The specimen that is invisible until its light is delayed

The image that is a diffraction pattern twice followed Ernst Abbe’s account of a microscope: the objective turns the light from the specimen into its spectrum of spatial frequencies in the back focal plane, and the eyepiece side turns that spectrum back into an image. The essay noted in a sentence that a plate placed in that plane can make an invisible change in optical path visible, and moved on. The sentence describes one of the most used inventions in biology, and it rests on an argument about light that is easy to state and was missed for half a century after Abbe: the eye records brightness, a transparent specimen changes only phase, and phase can be turned into brightness only by interfering it with something.

A cell that light passes straight through #

A cell in water is mostly water. Its contents — proteins, membranes, the denser material of its nucleus — have a refractive index a few per cent higher than the water around it, and they absorb almost no visible light. Light passing through the cell comes out as bright as light passing beside it, only slightly later: a cell ten micrometres thick with an index 0.02 higher than water delays the light by a third of a wavelength, two radians of phase. Thinner parts delay it less. The first figure takes a cell twenty micrometres across whose middle delays the light by 0.3 radians, with a nucleus delaying it a third as much again, and computes its image through a microscope objective.

With a plain objective, perfectly focused, the image is uniform to within a few parts in a thousand. The tiny wiggles that remain come from the objective’s finite aperture, which cuts off the finest details of the phase and turns a small part of the phase change into a ripple of brightness. Better optics would remove even those and leave nothing. This is not a failure of resolution. The image is exactly right; it is a faithful copy of a specimen whose only property is a delay, and brightness, which is what an image records, is blind to a delay.

Why no detector sees a delay #

The eye’s blindness to phase is shared by every detector of light, and for a reason that has nothing to do with eyes. Visible light oscillates about six hundred million million times a second. No retina, film or camera pixel can follow that; each responds to the energy the light delivers, averaged over very many cycles, which is proportional to the square of the wave’s amplitude. A delay shifts every cycle by the same fraction and leaves that average untouched. Detectors record intensity, intensity is blind to absolute phase, and a phase can be measured only relative to another wave, by letting the two interfere so that their relative phase decides how they add.

That is why every method of seeing a transparent object is an interference method at heart, whether it says so or not. Something must supply a reference wave and arrange for it to meet the light from the specimen with the right relative phase. Zernike’s insight was that a microscope already contains the reference: the undiffracted light, passing straight through, is a uniform wave of exactly known phase, arriving at every point of the image together with the light the specimen diffracted. All that was missing was the quarter-cycle shift that would make the two add in line rather than at right angles.

Two kinds of light, a quarter cycle apart #

Write the light leaving one point of the cell as $e^{i\varphi}$, a wave of unchanged amplitude delayed by φ. For a small delay that is $1 + i\varphi$: the wave the cell would have let through had it not been there, plus a small extra wave, $i\varphi$, a quarter of a cycle out of step with it. The first part is the undiffracted light, the light that passes straight on as though no cell were present. The second is the light the cell diffracts: its pattern carries the cell’s shape.

The brightness is the square of the total, and the phasor diagram of the second figure shows why it does not change. The diffracted arrow stands at right angles to the undiffracted one, so adding it changes the total’s direction, not its length — to first order in φ. The point is as bright as its surroundings. Only at second order, in $\varphi^2$, does the length change at all, which for the 0.1-radian phases of a thin cell is a change of a part in a hundred, below what the eye notices.

Zernike’s step was to turn one of the arrows a quarter of a turn. If the undiffracted light is delayed by a further quarter cycle relative to the diffracted light, the two arrows point the same way and their lengths add: the brightness becomes $(1+\varphi)^2 \approx 1 + 2\varphi$, changing in proportion to the phase. Delay it the other way and they subtract, $1 - 2\varphi$. Either way the cell’s phase is written into brightness, linearly, and the cell appears.

The one place the two kinds of light are apart #

To delay one kind of light and not the other, they must be separated, and in an ordinary image they are not: at every point of the image both arrive together. But in the back focal plane of the objective they are in different places. What a lens is doing found that a lens maps each direction of travel of the light arriving at it to one point in its focal plane: the field there is the Fourier transform of the field at the specimen, with each spatial frequency of the specimen sent to its own position.

