The limit a doughnut gets round

How far apart two things have to be reached the result that has shaped every microscope since 1873. Two points closer than about $\lambda/2\mathrm{NA}$ cannot be separated by a lens, whatever the quality of the glass, because the light they send out cannot carry finer detail than that through an aperture. The image that is a diffraction pattern twice showed the argument in Ernst Abbe’s own form: finer detail diffracts at angles the lens does not collect. For green light and the best oil-immersion objective, a spot is about 200 nm wide. A virus is about 100 nm across and a synaptic vesicle 40. So for a century and a quarter, the fine structures of a living cell were visible to an electron microscope, which needs the cell dead, and not to light.

Stefan Hell, working in the mid-1990s on a topic that his supervisors had no great hopes for, noticed a gap in the argument. Abbe’s limit concerns how sharply light can be focused, or how fine a pattern a lens can transfer. It says nothing about how much of what is at the focus has to shine. A molecule that absorbs light will shine only if it is in a state that can shine. If some of the molecules in a focused spot could be prevented from shining, by some means that did not need a sharper spot, what remained could be smaller than the spot. That was the idea, and it took him until 1999 to build it. In 2014, he shared the Nobel Prize in Chemistry for it. The trick needs a second beam of light, and that beam has a hole in it.

A ring whose middle is exactly dark #

A fluorescent molecule absorbs a photon and is lifted to an excited state. There it stays for a few nanoseconds, after which it emits a photon of a longer wavelength. But, as the third process equilibrium demands showed, an excited atom can also be persuaded to emit by a passing photon of the right colour. That is stimulated emission, and it goes to a wavelength the detector can be set to ignore. So a second beam tuned to the red tail of the molecule’s emission, arriving just after excitation, can return molecules to the ground state before they fluoresce. It does not need to do this in a sharp way. It needs only a place in the focus where it is dark.

The place comes from interference. Put a phase plate in the path of the second beam, a spiral ramp that delays light by one full wavelength around its circumference. Each part of the beam then arrives at the centre of focus with a phase equal to its angle round the aperture. Opposite parts of the beam are out of step by half a wavelength and cancel. The cancellation at the exact centre is complete, by symmetry alone, no matter how imperfectly the rest of the focus is formed. What arrives elsewhere makes a ring around the zero, a doughnut.

Look at the ring in the picture: it has the same blur as the spot. Nothing about it is finer than Abbe allows. It rises from zero at the centre to a peak about 200 nm off to the side, as smooth a curve as a lens can draw. If a molecule’s response to it were simply proportional to the light, the result of multiplying spot by ring would be just another blurred shape, and there would be nothing to gain. All the gain comes from a nonlinearity.

The exponential does the work #

The fraction of molecules that still fluoresce after a depleting pulse of intensity $I$ falls as an exponential in the number of photons offered per molecule:

$$\eta = \exp\!\left(-\ln 2\;\frac{I}{I_s}\right),$$

where the saturation intensity $I_s$ is the one that depletes half. An exponential, once its argument is large, has the property that it is near one only where its argument is near zero. It is a steep and very sharp cutoff. So where the ring is at many times $I_s$, fluorescence is completely gone. The same holds on most of the ring’s rising flank, where the ring is many times $I_s$ long before it has reached its peak. The surviving region is only the immediate neighbourhood of the zero, where $I$ is less than about $I_s$.

Near its centre, the ring is close to a parabola in intensity, with a curvature fixed by the lens: $I(r) \approx I_{\max}(r/w)^2$ for a width $w$ of the order of the Abbe distance. So the hole’s half-maximum radius is where $I$ equals $I_s$, with $r \approx w\sqrt{I_s/I_{\max}}$. The width of the hole falls as the inverse square root of the peak strength. The result has become a slogan in that field:

$$d \approx \frac{d_0}{\sqrt{1 + c\,\zeta}},$$

where $\zeta = I_{\max}/I_s$ and $c$ records how steep the ring is at its centre.

The picture is computed with actual Bessel functions from the pupil integral of a uniformly filled aperture with the spiral phase, and the square-root law is an approximation to it good to about a tenth. No limit to resolution appears, and that remains surprising. Diffraction still acts as strongly as it ever did, on every photon. A diffraction-limited pattern can have an infinitely sharp feature, since a zero is a point and can be located with arbitrary precision. It cannot be used to produce a sharp bright feature. What the molecule’s nonlinear response does is convert the sharp zero of a dark pattern into a sharp bright region for the fluorescence.

