The crystal that answers twice

A refractive index is written as a number, and a number is what it is for glass and water and air — enough to fix the whole of refraction when the medium has no preferred direction. It is not what it is for most solids. A crystal is an arrangement of atoms that is not the same in every direction, and its response to an applied electric field need not be parallel to that field — so the quantity relating the two is a matrix, and the speed of light in the material depends on which way the light is going and which way it is shaking.

For the simplest case — a uniaxial crystal, with one special direction — that costs exactly one extra number. One wave, called ordinary, always has its field perpendicular to the optic axis whatever direction it travels, so it sees the same index $n_o$ always. The other, called extraordinary, sees an index that depends on the angle $\theta$ between its wave normal and the axis:

$$\frac{1}{n(\theta)^2} = \frac{\cos^2\theta}{n_o^2} + \frac{\sin^2\theta}{n_e^2}.$$

That is the equation of an ellipse, which is why the effect is described by an index ellipsoid rather than by a pair of numbers.

Which wave is which #

Two waves can travel in a given direction through a crystal, and which of them a particular beam becomes is decided by its polarisation. The rule is geometric. For a wave normal at angle $\theta$ to the optic axis, there is a plane containing both the normal and the axis — the principal plane — and the two allowed states are: field perpendicular to that plane, and field in it. The first never has any component along the axis, whatever $\theta$ is, so it always sees $n_o$; that is the ordinary wave, and it is ordinary in the precise sense that Snell’s law with a single index describes it completely. The second has a component along the axis that grows with $\theta$, so the index it sees is a mixture, and it is the extraordinary one.

Unpolarised light entering a crystal is therefore not split by anything doing sorting. It is resolved, in the same sense a polariser resolves it: the incoming field is written as a sum of the two allowed states, each of those propagates according to its own index, and what emerges is two beams with orthogonal polarisations and no light lost. Putting a polariser in front of the crystal and turning it makes each image vanish in turn, which is the standard demonstration that the two beams are polarised and the standard way of proving that this is a resolution and not a filtering.

What is not the reason for the double image #

Calcite’s two images are the effect everybody meets first, and the usual explanation — two indices, so two refracted rays — is wrong in an instructive way.

It is worth ruling out the obvious explanation first. Ordinary refraction at a boundary bends a ray by an amount set by the ratio of the indices — and at zero incidence it bends nothing, whatever the ratio. So a beam entering a crystal face square on should not split at all, and in calcite it does. Whatever produces the second image, it is not the boundary doing something twice.

So a beam entering a calcite slab at normal incidence has both of its wave normals going straight through. Snell’s law has nothing to say. And yet the two images are separated, and the separation is a millimetre in a centimetre of crystal.

This is the one place in elementary optics where the distinction between a wave normal and a ray has consequences a reader can see with their own eyes. In an isotropic medium the two coincide and the distinction is pedantry. In a crystal they differ by up to six degrees, and turning the crystal turns the extraordinary ray’s offset with it while the ordinary ray goes straight — which is why one image goes round the other.

Where the anisotropy comes from #

The index tensor has been taken as given, and it is worth one paragraph on why a crystal has one, because the answer explains why calcite’s birefringence is so much larger than everything else’s.

A refractive index is a statement about how readily the material’s charges are displaced by a field — the polarisability, averaged over the structure. In an isotropic material that displacement is parallel to the field and its size is one number. In a crystal, the electrons are held by bonds that point in particular directions, so a field along a direction the bonds favour displaces them further than a field across it, and the induced polarisation is neither parallel to the field nor the same size in every direction.

Calcite is the extreme case among common minerals, and the reason is visible in its structure. It is built of flat, triangular carbonate groups, all lying in parallel planes stacked perpendicular to the optic axis. The electrons in such a group are easy to push around within the plane of the triangle and hard to push out of it, so a field lying in those planes sees a large polarisability and a field along the axis a much smaller one. The result is an ordinary index of 1.658 and an extraordinary one of 1.486 — a difference of 0.172, which is two hundred times quartz’s.

The sign follows from the same picture. Calcite is called negative uniaxial because $n_e < n_o$, and that is the direct consequence of the axis being the hard direction; a crystal whose special direction is the easy one is positive, and quartz, whose structure is a helix of linked tetrahedra rather than a stack of flat groups, is.

Which makes the whole of this essay a consequence of the dielectric response being a tensor rather than a scalar, and the tensor being a tensor because a crystal’s bonds point somewhere.