The undiffracted light has no spatial structure — it is the uniform wave that would be there without the cell — so in the back focal plane it comes to a single point at zero frequency. The diffracted light carries the cell’s structure, a range of spatial frequencies set by its size and its edges, and spreads across the plane. The third figure draws the spread for the cell of the first: one spike holding ninety-nine per cent of the light, and around it a faint skirt holding the cell.

So a small plate placed exactly on the spike, a quarter-wave thicker or thinner than its surroundings, delays the undiffracted light and leaves the diffracted light alone. Zernike made such plates in the 1930s, first to test the figure of telescope mirrors and then for the microscope, and was awarded the Nobel Prize in physics in 1953 for the phase-contrast method. In practice the specimen is lit not by a single parallel beam but by a hollow cone of light from a ring-shaped aperture in the condenser, and the undiffracted light then comes to a ring rather than a point in the back focal plane; the phase plate is a matching ring. The principle is the same: put the delay where the undiffracted light, and only it, goes. The plate itself is a film a fraction of a micrometre thick. To delay green light by a quarter of a cycle a transparent film of refractive index 1.38 must be λ/4(n − 1) thick, about a third of a micrometre, deposited as a ring on a glass surface inside the objective — a film thinner than the cells it reveals, and one that works only for light near the wavelength it was made for, which is why phase-contrast microscopes are usually used with a green filter.

The method began not with cells but with mirrors. Telescope makers tested the figure of a mirror with Léon Foucault’s knife-edge, which blocks half the light at the focus and makes slopes in the mirror’s surface visible as shading; Zernike, an astronomer’s son working on optics at Groningen, replaced the knife-edge with a small plate that shifted the phase of the focused light, which made the surface’s height errors themselves visible rather than their slopes. He saw that a microscope specimen is an object of the same kind as a mirror’s surface — a pattern of small phase errors on a smooth wave — and built the microscope version in the early 1930s. The optical industry was slow to take it up; phase-contrast objectives were first made commercially in Germany during the Second World War, and only after the war did the method spread into every biology laboratory.

How much brightness a delay buys #

The fourth figure follows the contrast as the phase grows. For small phases it is very nearly $2\varphi$: a cell delaying its light by a tenth of a radian is a fifth brighter or darker than its surroundings, easily seen. For larger phases the relation bends, because the phasors are no longer small and the diffracted arrow is no longer nearly perpendicular; past about one radian the brightness stops following the thickness and can even reverse. Phase contrast is a method for thin, weakly delaying specimens, which is what most living cells are.

The figure also shows the refinement that every commercial phase ring includes. A thin cell’s diffracted light is weak, so even with the quarter-wave delay it changes the brightness only modestly against a bright background. Dim the undiffracted light as well as delaying it — a ring of thin metal film that passes a few per cent of the light that strikes it — and the background darkens while the diffracted light is untouched. The change in brightness is then $2\varphi/a$ relative to the dimmed background, for an undiffracted amplitude a. Dimming it to a quarter of its amplitude, six per cent of its intensity, makes a tenth of a radian nearly double the background brightness. Biologists’ phase objectives use rings that pass between a few and twenty-five per cent of the direct light.

The halo and the fading middle #

The argument so far assumed the plate covers only the undiffracted light, and a real plate cannot. It must have some width, both to pass a usable amount of light from a lamp of finite size and because the undiffracted light comes to a ring of finite width rather than a geometric point. Any width catches, besides the undiffracted light, the lowest spatial frequencies of the diffracted light: the slowly varying part of the specimen, its broad, flat regions. Those get the same quarter-cycle delay, which puts them back in step with the undiffracted light, and so they lose the very contrast the plate was meant to give.

