The index that falls below one

The bend at the boundary derived Snell’s law from the statement that light changes direction when it changes speed, and the law that only asks about one component rebuilt it from phase matching. The angle that is two angles gave the index an imaginary part, the ray on the wrong side of the normal gave it a negative sign, and the reflection that needs no surface and the crystal made of moments let it change in time. Every one of those assumed, in its examples, what the everyday world supplies: an index somewhat greater than one, the light slower in the material than outside it.

That assumption fails for an entire region of the spectrum, and not by a little in a few materials but by a little in all of them. For X-rays the refractive index of every substance is less than one. The difference is tiny — a few parts in a million — and it has consequences for every optical element that can be built, all of which come out, so to speak, backwards.

An index made by free electrons #

The refractive index of a material measures how its electrons respond to the oscillating field of a light wave. For visible light, whose frequency lies below the natural frequencies at which electrons in atoms resonate, the electrons oscillate in step with the field and add a wave that delays the light: the index is above one. X-rays have frequencies far above almost every resonance in the atom. At such frequencies an electron responds as if it were free, oscillating a half-cycle out of step with the field, and the wave it re-radiates advances the light’s phase instead of delaying it.

The frequency below which nothing gets in worked out that response for the free electrons of a plasma: the index is $\sqrt{1 - \omega_p^2/\omega^2}$, where the plasma frequency $\omega_p$ is set by the density of electrons. Below $\omega_p$ radio waves cannot enter an ionosphere; above it they pass with an index a little less than one. For a solid, the electron density is some $10^{30}$ per cubic metre, the plasma frequency lies in the ultraviolet, and X-rays are far above it. Expanding the square root, the index falls short of one by

$$\delta = \frac{\omega_p^2}{2\omega^2} = \frac{r_e \lambda^2 n_e}{2\pi},$$

with $r_e$ the classical radius of the electron and $n_e$ the density of electrons.

The figure plots $\delta$ for three solids. It falls as the square of the photon energy, because it goes as the square of the wavelength. It is largest for gold, whose atoms each bring seventy-nine electrons and are packed densely, and smallest for beryllium, the lightest structural metal. At the eight kiloelectronvolts of a copper X-ray tube it is a few parts in a hundred thousand for gold and a few parts in a million for the others. For comparison, the index of window glass for visible light exceeds one by a half. The same materials that bend visible light by tens of degrees at a surface bend X-rays by microradians.

That the phase travels faster than light in vacuum breaks nothing. The answer that cannot come first showed that the requirement that a material respond after it is pushed ties its index at every frequency to its absorption at every other, and that the same requirement forces the index towards one from below at frequencies high enough that nothing can follow. Energy and information travel at the group velocity, which in this regime is below $c$.

Total reflection from the outside #

Snell’s law applies to X-rays exactly as to light. Going from vacuum, index one, into a material of index $1 - \delta$ is going from a higher index to a lower one, and a ray bent towards the surface can be bent right out of the material. The angle past which light cannot leave found this for light inside glass meeting air: beyond a critical angle nothing is transmitted and the reflection is total. For X-rays the vacuum plays the part of the glass. Radiation arriving from outside at a grazing angle smaller than a critical angle is reflected completely.

The critical angle is small. Measured from the surface rather than from the normal, it is $\sqrt{2\delta}$ — an angle whose square is twice the index decrement.

The figure computes the reflected fraction for gold and silicon at eight kiloelectronvolts. Without absorption the reflection is exactly total below the critical angle, 0.57 degrees for gold and 0.22 for silicon, and falls away steeply above it, as the fourth power of the angle, to a few parts in ten thousand. Absorption — the imaginary part of the index, the angle that is two angles found in metals — rounds the edge and keeps the plateau slightly below one, because the reflected wave penetrates a few nanometres into the surface as an evanescent wave and some of it is absorbed there. Gold absorbs more and loses more, but reflects to a larger angle.

The practical meaning is that an X-ray mirror has to be met almost edge-on. A mirror that reflects X-rays at normal incidence, as a bathroom mirror reflects light, would return about $\delta^2/4$ of them — a part in $10^{10}$. At grazing incidence the same surface can return ninety per cent. The same physics, turned the other way, is why X-rays pass through a body at almost any angle with almost no deflection: the index barely differs from one, so nothing refracts them.