A cross-check on what this means is to draw two identical molecules.

Two molecules 70 nm apart are about a third of Abbe’s distance from each other. Under a normal microscope they are one fuzzy blob and nothing can be recovered by processing, since the separation cannot be told from one blob. But with the doughnut, when the zero sits over one of them, the other is at a point on the ring’s flank and forced dark. It is then possible, by scanning, to record each of them on its own, at a different moment. That is the deeper reason it works. It does not separate molecules by the sharpness of light. It separates them by their state, with the two switched on one after the other. Abbe’s theorem is about two things emitting at the same time, and these do not.

What the same idea buys elsewhere #

Once the principle was recognised, several different ways of using it appeared. All of them ensure that only one state of a molecule can emit and only a sparse or specified subset occupies it at a time. PALM and STORM, developed by Betzig and Hess, and by Zhuang, switch on a random sparse set of fluorophores in each camera frame. Each one is then alone in its blur, and its centre can be fixed to a few nanometres by finding the middle of its diffraction pattern, as the fringe that measures a star located a source of light from a pattern. Thousands of frames then add up to a picture. In these methods the state change is stochastic rather than steered by a beam, but what limits the precision is the same counting of photons.

A related technique, MINFLUX, published by Hell’s group in 2017, turned the argument around in a very elegant way. Rather than look for the brightest place it looks for the darkest, placing the zero of a doughnut on a molecule and measuring how few photons come back. Since the number of photons emitted goes to zero when the zero sits exactly on the molecule, location at a nanometre scale takes a much smaller number of photons than a bright peak needs. So the method wins both in resolution and in light dose.

There is a sibling of the doughnut in lithography, where one wishes to write features smaller than light’s limit into a chip’s resist. Similar depletion ideas have been shown, though extreme ultraviolet light has largely taken over the industry. There is also an optical tweezers version of the ring, as in the beam that does not spread, in which the zero itself is the useful feature. A spiral phase plate is also how an optical vortex, a beam with orbital angular momentum, is made. It is the same device doing very different things.

The bill #

The formula’s exactness conceals a practical limit, and the picture of it is simple.

A real doughnut does not reach zero. Lens aberrations, scatter in tissue and an imperfect phase plate all leave a little light in the centre. And since the signal at the centre goes as an exponential in the ring’s strength multiplied by the leak, turning up the ring to gain resolution will at some point extinguish the very molecules being looked at. Half a per cent is good for a modern instrument in a clear sample, but a few tens of micrometres into tissue, a scattered ring is already much worse than that, and resolution suffers well before the theoretical figure.

The other limit is that intensities are enormous. A typical saturation intensity in a stimulated emission experiment is around $10$ MW/cm², and a ring at 30 or 100 times that is delivering a dose of light that fluorophores do not survive for long. Some molecules bleach within a few scans. Living cells, which are not fond of intense red light either, can be damaged. This is why a good deal of effort has gone into fluorophores that switch by slower, gentler mechanisms, like the reversible switching in RESOLFT, which needs much weaker light; and why practical resolution in a living animal is around 50 nm and not a theorist’s 5.

Why Abbe’s proof does not cover this #

It is worth being exact about how the theorem is avoided, as it says something about what a theorem assumes. Abbe’s reasoning, in its modern form, treats a microscope as a linear system. Light at each point is proportional to the fluorophore density at that point multiplied by the excitation, and what is recorded is that product convolved with the lens’s response. In the language of the fringe and the spectrum are one measurement, a linear system can only multiply each spatial frequency by a transfer function. The lens’s transfer function vanishes above $2\mathrm{NA}/\lambda$. Anything above that frequency in the object is multiplied by zero, and so it is gone. No amount of data processing can get it back, because it was never recorded.

Now the object’s emission in the ring experiment is not proportional to the light. It contains the exponential, which is a nonlinear function of the ring’s intensity. The exponential of a function with a limited range of spatial frequencies has frequencies far beyond that range, in the way that a power series in $\cos x$ has all of the harmonics of $x$. So the effective excitation pattern, which is what scanning delivers, has a far wider spectrum than the ring or the lens could. Spatial frequencies above Abbe’s limit are present in the emission because the molecules created them. That is the whole of it.

