The lens that is a set of rings

A lens works by delay. Light through the middle travels further in glass than light through the edge, and the thickness is cut so that every path from object to image takes the same time. That is a statement about a material and a shape.

There is a second way, and it uses no material and no shape. Take a flat screen, work out which parts of it deliver light to the focus in step and which deliver it out of step, and paint out the second kind. What gets through is only light that arrives within a quarter-wave of being in phase, and it adds.

The rings, and why they are equal in area #

Fix a focus at distance $f$ and ask how far off-axis a point on the screen must be for its path to the focus to be half a wavelength longer than the axial one. That radius is the first zone’s edge. A whole wavelength longer gives the second, and so on:

$$\sqrt{r_n^2 + f^2} - f = \frac{n\lambda}{2}.$$

Solving gives $r_n$ exactly, and for a slow plate it is very nearly $\sqrt{n\lambda f}$ — the radii go as the square root of the zone number. The figure computes the exact radii and prints how far the approximation is from them, which for these numbers is under two per cent at the outermost ring.

The consequence worth noticing is not on the formula, it is on the drawing. Because the radii go as $\sqrt n$, the area of each annulus is the same. The rings get thinner outwards at exactly the rate that keeps their areas equal, to three parts in a hundred thousand in the figure.

Equal areas mean equal contributions. Every open ring sends about the same amount of light to the focus, and — by construction — all with the same phase.

Why blocking half of it helps #

That is the surprise, stated as arithmetic. With alternate rings blocked, $N$ open rings contribute $N$ equal amplitudes in phase, so the amplitude at the focus is proportional to $N$ and the intensity to $N^2$.

The open aperture is the comparison that makes the point. Take the same discs with nothing painted out and the intensity at the same distance does not grow with the aperture at all: it oscillates, between roughly four times the unobstructed value and nearly nothing, according to whether the outermost zone the aperture happens to contain is a contributing one or a cancelling one.

That oscillation is the Cornu spiral read as chords, and it is exactly what an unmodified aperture does. What the zone plate does is remove every contribution that would have subtracted, so the chord becomes a straight run rather than a spiral.

Thrown away: half the light. Gained: an amplitude proportional to the number of rings instead of bounded by the diameter of the spiral. For twenty zones that is a factor of a hundred in intensity over the unobstructed value, at the cost of half the transmitted power.

It is worth being explicit about what is and is not being claimed by “the axis gets brighter”. The total power reaching the screen is halved, since half the aperture is opaque. What has grown is the intensity at one point, and it has grown at the expense of everywhere else: the light that would have gone into the surrounding pattern is gone, and what remains is concentrated. That is the same bargain a lens makes and the same bargain a grating with many slits makes — sharpness bought by removing the light that was in the way of it — arrived at here by subtraction rather than by delay.

Not one focus, but a series #

The construction was carried out for one distance, and the same rings are a valid zone construction for other distances too.

At a third of the design distance, each open ring spans three half-period zones rather than one. Two of the three cancel, one survives, and there is a focus. At a fifth, each ring spans five and one survives. At a half, each ring spans two, they cancel exactly, and there is nothing.

So a zone plate has foci at $f$, $f/3$, $f/5$, $f/7$ and so on for ever, and none at the even fractions. It also has virtual foci at the negative values, which is why looking through a zone plate at a point source shows a diverging beam as well as a converging one.

The heights are worth reading carefully because they are not what the usual account predicts. The efficiency into order $m$ falls as $1/m^2$ — only a ninth of the light goes into the third order. But the third-order focus is three times shorter, so its numerical aperture is three times larger and its spot is nine times smaller in area. The two effects cancel exactly, and the on-axis intensity at $f/3$ is essentially the same as at $f$: the figure gives $99.9$ per cent.

That is a real nuisance in practice. An imaging system built round a zone plate has to reject the other orders with a stop, because they are not faint.

The construction has a history longer than its usefulness. Fresnel’s zones date from 1818 and the observation that blocking the alternate ones would brighten the axis was made by Lord Rayleigh in a notebook in 1871; Soret built and published one in 1875. For most of a century afterwards it was a lecture demonstration, because for visible light there was never any reason to prefer it to glass. It became an instrument only when a wavelength was reached at which glass does nothing.

Colour, the wrong way round #

The rings are cut once. A wavelength other than the design one finds its focus where its half-wave steps land, which is a different place.

Since the zone radii are fixed and the path condition scales with the wavelength, $f \propto 1/\lambda$: the focal length is shorter for red light than for blue. A glass lens does the reverse, because its index falls with wavelength and a weaker lens has a longer focus.

The size of the effect is the disqualifying part. The plate’s focal length changes by $62$ per cent across the visible; the singlet’s by $2.7$. A zone plate is a hundred times more chromatic than an uncorrected lens, and of the opposite sign.