A few nanometres of surface, lit and nothing else #

Total reflection is never quite total in the sense of the wave not entering. Below the critical angle the field inside the material is an evanescent wave, decaying away from the surface without carrying energy into the bulk, exactly the wave that the mirror that lights a tenth of a micrometre used to illuminate a thin sheet of a sample just beyond a glass surface for fluorescence microscopy. For X-rays the depth of that wave is set by the same geometry, the wavelength divided by $2\pi$ times the critical angle, and for silicon at eight kiloelectronvolts it comes to about six nanometres — some twenty atomic layers.

That makes grazing X-rays one of the most surface-sensitive probes there is. Diffraction patterns taken with the beam below the critical angle come only from the top few nanometres, so the structure of a surface, of a monolayer of molecules lying on water, or of the first atomic layers of a growing film, can be measured without the enormous signal from the bulk beneath swamping it. Raise the angle a little past critical and the wave enters, penetrating micrometres, and the same experiment sees the bulk. The grazing angle becomes a dial for depth.

Just above the critical angle, a thin film on a substrate returns a reflectivity curve striped with fringes, because X-rays reflected from the film’s top surface and from its interface with the substrate interfere. The spacing of the fringes in angle gives the film’s thickness, their decay gives the roughness of the interfaces, and the position of the critical edge gives the film’s density. X-ray reflectometry, as the method is called, measures the thickness of layers a few nanometres thick to a fraction of a nanometre without touching them, and it is used on every production line that makes the multilayer stacks of hard-disk read heads and the gate oxides of transistors. All of it rests on the index being a few parts in a million below one, so that the relevant angles are small and a laboratory goniometer can resolve them.

The angle sets the energy #

Because the critical angle is $\sqrt{2\delta}$ and $\delta$ falls as the square of the energy, the critical angle falls as one over the energy.

For any mirror, fixed in its mounting, there is therefore an energy above which it stops reflecting. The figure plots the critical angle for gold and silicon and marks where a mirror met at 0.85 degrees, close to the steepest angle in the mirrors of the Chandra X-ray Observatory, reaches its limit: about five kiloelectronvolts for gold, two for silicon. Chandra’s mirrors are coated with iridium, denser still, which reaches about ten.

That sets the shape of every X-ray telescope. To gather light from the sky a telescope needs collecting area facing the sky, and a mirror met at a grazing angle of half a degree presents to the sky only a hundredth of its surface. The solution, due to Hans Wolter in 1952, is a pair of nested surfaces, a paraboloid followed by a hyperboloid, each met at grazing incidence, which together bring rays to a focus with little distortion; to build up area, many such pairs are nested inside one another like the layers of an onion, each a thin barrel of glass or metal a metre long. Chandra has four nested pairs; the XMM-Newton observatory has fifty-eight in each of its three telescopes. Telescopes for higher energies, such as NuSTAR, which reaches eighty kiloelectronvolts, coat their shells with hundreds of alternating layers of heavy and light materials, which reflect by interference at angles beyond the critical one — the same idea as a Bragg reflector, the layer that makes a reflection vanish used the other way.

A lens that must be concave #

A lens focuses by delaying the middle of a wavefront relative to its edges, so that the wavefront comes out curved towards a point. Glass slows the phase, so a converging glass lens is thicker in the middle. A material for X-rays advances the phase, so a converging X-ray lens must be thicker at the edges than in the middle — concave.

The figure sets the two side by side. The rays converge in both; the lens shapes are opposite. The X-ray lens is also extraordinarily weak. A lens whose surfaces are paraboloids with an apex radius $R$ has a focal length $R/2\delta$ for X-rays, and for a beryllium lens with $R$ of fifty micrometres — a surface curved as tightly as a human hair — the focal length at eight kiloelectronvolts is 4.7 metres. A glass lens of that curvature would focus visible light within a tenth of a millimetre.

For most of the twentieth century this was taken to mean that X-ray lenses were impractical, and X-ray optics meant grazing mirrors and the diffraction of crystals and gratings. The remedy turned out to be trivial once somebody tried it.

In 1996 Anatoly Snigirev and colleagues at the European Synchrotron drilled a row of thirty holes, each a millimetre across, close together in a block of aluminium. Each gap between two holes is a biconcave lens, and thirty of them in a row have a thirtieth of one lens’s focal length. The figure shows the arithmetic for beryllium: thirty lenses bring eight-kiloelectronvolt X-rays to a focus sixteen centimetres away, a hundred to one under five. Compound refractive lenses, now made of beryllium or aluminium with parabolic profiles pressed or etched, are standard at synchrotrons for focusing beams to micrometres. Their weakness is chromatic: $\delta$ goes as one over the energy squared, so the focal length does too, and a stack designed for one energy is badly out of focus at another.