A related technique shows how clean the distinction is. Structured illumination microscopy projects a pattern of stripes on the sample. The stripes beat against fine detail, as in a moiré, moving high spatial frequencies into the range the lens can pass, in a way close to the pattern two gratings make that neither contains. Since the stripes themselves cannot be finer than the limit, this gains a factor of exactly two, and no more. That is a hard limit of a linear method. Saturated structured illumination, by Mats Gustafsson in 2005, pushed the stripes hard enough that the molecules’ response became nonlinear. Then harmonics of the stripes appeared, and the resolution went to about 50 nm. So the two forms of the same instrument show the difference: one linear with a factor of two, and one nonlinear with no bound in principle.

A zero has a second property that makes it useful. Phase singularities are robust, as the dark lines a wave cannot avoid showed: a point of zero amplitude with a phase circulating around it cannot be removed by a small perturbation. It can move but it cannot be filled in, by anything smooth and coherent. So the doughnut’s zero exists in an aberrated focus, shifted and distorted, and that is a much more forgiving requirement than a bright peak’s sharpness. Scatter, on the other hand, adds light incoherently from elsewhere. The zero is robust against one kind of error, and helpless against the other.

The lens has not improved. Nothing here has made use of what the condition a lens must meet requires for a good image, other than for the two beams to overlap. A microscope built this way needs two colours aligned to a few nanometres, and that has been harder in practice than any of the theory.

What the picture is not showing #

The figures are scalar: the field is treated as a single number. At a numerical aperture of 1.4, the actual vector nature of focused light matters, and for a doughnut made with a single circular polarisation sense in one direction, the central zero is preserved, while using the wrong sense fills it in. The calculation here does not know that. It would give a clean zero with any polarisation, and an experiment would not.

The same goes for the assumption that every molecule is excited by a uniform distribution of light, and that a single exponential describes depletion. Molecules have orientations. A dipole aligned along the ring’s field responds at full strength, and a perpendicular one much less. Time matters also, as an excited molecule can emit before the depleting pulse gets to it, which is the source of a tail in actual images, and how the pulses are timed in practice is a large part of a real microscope’s design. The picture has one lateral dimension, too. In the axial direction, a focus is much longer than across, and an image needs a different arrangement to sharpen it, with a second ring having its zero along the axis. All these lie outside the figures, and they are what an optical engineer would mainly be working on.

Nothing in the argument has made a lens better. The light sent back from a molecule at the centre of the hole is blurred by the same 220 nm in the detector as before. The microscope knows where that light came from not because it can see that place sharply but because it knows where the hole was when the light arrived. So this is a scanning method. Its image is built up one point at a time. It cannot produce a snapshot of a whole field with any single exposure at that resolution, and for fast dynamics that is a real limit.

Still open: how close a light microscope can get to a molecule #

Since MINFLUX began reporting localisations at about a nanometre, the limit has moved to places where the label is the main problem. A fluorescent molecule attached by an antibody can sit ten or more nanometres from what it is meant to mark, and in that case a resolution of one nanometre is locating the wrong thing. Smaller labels are being developed, and there are methods that attach them site-specifically. Whether structural biologists will get from fluorescence the single-molecule distance measurements inside a living cell that would rival cryo-electron microscopy of dead ones is a live question. Two fluorophores on a single protein have now been resolved at a separation of about a nanometre, using cooled samples and the same logic. The optics have stopped being what limits the method. The chemistry of what shines, and for how long before it fails, has taken its place.

A ring of light with a zero at its centre, itself as blurred as any focus, turns an exponential depletion into a bright spot of width $d_0/\sqrt{1 + c\zeta}$: 220 nm falls to 38 nm at 30 times saturation, separating two molecules 70 nm apart by switching them on in turn, at a cost measured in how dark the middle can be kept. Abbe’s limit stands. It is a limit on patterns of light, and it was only assumed to limit what molecules do in response. The habit worth keeping is to read a limit’s own terms: a theorem about what a lens can transmit says nothing about what can be switched.