Two components, one delay #

The other use of a crystal is not to separate the two waves in space but to delay one against the other in time. Cut a slab with its optic axis lying in the surface, and light entering at normal incidence has both waves travelling straight down the same path — no walk-off, because the wave normal is at right angles to the axis and the walk-off vanishes there — at different speeds. They emerge together, out of step.

The phase difference accumulated in a thickness $d$ is

$$\Gamma = \frac{2\pi d\,|n_o - n_e|}{\lambda},$$

and choosing $d$ makes $\Gamma$ whatever is wanted.

What the delay does to the light is best seen by tracing the electric field.

The last of those panels is the half-wave plate, which turns linear polarisation through twice the angle between the input and the crystal’s axis. Set the plate at 45° and the polarisation is turned by 90°, which is the cheapest way to rotate a beam’s polarisation without rotating anything mechanical through the beam.

What circular polarisation is not #

The natural reading of that circle is that a third kind of light has been produced. It has not.

Polarisation is a statement about the electric field’s behaviour over a cycle, not about a second field or a second wave. In an electromagnetic wave the electric and magnetic fields are at right angles to each other and to the direction of travel, and what distinguishes linear from circular is only how the electric vector moves as the wave goes by — steady in one direction, or turning. There is one wave in both cases.

Circular light is two linear oscillations of equal amplitude, a quarter cycle apart, in perpendicular directions. It could equally be described as a single state in a basis of left- and right-circular components, in which case linear light is the sum of two circular ones. Which description is the real one is not a question a measurement can answer, and both are used: chemists working with sugars think in circular components because a chiral molecule treats them differently, and everybody working with polarisers thinks in linear ones because a polariser is a linear device.

The instruments this makes possible #

Almost every use of birefringence is one of the two effects above put to work.

Separating two polarisations in space uses the walk-off. A calcite beam displacer is a slab with its axis cut at 45°, and it takes one beam in and gives two parallel beams out, laterally displaced, orthogonally polarised and both at full brightness. A polarising beamsplitter built out of a polariser throws half the light away; one built out of a crystal throws none of it away, and that difference matters wherever the light is scarce. The Nicol prism, which was the standard polariser for a century, goes further: it splits the two rays inside calcite and then gets rid of one of them by total internal reflection at a cemented joint cut so that the ordinary ray exceeds the critical angle and the extraordinary one does not.

Changing a polarisation state in time uses the retardance. A quarter-wave plate in front of a mirror is an optical isolator — though not a non-reciprocal one, which needs a rotation a return trip doubles: light goes in linear, comes back circular, passes the plate again and returns linear at 90° to where it started, so a polariser that let it out will not let it back in. That arrangement protects lasers from their own reflections, and it is why a plate whose whole function is to delay one component by a quarter of a cycle turns up in equipment that has nothing else optical about it.

And making the retardance controllable turns the plate into a display. A liquid-crystal cell is a layer whose birefringence responds to an applied voltage, sandwiched between crossed polarisers: at one voltage it is a half-wave plate and the pixel is bright, at another it retards nothing and the pixel is dark. Every screen this essay might be read on is several million of the arrangement in the retardance figure, addressed individually.

Thirty micrometres, and why #

The colours between crossed polarisers are usually treated as a decoration, and one whole science uses them as a measurement instead.

A geological thin section is a slice of rock ground to a standard thickness of thirty micrometres, mounted on a slide and examined between crossed polarisers on a rotating stage. Every mineral in it retards light by its own birefringence times that thickness, and the retardation decides the colour — the same subtraction of one wavelength from a white spectrum a thin film performs: white where every wavelength gets through, and successively more saturated colours as particular wavelengths are extinguished and the transmission curve of the retardance figure is climbed.

The standard thickness exists because it makes the colours a reference. Quartz has a birefringence of about nine thousandths, so at thirty micrometres it retards by around 270 nanometres — a pale grey at the top of the first order, which is instantly recognisable and present in almost every rock. The section is ground until the quartz looks right, and every other mineral in the field is then read against a chart of colour against retardation, which converts a colour directly into a birefringence and a birefringence into an identification.

That is an unusually direct instrument. A property of a material — the difference of two indices — is read off as a colour by eye, at a standardised thickness, with no photometry and no calibration beyond the grinding. And the sensitivity is remarkable: minerals differing by a few thousandths in birefringence are told apart at a glance, which is a measurement of an index difference to a part in a thousand made without measuring an index at all.

Rotating the stage adds the second half of the information. As the crystal turns, its axes sweep past the polariser’s direction, and the pixel goes dark four times per revolution — at the angles where one of the crystal’s own axes lines up with the polariser and there is nothing to resolve into two components. Where those extinctions fall relative to a visible crystal face is a statement about the orientation of the optic axes in the crystal, which is a further identification.

