The image that is a diffraction pattern twice

The ray picture of a lens is a construction: three rays leave a point on the object, the lens bends them, they meet again, and the meeting is the image. It works, it is drawn in every textbook, and it has nothing to say about the one question that decides what a microscope can do.

The ray construction uses three rays because three can be drawn without calculation, and nothing in it involves a wavelength — so nothing in it can say anything about resolution. That is the gap this essay fills: geometrical optics locates an image perfectly and is silent about how much detail the image contains, and the answer to the second question turns out to be a statement about diffraction rather than about focusing.

Ernst Abbe, working for Carl Zeiss in the 1870s, replaced the question. Instead of asking where the rays from a point go, he asked what happens to a periodicity.

Two transforms, and where they happen #

A lens performs a Fourier transform between its front focal plane and its back focal plane. That is a statement about the geometry rather than a metaphor: parallel light arriving at angle $\theta$ converges to a point at $f\tan\theta$ in the back focal plane, so direction at the object becomes position there.

An object illuminated coherently sends out a set of plane waves, one for each spatial frequency in it, at angles given by the grating equation. Each arrives at its own place in the back focal plane. The lens then transforms a second time, from that plane to the image, and the second transform undoes the first — except for whatever the aperture threw away in between.

Two sources produce a set of directions in which their contributions arrive in step, and a periodic object is a set of such sources. So the light leaving a specimen is sorted by angle according to how fine the detail is that produced it — the first transform — and the lens then collects those angles and reassembles them, which is the second. An image is a diffraction pattern of a diffraction pattern.

The general statement the optics is an instance of is that a shape is a sum of periodic components, and a narrow feature needs high frequencies. What an aperture does is exactly what a low-pass filter does in any other subject: it admits components up to some frequency and discards the rest, and the sharpest feature the output can contain is set by the highest frequency admitted.

What one order is worth #

The strongest way to see that the image is a filtered spectrum is to admit almost nothing and look at what is left.

That is the result the whole theory rests on. The zeroth order carries how much light the object passes and nothing else; the information about structure lives entirely in the diffracted orders, and at least one pair of them must be caught.

So the condition for resolving a pitch $d$ is that the first order fit inside the aperture: $\sin\theta_1 = \lambda/d \le \mathrm{NA}$.

Two things about that condition are worth noticing before its consequences. It is a condition on the object, not on the image: whether a structure can be seen is decided at the front of the objective, and everything after that point can only lose information. And it is a threshold rather than a gradual falling-off — an order either fits inside the aperture or it does not, and the transition as an object is made finer is not a fading of contrast but the abrupt disappearance of everything except a uniform field.

An immersion objective is not gathering more light in any useful sense. It is being given access to angles that would have been totally internally reflected at a glass-to-air surface.

The angle past which light cannot leave a coverslip into air is 41 degrees, so an objective in air can never see an order leaving at more than that — however large its lens. Immersion oil removes the boundary rather than the limit, which is why oil buys resolution and why the gain is exactly the index ratio and no more.

The condenser is half the aperture #

The oblique-illumination remark above is usually read as a trick, and it is the ordinary operating condition of every transmitted-light microscope. Stating it properly changes what the resolution formula’s numerical aperture refers to.

Illuminate the specimen straight on and the zeroth order goes up the axis, so an order fitting inside the aperture has to be within $\mathrm{NA}$ of the axis. Illuminate it obliquely and the whole diffraction pattern tilts with the illumination: the zeroth order comes in at one edge of the aperture and the first order can now be almost $2\,\mathrm{NA}$ away from it. Illuminate it from every direction at once, which is what an open condenser does, and every azimuth gets that benefit.

So the resolution bound is properly

$$d = \frac{\lambda}{\mathrm{NA}_{\text{obj}} + \mathrm{NA}_{\text{cond}}},$$

and the familiar $\lambda/2\mathrm{NA}$ is the special case where the condenser matches the objective. A microscope with a 1.4 objective and a dry condenser stuck at 0.9 is not a 1.4 instrument; it is a 1.15 one, and the missing resolution is in a part of the instrument that nobody looks through.

This has a practical edge that every microscopist meets and few connect to Abbe. Closing the condenser iris increases contrast — a specimen that was washed out becomes crisp and dark — and it decreases resolution, because it is removing exactly the oblique illumination that the formula above is about. The tempting adjustment and the correct one are opposed. A specimen photographed with the condenser closed down looks better and contains less, and the extra apparent sharpness is edge ringing from a narrowed spectrum rather than detail.

The demonstration Abbe used #

Abbe’s own lecture demonstration is still the sharpest available argument, and it consists of interfering with the back focal plane on purpose.

The frequency doubling is what makes the demonstration decisive. A picture with twice as many lines as the object is not a degraded image or a noisy one; it is a correct image of something else, and there is no account of it at all in a theory where a lens projects an object.

The pattern a single opening produces is the same physics one step earlier: a narrower opening spreads light through larger angles. Abbe’s demonstration inverts that — he put masks in the back focal plane of an objective, removing orders one at a time, and watched the image lose exactly the detail those orders carried. It is the most direct evidence there is that an image is assembled from angles.