The opposite sign is also an opportunity. A weak zone plate cemented to a glass lens can cancel the glass’s chromatic aberration exactly at two wavelengths, using one element instead of two, and diffractive surfaces are used for this in camera lenses and in eyepieces. It is the same trick as an achromatic doublet with a diffractive surface standing in for the flint glass.

The hologram hiding in it #

There is a fact about zone plates that turns them from a device into a principle, and it is worth arriving at by asking what the ring pattern is.

Record the interference of two point sources on a screen at right angles to the line joining them and the pattern of bright and dark is a set of rings whose radii go as the square root of the ring number — which is a zone plate. That is not a resemblance: a zone plate is the recorded interference between a point source and a plane wave, and a hologram is the same recording of a more complicated object. The lens and the hologram are one idea at two levels of complexity.

Take a point source of monochromatic light and a plane wave of the same wavelength, and let them interfere on a flat plate. Where they are in phase the plate is exposed; where they are out of phase it is not. The resulting pattern of rings is — exactly — a zone plate, with the point source’s distance as the focal length, because the condition for constructive interference is the same half-wave path condition the zones were constructed from.

So a zone plate is the recorded interference pattern of a point and a plane wave. Illuminating it with the plane wave afterwards reconstructs the point: the rings diffract the light into a converging beam that comes to a focus where the source was.

That is holography, in its simplest possible case. A hologram of an arbitrary object is the superposition of the patterns from every point of it, and playing it back reconstructs every point at once. Gabor arrived at the idea from the other end — he was trying to correct electron microscope aberrations — and the zone plate is what the theory reduces to for a single object point.

It also explains the virtual focus. A zone plate produces a converging beam and a diverging one because the recorded pattern cannot tell which of the two waves was which, and playing it back reconstructs both the real image and its conjugate. Every hologram has the same problem, and separating the two is what off-axis holography is for.

Where it is the only option #

For visible light a zone plate is a curiosity, because glass exists. For X-rays it is the instrument.

At a wavelength of a nanometre every material has a refractive index within a few parts in ten thousand of one, so a refracting lens would need a focal length measured in kilometres. Mirrors work only at grazing incidence and cannot be made to converge steeply. What is left is diffraction.

An X-ray zone plate is a set of rings of gold or nickel a few hundred nanometres thick, written by electron-beam lithography, with an outermost ring width that decides the resolution — because the numerical aperture is $\lambda/2\Delta r_N$, so the finest achievable spot is about the width of the outermost ring and has nothing to do with the wavelength directly. That single fact sets the whole technology: making better X-ray optics means writing narrower outer rings, and the current limit is around ten nanometres.

The same construction is used at radio wavelengths for the opposite reason. A zone plate antenna is cheap because it is a flat sheet with holes in it rather than a machined paraboloid, and at metre wavelengths a “ring” is a metre wide.

What the resolution actually depends on #

The single most useful statement about a zone plate is not about its focal length, and it is worth deriving rather than quoting.

The numerical aperture is $r_N/f$, and using $r_N^2 = N\lambda f$ together with the width of the outermost ring, $\Delta r_N = r_N/2N$, gives

$$\mathrm{NA} = \frac{r_N}{f} = \frac{\lambda}{2\Delta r_N}.$$

The wavelength cancels out of everything that follows. The resolution, which is about $0.61\lambda/\mathrm{NA}$, becomes about $1.22\,\Delta r_N$: the finest detail a zone plate can resolve is a little over the width of its outermost ring, whatever the wavelength.

That is an unusual situation in optics and it has an unusual consequence. Going to a shorter wavelength does not improve the resolution of a zone plate at all; it only makes the plate’s focal length longer. Improving the resolution means writing narrower rings, which is a lithography problem rather than an optics one, and the history of X-ray microscopy is a history of ring widths: a micrometre in the 1970s, a hundred nanometres in the 1990s, ten today.

The number of zones then decides something else entirely — the monochromaticity required. A plate of $N$ zones needs a source with a bandwidth better than $1/N$, because outside that the different colours’ foci are separated by more than the depth of focus. A thousand-zone plate needs a part in a thousand, which is a synchrotron with a monochromator on it.

The spot that was offered as a refutation #

The zone construction has a complement, and it settled the wave theory of light.

Block only the central zones — put an opaque circular disc in the beam — and the zones that remain are the outer ones, which are still equal in area and still deliver to the axis in the ordinary sequence. The first surviving zone dominates as always, so the intensity on the axis behind the disc is essentially what it would have been with no disc at all. The centre of a circular shadow is bright, and about as bright as the surrounding light.

