth, Gaussian shape, centred at wavelength $\lambda_0$ with full width $\Delta\lambda$, the burst’s full width at half maximum, in mirror displacement, is

$$\Delta z = \frac{2\ln 2}{\pi}\,\frac{\lambda_0^2}{\Delta\lambda}.$$

The resolution improves in proportion to the spectral width and worsens as the square of the centre wavelength. At 840 nanometres a fifty-nanometre band resolves 6.2 micrometres and a hundred-and-fifty-nanometre band 2.1; at 1,310 nanometres, a wavelength chosen because it penetrates tissue further, the same fifty-nanometre band resolves only 15. Inside tissue, whose refractive index is about 1.38, every figure shrinks by that factor. A source for coherence tomography is therefore chosen by its bandwidth before its brightness: superluminescent diodes with bands of fifty to a hundred nanometres, and for the finest work femtosecond lasers or supercontinuum sources spanning hundreds.

That is the reverse of the usual preference in optics. Almost every other instrument wants light as monochromatic as possible, to keep its fringes and focus sharp. Here what is wanted is light as incoherent in time as possible — a short correlation, a broad spectrum — while still spatially coherent enough to interfere at all, which means a source that is both a single tight beam and a rainbow. Sharpness has to be paid for found that a short pulse needs a broad spectrum; the same trade appears here with the pulse replaced by a correlation.

A scan into a layered sample #

Now replace the sample-arm mirror with something translucent. Light enters it and is reflected weakly from every boundary where the refractive index changes: the surface, the layers of the retina, the interfaces between skin’s layers, the coats of a painted surface. The sample arm now contains many reflectors at many depths, and each makes its own burst of fringes when the reference mirror matches its depth.

Scanning the reference mirror through a range of depths and recording the envelope of the fringes gives a depth profile — an A-scan, in the language of ultrasound, from which the method borrowed its vocabulary. The figure computes one for a sample with five boundaries. The fifty-nanometre source resolves all five, including two pairs only eight micrometres apart. A ten-nanometre source, with a resolution of 31 micrometres, merges each pair into one blurred echo. Moving the beam sideways and repeating builds up a cross-sectional image, a B-scan, and a stack of those a three-dimensional volume.

The retina is the ideal subject. It is transparent enough for light to reach its back, its layers are a few tens of micrometres thick, and the diseases that matter most — macular degeneration, glaucoma, diabetic swelling — change their thickness and structure. Coherence tomography shows those layers directly, without contact and in seconds, and it has changed how eye disease is diagnosed and treated. It is also used on skin, on arteries through fibre-optic catheters, on teeth, on paintings to read the layers of varnish without sampling them, and in manufacturing to measure the thickness of films.

Why a weak echo is still measurable #

Tissue reflects very little. A typical boundary returns a millionth of the light that falls on it, or less, and light that has scattered on its way in and out is lost from the measurement. Detecting such weak echoes is possible because the method is interferometric in exactly the sense the faint light a bright one makes measurable found: the weak light from the sample is mixed with the strong light from the reference arm, and the fringe term, proportional to the product of their field amplitudes, is lifted far above the detector’s own noise. The reference arm is a local oscillator, and coherence tomography is heterodyne detection that listens only at one path length.

The result is a sensitivity of about a hundred decibels — a reflectivity of ten to the minus tenth is detectable — which is what makes the faint boundaries inside tissue visible at all. The limit on depth is not sensitivity but scattering. Light that has been scattered several times inside tissue arrives with its path length scrambled and contributes no useful signal; the cloud light has to walk through found how quickly multiple scattering randomises light, and in skin it limits coherence tomography to one or two millimetres. The eye’s transparency is why its images go so deep.

Reading depth off the colours #

Moving a reference mirror is slow. The Fourier relation between spectrum and fringes offers another way, and it is how nearly all modern instruments work.

Keep the reference mirror still and send the combined light into a spectrometer. A reflector at depth $z$ beyond the reference adds a ripple across the spectrum, $\cos 2kz$, whose frequency in wavenumber is proportional to $z$: shallow reflectors make slow ripples, deep ones fast. Several reflectors make several ripples superposed, beating against each other in the recorded spectrum. A Fourier transform of the spectrum sorts them out, returning a peak at each reflector’s depth, with a height set by its reflectivity and a width set by the spectrum’s width. This is the fringe and the spectrum are one measurement run in reverse: there, a moving mirror measured fringes and the Fourier transform gave a spectrum; here, a spectrometer measures the spectrum and the Fourier transform gives the positions of the mirrors.