The theory arrived from a factory floor #

Abbe was not a microscopist looking for a theory. He was a physicist hired in 1866 by Carl Zeiss, whose workshop in Jena made microscopes by a method Zeiss had come to distrust: build a lens, look through it, adjust, build another. Instruments were improved by iteration, the improvement had stopped, and nobody could say why one objective was better than the next.

What Abbe supplied first was a way of specifying an objective. The numerical aperture is his — the phrase and the quantity — and its point is that it is the number that decides performance, so two objectives with the same numerical aperture resolve the same detail whatever their focal lengths, magnifications or glass. That converted the trade from an art into a set of numbers a workshop could aim at.

The theory came next, and it was resisted. Microscopists had spent decades interpreting fine structure in specimens near the limit, and a theory that said the finest structures they were reporting could not be images of anything was unwelcome. Abbe’s demonstration with the stop in the back focal plane — a picture with twice as many lines as the object, produced deliberately — was the argument that ended the dispute, because it exhibited a photograph of a periodicity that was not in the specimen.

The commercial consequence followed quickly. Oil immersion, homogeneous immersion with a coverslip of the same index, and the apochromatic objective all came out of Jena in the following decade, and every one of them was designed from the theory rather than found by iteration. It is one of the clearer cases in the history of instruments where knowing why something worked changed what could be built.

Two theories of the same limit #

The Rayleigh criterion and the Abbe limit are usually presented as rivals or as the same thing, and they are neither.

Rayleigh’s account starts from two self-luminous points, each producing an Airy pattern, and asks when the sum shows a dip. Abbe’s starts from an illuminated grating and asks which orders get in. The two give nearly the same number for different reasons, and which applies depends on whether the specimen makes its own light — a distinction that matters in fluorescence microscopy and nowhere else.

The difference matters in practice. A fluorescent specimen is self-luminous, its points are incoherent, and the Rayleigh account is the right one. A transmitted-light specimen is illuminated by one source, its parts are coherent with each other, and the Abbe account is the right one — which is why the effective resolution of a transmitted-light microscope depends on how the condenser is set as well as on the objective.

The thin-lens equation runs away near the focus and says nothing about a wavelength: it decides where an image forms and not what is in it. That is worth stating once more at the end, because the two questions are so easily run together — magnification is a geometric quantity and resolution is a wave one, and no amount of the first buys any of the second.

Some numbers this fixes #

The bound is worth putting into the units the instruments are actually specified in, because the answers are all familiar objects.

A good light microscope. Green light at 550 nanometres, an oil objective at NA 1.4: the finest resolvable pitch is 196 nanometres. That is the number every account of light microscopy quotes as “about two hundred nanometres”, and it has not moved since 1880 because there is no larger numerical aperture available — 1.4 is already 0.92 of the index of immersion oil.

Ultraviolet. Dropping the wavelength to 250 nanometres halves the bound, and microscopes were built to do it. They need quartz optics, the specimen has to survive the light, and nothing can be looked at directly; the gain of a factor of two cost enough that the technique never became routine.

An electron microscope. An electron accelerated through 100 kilovolts has a wavelength of about 4 picometres, five orders of magnitude below visible light. The bound is then not the wavelength at all but the aberrations of magnetic lenses, which limit the usable aperture to a few milliradians — so a modern instrument reaches about 50 picometres, ten thousand times better than light and ten times worse than its own wavelength would allow.

A telescope. The same expression with the object at infinity gives an angular resolution of 1.22λ/D, and for a 2.4-metre mirror in green light that is 0.05 arcseconds. What limits a ground-based telescope is not that number but the atmosphere, which is why the same aperture in orbit is worth building.

The list has one shape. In every case the formula says one thing, and the instrument is limited by whichever of the wavelength, the aperture and the medium is worst — and knowing which of the three is binding is what decides whether an improvement is worth attempting.

What magnification is actually for #

The second refutation says magnification cannot recover what the aperture threw away, which is true and leaves the obvious question of what magnification is for. It has a precise answer, and the answer is about the detector rather than the optics.

The objective delivers an image whose finest structure is $\lambda/2\mathrm{NA}$ across at the specimen. Whatever records it — an eye, a film, an array of pixels — has its own finest resolvable spacing, and the image is only fully captured if the optical detail arrives spread across at least two of the detector’s elements. Magnification is the factor that matches the two.

For an eye, which separates about a tenth of a millimetre at a comfortable viewing distance, capturing a 200-nanometre detail needs the image to be at least a few tenths of a millimetre across — a magnification of something like a thousand. That is where the old rule comes from that useful magnification runs to about a thousand times the numerical aperture, and where the phrase empty magnification comes from for anything beyond it: a bigger picture of the same information.

For a camera the arithmetic is the same with different constants. A sensor with pixels six and a half micrometres apart needs the 200-nanometre detail magnified until it spans two of them, which is a magnification of about sixty-five. Anything less throws information away in the detector that the objective successfully delivered; anything much more spreads the same photons over more pixels and costs signal for nothing.