Birefringence without a crystal #

Nothing in the argument requires a crystal, and two cases where there is none are worth having.

The first is a material with no anisotropy in its molecules at all, made anisotropic by its geometry. A stack of alternating layers, each isotropic and each much thinner than a wavelength, behaves as one medium — the light averages over many layers and cannot resolve them — and that averaged medium has different indices for a field along the layers and across them. It is form birefringence, and it is why some biological tissues, which are stacks of membranes, show between crossed polarisers even though nothing in them is crystalline.

The second is stress. Squeeze an isotropic solid and its bonds are no longer equivalent in every direction, so it becomes weakly birefringent, with the axes along the principal stresses and the retardation proportional to the difference between them. A transparent model of a structure, loaded and viewed between crossed polarisers, therefore displays a map of its own internal stress as a pattern of fringes, each fringe marking a contour of constant stress difference.

That was the standard method of stress analysis before anything could be computed, and it is still the quickest way to see where a load concentrates. It also means that a moulded plastic object viewed through polarised light shows the stresses frozen into it when it cooled — which is why a clear plastic lid seen through polarised sunglasses is covered in colours nobody put there.

Where the model stops #

Uniaxial is the easy case. Most crystals are biaxial: three different principal indices, two optic axes rather than one, and a surface of wave normals that has four conical points where extremely odd things happen — a single ray entering along one emerges as a hollow cone of light, which Hamilton predicted in 1832 and Lloyd found within two months. None of that appears in the equation above, which has one special direction in it by assumption.

The indices depend on wavelength. Every number quoted here is at the sodium D line, and a quarter-wave plate cut for 589 nm is not a quarter-wave plate for 450 nm. That is why a plate between crossed polarisers shows colours in white light rather than simple brightness: the transmission is the square of a sine whose argument carries $1/\lambda$, so different colours are at different points on the curve.

The plate is assumed thin enough to ignore the beam’s own spread. A wave plate delays one component against the other by a fixed phase only for a beam travelling exactly along the design direction. A converging or diverging beam contains rays at a range of angles, each seeing a slightly different index difference, so a plate placed in a focused beam retards different parts of it differently. This is why high-quality plates are made from two pieces of crystal with their axes crossed, so that the bulk of the retardance cancels and what is left is a small difference that is far less sensitive to angle.

And the crystal is assumed transparent and unstressed. Absorption that depends on polarisation — dichroism — is a different effect, and it is what a sheet polariser is made of. Stress makes an isotropic material birefringent, which is what turns a plastic ruler between crossed filters into a map of its own stresses, and it makes the effect a function of position rather than a property of the material.

One more limit is worth naming because it is the reason the equations here have any use at all. Everything above treats the crystal as a continuous medium with a smooth dielectric tensor, and a crystal is an arrangement of discrete atoms. The continuum description works because the wavelength of light is thousands of times the spacing between atoms, so the light samples an enormous number of them and sees only their average. At X-ray wavelengths the ratio is reversed, the averaging stops, and the same crystal is no longer a birefringent medium but a diffraction grating in three dimensions — a different phenomenon, from the same atoms, because the probe got smaller than the structure.

What the pictures cannot show #

The polarisation-state figure draws the field’s path over one cycle, and a real optical cycle is $5\times10^{14}$ per second. Nothing has ever seen that path. What is measured is always an average — an intensity through an analyser, at several analyser angles, from which the state is reconstructed — so the ellipse in the figure is an inference, and a very secure one, rather than an observation.

The walk-off figure draws two rays and no fields. What is actually different between the two rays is the direction the electric field points, and that is what selects which index each one sees; the drawing shows the consequence and cannot show the cause.

Where the ladder goes next #

The polarisation ladder began with the direction of the shaking and the filter that asks about it and continued with the angle at which a reflection picks a side. This rung makes the medium itself directional. The rungs above it: optical activity, where a solution of chiral molecules rotates the plane continuously with no crystal axes anywhere; the electro-optic effect, where an applied voltage changes the indices and a wave plate becomes a switch running at gigahertz; the Fresnel rhomb, which makes circular light out of two total internal reflections and no birefringence at all; and the Jones and Mueller calculus, which turns all of this into matrix multiplication and handles partial polarisation, which none of the pictures here can.

The habit worth carrying away is what happens when a constant becomes a matrix. A scalar relation assumes the response is parallel to what caused it, and dropping that assumption is not a small correction — it produces two waves where there was one, an energy flow that is not along the wave normal, and a device whose whole function is that the two do not keep in step.