Why beryllium #

The choice of material shows the trade that sets all X-ray optics. A lens needs a large $\delta$, which means many electrons, but every electron in a heavy atom also contributes to absorption, which grows much faster with atomic number. The photoelectric absorption that the steps in an absorption curve followed rises roughly as the fourth power of the atomic number, while $\delta$ rises only as the electron density. Beryllium, with four electrons per atom, is the lightest metal that can be machined, and it gives the most bending per unit of absorption; lithium and carbon are used for the same reason. A gold lens would bend ten times as strongly and absorb the beam in a few micrometres.

Mirrors reverse the trade. Absorption barely matters at grazing incidence, where the X-rays penetrate only a few nanometres, and what matters is the critical angle, which grows with $\delta$. Heavy metals — gold, iridium, platinum — are therefore the coatings of choice, deposited a few tens of nanometres thick on a light substrate.

The same optics for neutrons #

X-rays are not the only waves for which matter has an index just below one. A slow neutron is a wave too, and it refracts at a surface because the nuclei it passes present an average potential energy — the Fermi pseudopotential — set by their density and by how strongly each scatters neutrons. For most nuclei that potential is repulsive, which for a matter wave means an index less than one, by a few parts in a million for thermal neutrons: the same size as for X-rays, for an entirely different reason. Neutron guides that carry beams tens of metres from a reactor to an instrument are therefore glass tubes coated with nickel, met by the neutrons at grazing incidence and totally reflecting them exactly as X-ray mirrors do; multilayer “supermirrors” extend the angle in the same way as multilayer X-ray coatings.

For very slow neutrons the index departs so far from one that the critical angle reaches ninety degrees. Ultracold neutrons, moving at a few metres per second, have kinetic energies below the potential of materials such as beryllium or stainless steel, and are reflected at every angle of incidence. They can be poured into a bottle and kept there for minutes — which is how the neutron’s lifetime is measured by counting how many survive — and they are slow enough that gravity stops them rising more than a metre or two, the regime the fall that leaves the mass in the phase explored.

Where the free-electron picture stops #

The formula for $\delta$ treats every electron as free, and near an absorption edge that is wrong. When the photon energy approaches the binding energy of an inner shell, the electrons of that shell resonate, and the index departs from the free-electron value in the way the dispersion near any resonance does: $\delta$ dips and then overshoots, and just below an edge it can even change sign. Crystallographers use exactly this, tuning the energy of a synchrotron beam near an element’s edge to change how strongly that element scatters. The values in the figures are the free-electron ones, accurate to a few per cent away from edges; the tabulated values for gold at eight kiloelectronvolts, near its L edges, are about six per cent lower.

The figures also take surfaces to be perfectly smooth. Real mirrors have roughness, and a surface rough on the scale of a nanometre scatters X-rays out of the specular beam at these tiny angles, reducing the reflectivity by a factor that depends exponentially on the roughness divided by the wavelength. The mirrors of X-ray telescopes are polished to a few tenths of a nanometre for that reason, among the smoothest surfaces ever made.

Still open: how small a focus X-rays can reach #

Refractive lenses, grazing mirrors and diffractive zone plates now focus hard X-rays to spots of a few tens of nanometres, and the question of how far that can go is active. The ultimate limit for a refractive lens is set by absorption: a strong enough lens has so much material at its edges that it absorbs the rays it most needs, and the numerical aperture it can reach is limited to roughly $\sqrt{\delta}$ times a factor that depends on the ratio of $\delta$ to $\beta$. Designs that remove the unnecessary material — lenses cut into steps like a Fresnel lens, or made of many thin plates — push the limit, and whether hard X-rays can be focused to a single nanometre, where the images would resolve individual atomic columns in thick samples, is being pursued with lenses, mirrors and zone plates in parallel.

The habit worth carrying away is to ask which side of a resonance a wave is on. Below the natural frequencies of the electrons, matter delays light and bends it towards the normal; far above them, as X-rays are, it advances the phase by a few parts in a million, and every optical element has to be reversed — lenses concave and stacked by the dozen, mirrors met at a glance from outside, telescopes built as nested barrels. The same electrons do the bending in both regimes; what changes is whether they can keep up.