Every depth is measured at once, from one spectrum recorded by a camera in microseconds. Fourier-domain instruments, introduced in the early 2000s, made scans a hundred times faster than time-domain ones, and they are also more sensitive, because every depth’s signal is collected for the whole exposure instead of only while the moving mirror passes it. A later variant replaces the spectrometer with a laser whose wavelength sweeps rapidly across a broad band, recording the same spectrum one colour at a time, and reaches millions of depth scans a second.

Why the shape of the spectrum matters #

Because the depth response is the Fourier transform of the spectrum, every feature of the spectrum’s shape reappears in depth.

A Gaussian spectrum transforms into a Gaussian response that falls smoothly to nothing on either side. A flat-topped spectrum of the same width — the natural output of some broadband sources, and what a naive filter would produce — transforms into a response with a central peak about a third wider and, far worse, side lobes: echoes at regular spacings on either side, the largest at 22 per cent of the main peak. In an image, every strong reflector acquires faint ghost copies of itself above and below, at depths where there is nothing. Sharp edges and ripples in a spectrum are the source of such artefacts, and instruments either choose sources with smooth spectra or reshape the recorded spectrum numerically, trading a little resolution for clean depth profiles.

This is the same trade that windowing makes in any Fourier analysis, and the same that shapes the aperture of a telescope to suppress the rings round a star. The coherence envelope is the depth-domain counterpart of a lens’s point-spread function, and it is apodised for the same reason.

Depth and breadth, decoupled #

One more property sets coherence tomography apart from microscopy. In an ordinary microscope, depth resolution comes from the focusing lens: a strongly focusing lens makes a thin focal slab, and the focus that is a slab, not a plane found that the slab’s thickness shrinks as the square of the numerical aperture, while the depth over which an image stays sharp shrinks with it. Fine depth resolution in a microscope therefore comes with a shallow field of view.

In coherence tomography the depth resolution comes from the spectrum, not the lens. A weakly focused beam, with a long depth of field and only moderate sideways resolution, can still resolve a few micrometres in depth, and image through a millimetre of tissue in one scan. The two resolutions are independent, and that decoupling is what lets the method take a cross-section of the retina through the pupil of the eye, with the eye’s own lens doing the focusing. When both are wanted — fine depth and fine sideways resolution together — instruments combine a broad spectrum with a strong lens and scan the focus, but they pay for it in the depth of field that the decoupling otherwise gives away for free.

Where the ruler stops #

The resolution formula assumes a smooth spectrum and a sample with no dispersion. Tissue and glass are dispersive — their refractive index varies with wavelength — so different colours travel the same depth in slightly different times, the colours go out of step, and the coherence envelope broadens with depth. Instruments compensate by matching the dispersion of the reference arm to the sample’s, or by correcting the recorded spectrum, and the broader the spectrum the more careful the compensation must be. The images are also speckled: light scattered from many unresolved particles within one resolution volume interferes with itself, and the grain that is in the light found that such a pattern has grains the size of the resolution. Speckle limits the contrast of fine structure and is reduced by averaging scans taken with slightly different angles or wavelengths.

What the pictures cannot show #

The figures show the envelope of the fringes, which is what an image displays, and not the fringes themselves, whose phase carries additional information: tiny motions of a reflector shift the phase long before they move the envelope, and phase-sensitive coherence tomography measures blood flow and the vibration of the eardrum at nanometre scales. The figures treat every boundary as a mirror, when real tissue reflects diffusely from many small structures. And they cannot show what an image of the retina actually looks like — ten layers of cells distinguished by their faint differences in reflectivity, in a cross-section taken in a fraction of a second through a pupil a few millimetres across.

Still open: how far into scattering tissue light can be made to see #

Coherence tomography reaches one or two millimetres into skin and other opaque tissue, and the limit is multiple scattering, which scrambles paths faster than the coherence window can select them. Several approaches aim to push deeper: longer wavelengths, around 1.7 micrometres, where scattering is weaker; shaping the incoming wavefront to undo some of the scattering, using the same ideas as adaptive optics; and computational methods that separate singly scattered light from the multiply scattered background by exploiting its different statistics. How far each can extend useful imaging, and whether any can reach the centimetre depths where ultrasound and X-rays are now needed, is an open question.

The habit worth carrying away is to ask what a limitation could measure if it were turned round. A broad spectrum destroys interference away from equal paths — and so locates equal paths, to a precision $(2\ln 2/\pi)\lambda_0^2/\Delta\lambda$ set by the spectrum alone: 6.2 µm for 50 nm of light at 840 nm, fine enough to count the layers of a retina through the eye’s own lens. The same Fourier relation that turned fringes into a spectrum turns a spectrum into a depth profile, ghosts and all.