In 1818 the French Academy set a prize competition on diffraction, expecting to see the corpuscular theory vindicated. Fresnel submitted his integral. Poisson, judging, worked out this consequence and presented it as a reduction to absurdity: the theory predicts a bright spot in the middle of a shadow, therefore the theory is wrong. Arago went and looked, and the spot was there. The prize went to Fresnel.

Two details make the story better than its usual telling. The spot had been seen before — by Delisle and by Maraldi, around 1723 — and forgotten, because there was no framework that made it worth reporting. And it is unusually robust: a ragged or imperfectly circular disc still produces it, because the contributions come from all the way round the rim and roughness in one place is compensated elsewhere. That robustness is why the demonstration works with a ball bearing glued to a microscope slide, and why the spot is a nuisance in any instrument with an obstruction in it — a particle on a lithography mask, a secondary mirror’s support, a coronagraph’s occulting disc, all of which put light back exactly where it was meant to be excluded.

The wall the technology has run into #

The resolution argument says that a better X-ray microscope means a narrower outermost ring. What stops that from being merely a lithography problem is a second requirement pulling the other way.

A zone plate for X-rays must absorb, and X-rays are hard to absorb. Stopping a useful fraction of a kilovolt beam needs a few hundred nanometres of gold; stopping ten kilovolts needs micrometres. So the rings must be tall. The outermost ring is meanwhile ten or twenty nanometres wide, and the ratio of the two — the aspect ratio of a structure that must be written, plated and left standing — is where the technology is stuck. Fifty to one is difficult and a hundred to one is at the edge of what exists.

The physics closes in on the same number from the other side. Everything in this essay treats the plate as infinitely thin: a ray enters a ring and leaves it in the same ring. That holds only while the plate’s thickness is below about $2\Delta r_N^2/\lambda$, and a moment’s arithmetic shows that quantity is exactly the depth of focus of the plate itself. So the thin-element description survives precisely as long as the plate is thinner than its own depth of focus, and the finest plates being made now sit at that boundary rather than comfortably inside it.

Past it a zone plate stops being a screen and becomes a volume grating: the rings have to be tilted so that each one points at the focus, satisfying a Bragg-like condition through their depth, and the design problem changes character entirely. Tilted and stacked zone plates are made for hard X-rays for exactly this reason, and they are a different device wearing the same name.

Where the model runs out #

Everything here is scalar and paraxial. The Fresnel integral treats light as a scalar amplitude and the zone radii as small compared with the focal length, which is fine for the twenty-zone plate drawn and fails for the fast plates X-ray microscopy uses. There the exact radii differ from $\sqrt{n\lambda f}$ noticeably, which the generator refuses to draw as though they did not.

A zone plate’s resolution is governed by the same criterion as any other aperture of the same numerical aperture — the outermost zone width is what sets it, and the number of zones sets nothing except the efficiency. That is worth stating because it removes a common confusion: more zones make a brighter image and not a sharper one.

The plate is an amplitude plate. Blocking half the light costs half the light, and the efficiency into the first order is $1/\pi^2$, about ten per cent. A phase zone plate, which delays the alternate zones by half a wave instead of blocking them, uses all of the light and reaches about $41$ per cent; a blazed one, which imposes a continuous phase ramp, reaches nearly all of it and has only one order. Each is a different device, and the arithmetic of the odd-order series changes with them.

The source is a point at infinity. For a finite conjugate the zone construction has to be redone with both distances, which moves the radii; a plate cut for a collimated beam is wrong for a nearby source in exactly the way a lens is not, since a lens’s focal length is a property of the lens and a plate’s zones are a property of the geometry they were computed for.

And the light is assumed coherent across the whole plate. A zone plate works by adding contributions from rings millimetres apart, so it needs a coherence length and a coherence width exceeding those separations. With a broad thermal source the outer zones contribute incoherently and the gain is lost, which is another reason the technique belongs to synchrotron beamlines rather than to laboratory X-ray tubes.

Everything a refracting lens does about colour comes from the slope of the index against wavelength, which is small and awkward to control. A zone plate’s chromatic behaviour comes from geometry instead — its focal length is inversely proportional to the wavelength exactly — so it is far more dispersive than any glass and dispersive in a way that can be computed rather than measured.

The ladder from here #

Later rungs on this anchor: the phase zone plate and the blazed one, where the efficiency argument is redone and the order series changes; photon sieves, which replace the rings with an arrangement of holes and suppress the higher orders by their statistics; the Fresnel lens, which is a refracting device that looks like a zone plate and works by an entirely different principle; and holography, where the zone plate turns out to be the recorded interference pattern of a point source and a plane wave — so every hologram is a superposition of them.

The neighbouring ladders are the spiral that says how much arrives, which is where the zones come from, what a lens is doing, which is the other way of building a focus, and what a thousand slits buy, where the same adding of many equal contributions in phase is done in one dimension.