So magnification has an optimum rather than a direction, and it is set by the ratio of two resolutions that have nothing to do with each other. That the instrument’s optical limit and its sampling limit are independent is the reason a microscope is specified by two numbers rather than one.

The threshold is sharper on a grating than on anything else #

One honest qualification to the picture of orders either fitting or not. A grating has a discrete spectrum, so the cut-off really is abrupt: an order is in or out, and the image changes character suddenly. A general object has a continuous spectrum, and what an aperture does to it is smoother.

Under incoherent illumination the useful description is a transfer function — how much contrast survives, as a function of spatial frequency — and it does not drop off a cliff. It falls steadily from one at zero frequency to zero at a cut-off of $2\mathrm{NA}/\lambda$, so the frequencies just below the limit are transmitted with very little contrast rather than with full contrast. A structure at nine tenths of the cut-off is present in the image at a few per cent of its true contrast, which is indistinguishable from absent if there is any noise at all.

That is why quoted resolutions are softer numbers than the derivation suggests, and why a specimen’s own contrast matters. A high-contrast test target resolves closer to the theoretical bound than a faint biological specimen does, using the same objective, because the same few per cent of transmitted contrast is visible in one case and buried in the other.

The grating demonstration is therefore the right experiment and an unrepresentative one. It was chosen because a discrete spectrum makes the mechanism unmistakable, and the price of that clarity is a sharper edge than any real specimen shows.

What it costs #

Coherent illumination has been assumed. Everything above treats the object as illuminated by a single plane wave. Real condensers illuminate over a cone of angles, which is a superposition of many such calculations and gives a resolution between the coherent and incoherent limits — and, awkwardly, an image whose intensity is no longer a simple filtered version of anything.

The lens is perfect. Aberrations displace and distort the spectrum in the back focal plane, and the second transform then reassembles something else.

A spherical surface does not bring parallel light to a point, and the departure grows as the cube of the aperture while the diffraction limit falls as its reciprocal. So the two limits cross, and above that diameter the instrument is aberration-limited and the extra aperture is wasted — which is why the sine condition and this essay’s bound are the two halves of one design problem.

And the transform is only exact under conditions nobody quite meets. The back focal plane holds the object’s Fourier transform exactly when the object sits in the front focal plane; elsewhere it holds the transform multiplied by a quadratic phase, which matters for what is done to the plane and not for what comes out of the second transform.

Where the model stops #

The limit is a limit on a linear image and not on knowledge. If something is known about the object in advance — that it consists of isolated point emitters, that it is sparse, that it can be made to blink — the positions of those emitters can be recovered far below λ/2NA, because the problem has stopped being one of forming an image and become one of fitting a model. Every super-resolution technique is that substitution.

Evanescent components never propagate at all. Spatial frequencies above 1/λ correspond to $\sin\theta > 1$, which is not a direction: those components decay within a wavelength of the object and no lens anywhere can catch them. Reaching them requires putting something within a wavelength of the specimen, which is what a near-field probe is.

And nothing here is about photons. The whole argument is classical wave optics, and the same bound applies to a single photon at a time — because what has been computed is which spatial frequencies reach the image plane, and a photon that arrives is distributed according to exactly that.

The same theory holds for a mirror, an antenna and an ultrasound array. What is being computed in each case is which angular components of a field an instrument of a given size can collect, and the answer never depends on what the instrument is made of — only on how wide it is in wavelengths. That is why the same formula sizes a radio dish and an oil-immersion objective.

What the pictures cannot show #

The back focal plane figure draws bars at heights equal to the object’s Fourier coefficients, and those coefficients are complex numbers. Only their magnitudes are on the axis. The phases are what carry the position of every feature in the object, and an experiment that measured the back focal plane’s intensity alone and tried to reconstruct the image would fail for exactly that reason — which is the phase problem, and it is the central difficulty of crystallography for the same cause.

Nor can the reconstruction figures show what a truncated spectrum does to a two-dimensional object. A grating is one-dimensional and its orders are dots on a line; a real specimen has a two-dimensional spectrum, the aperture is a disc cut out of it, and what the image loses is a ring of directions at a time rather than a pair of orders.

Where this ladder goes next #

Three rungs stand on imaging. The first drew a lens as a construction with three rays; the second found that a spherical surface cannot focus and that the paraboloid which can, cannot do it off axis. This one abandons the rays entirely and replaces the question: not where light goes, but which of an object’s periodicities survive the journey.

The habit worth carrying away is about which variable an instrument is really acting on. A device that seems to act on positions often acts on frequencies, and its limitations are then statements about a passband rather than about accuracy. An aperture is a low-pass filter; a finite exposure is a low-pass filter in time; a finite crystal is a low-pass filter in reciprocal space. Once a limit is written as a cut-off, what to do about it is usually obvious and what cannot be done about it is provable.

What is left on this ladder is what the back focal plane can be used for once it is understood as a spectrum. Putting a phase plate there converts an invisible variation of optical path into a variation of brightness, which is phase contrast; putting a mask there measures a specific spatial frequency; and putting a hologram there performs a correlation. All of them are operations on a Fourier transform that a lens has laid out on a physical plane.