The fifth figure shows what happens to a broad, flat object, fifty micrometres across with a uniform phase. With a plate on the undiffracted light alone, it appears as a flat, bright step of the right height. With plates three and fifteen times wider, its middle fades almost to the background, and its edges, where the phase changes sharply and the spatial frequencies are high enough to escape the plate, ring with bright and dark bands: the halo and the shading-off that every user of a phase-contrast microscope learns to read through. Large, flat objects in phase contrast are drawn as their outlines.

That is not a defect to be engineered away so much as a trade. A narrower ring gives truer images and less light; a wider one gives brighter images and larger haloes. Modern alternatives — differential interference contrast, which interferes the light with a slightly displaced copy of itself, and quantitative phase imaging, which records the phase directly by interference with a reference beam and computes the image — avoid the halo at the price of more complicated optics.

A cousin: making the invisible visible by turning it #

Phase contrast is one member of a family of methods that make an invisible property of light visible by interfering it with a reference. The stress that polarised light can see found the same strategy at work with polarisation: a transparent plastic under stress delays one polarisation more than the other, which no eye can see, until crossed polarisers turn the delay into colour. Defocus is a cruder member. The half cycle a focus adds found that a focused wave picks up an extra phase as it passes through focus, and moving the focus slightly changes the phase relation between undiffracted and diffracted light by a small amount that depends on spatial frequency. The first figure includes a defocused image: the cell appears, faintly, but only where its phase changes sharply — at its edge and around its nucleus — and as fringes whose sign flips as the focus passes through the specimen. Defocus makes edges visible. Zernike’s plate makes the phase itself visible, in focus.

Where the model holds #

The figures are one-dimensional: a slice through a cell rather than a cell, imaged with perfectly coherent light through a perfect objective whose only limit is its numerical aperture. Real phase-contrast microscopes use partially coherent light from an annular condenser, which changes the shape of the halo and smooths the ripples at sharp edges, and a real cell has structure in two dimensions, which spreads its spectrum into a disc rather than a line. The specimen is also taken to be thin, so that the light through it is delayed but not bent; a thick specimen refracts the light as a lens would and its image depends on where in its depth the microscope is focused. None of these changes the central result, that a quarter-wave delay of the undiffracted light turns phase into brightness in proportion, but each changes the details of the halo and the limits of linearity.

What the figures cannot show #

The figures show brightness. They cannot show what a biologist sees through a phase-contrast microscope, which is a picture full of structure the figures leave out: organelles, the boundaries of the cell membrane, the strands of the dividing chromosomes, all appearing because each has a slightly different refractive index. They cannot show the advantage that made the method important, which is that the cells are alive. Before phase contrast, seeing a cell’s interior meant killing and staining it; afterwards, cell division could be filmed. Nor can the figures show how the delay is measured rather than merely seen: phase contrast gives a picture whose brightness follows the phase only approximately and with haloes, so it shows structure but does not measure thickness, which is the job of the interference methods that followed it.

Still open: what the phase alone can say about a cell #

Quantitative phase imaging now records the phase delay at every point of a living cell directly, by interfering the light through it with a reference beam and computing the delay, and from the delay — knowing how refractive index depends on concentration — it computes the cell’s dry mass, how much protein it holds, to a fraction of a picogram. Watching that mass change over hours measures how fast a cell grows, without any stain or label. What is still being worked out is how far the phase alone can go: whether combining delays measured from many angles can reconstruct the cell’s refractive index in three dimensions reliably enough to identify its organelles without staining them, how scattering in thick tissue can be undone, and whether the measured index can distinguish healthy cells from diseased ones as reliably as a stain. Each is pursued with instruments whose principle is still Zernike’s: the information is in the phase, and getting it out means interfering the light with a reference.

The habit worth carrying away is to ask which property of a wave a detector records. A transparent specimen delays its light without dimming it, so a perfect image is uniform — 0.998 to 1.001 for a cell delaying its light by 0.3 rad — because the light it diffracts, about φ, stands a quarter cycle off the undiffracted light and adds at right angles; delaying the undiffracted light by a quarter cycle where a lens has gathered it to a single spike, 99% of the light at zero spatial frequency, lines the two up and makes the brightness 1 ± 2φ, at the price of a halo round anything large. The cell was always in the light. The eye needed a reference to see it.