The retarder with no crystal in it

Past the critical angle a boundary returns every photon. The reflection is total, the coefficients have modulus exactly one, and it is tempting to conclude that nothing has happened.

Something has. The amplitude is fixed at one, so it can carry no information, and everything the angle does is carried in the phase — which is different for the two polarisations, and whose difference is a retardation of exactly the kind a wave plate exists to produce.

Where a phase comes from when the amplitude cannot change #

Past the critical angle Snell’s law gives a sine greater than one for the transmitted beam, so the transmitted cosine is imaginary. Substituting an imaginary cosine into the Fresnel coefficients turns them from real numbers into complex ones of the form $(a - ib)/(a + ib)$, whose modulus is one and whose argument is $-2\arctan(b/a)$.

Two coefficients, two different arguments:

$$\delta_s = -2\arctan\!\frac{\sqrt{n^2\sin^2\theta - 1}}{n\cos\theta}, \qquad \delta_p = -2\arctan\!\frac{n\sqrt{n^2\sin^2\theta - 1}}{\cos\theta}.$$

They differ by a factor of $n^2$ inside the arctangent, which is the whole of it. At the critical angle the square root is zero and both phases vanish; at grazing incidence the cosine is zero and both reach $\pi$; in between they take different routes, and the gap between the routes is the retardation.

The gap peaks. For borosilicate at $n = 1.5185$ the maximum is $46.53^\circ$, and it is reached at one particular incidence. The closed form for that maximum, $\tan(\delta_{\max}/2) = (n^2-1)/2n$, contains only the index — no wavelength, no thickness, no material property beyond that one number.

It is worth pausing on why the difference exists at all, since the two polarisations meet the same boundary at the same angle. The reason is that the boundary condition they have to satisfy is different: the $s$ field lies wholly in the surface, so its continuity is a single statement, while the $p$ field has a component perpendicular to the surface, whose continuity involves the ratio of the two permittivities. That factor of $n^2$ is the same one that produces the Brewster angle below the critical angle, where it makes one reflectance vanish rather than making one phase run ahead. The asymmetry is the same asymmetry, appearing as an amplitude in one regime and as a phase in the other.

Two reflections make ninety degrees #

A quarter-wave retardation is $90^\circ$, and one reflection cannot supply it for ordinary glass. Two can.

The condition is that each reflection give $45^\circ$, and since the retardation rises to a peak and falls again there are two angles at which it does: $47.70^\circ$ and $55.35^\circ$ for this glass. A rhomb is a parallelepiped cut so that a beam entering normally strikes two faces at one of those angles and leaves normally at the far end, with the two reflections adding.

A beam entering linearly polarised at $45^\circ$ to the plane of incidence has equal $s$ and $p$ components; after two reflections they differ in phase by a quarter cycle, and the beam leaves circularly polarised. Reverse it and circular becomes linear.

That is a quarter-wave plate made out of an angle. There is no birefringent material in it, no optic axis, and nothing cut to a thickness.

There is a further practical consequence of the peak worth naming. Because the retardation has a maximum, the two solutions merge as the index falls towards the threshold, and near the merge the retardation is flat in angle — the derivative vanishes at the peak itself. A rhomb cut for a glass whose peak is only just above 45° is therefore very tolerant of angular error and very intolerant of index error, and one cut for a high-index glass is the reverse. Choosing the glass is choosing which of the two tolerances to spend.

Why the colours do not separate #

The comparison in the figure is the point of the whole device, and the reason is a difference of kind.

A conventional wave plate retards by making one polarisation travel through a slightly larger index for a thickness $d$, giving a path difference $\Delta n\,d$ and a phase $2\pi\Delta n\,d/\lambda$. The wavelength is in the denominator. A plate cut for $550$ nanometres retards by $90^\circ$ there, by $124^\circ$ in the violet and by $73^\circ$ in the deep red — a swing of $62^\circ$ across the visible, and the figure draws it.

The rhomb’s retardation contains no thickness and no wavelength. It depends on the index, and a glass’s index changes by about one per cent across the visible. The swing is $4.05^\circ$.

The factor of fifteen is not the whole of the difference either. A plate’s error is proportional to the wavelength shift and grows without bound outside the design band; a rhomb’s is bounded by the glass’s whole dispersion, so a rhomb cut for the green still works in the near infrared where a plate is useless.

A crystal does the same job with two indices instead of two reflections, and the comparison is where the colour problem comes from. Everything a wave plate does comes from the gap between the two indices; everything it does wrong with colour comes from the wavelength in the denominator of $2\pi\,\Delta n\,d/\lambda$. Calcite, quartz and lithium niobate differ enormously in $\Delta n$ and not at all in that structure — so a crystal retarder is a quarter-wave plate at one wavelength and something else everywhere.

Fresnel built one before the theory was believed #

The rhomb is from 1817, which is worth stating because of what was and was not known then.

Circular polarisation had been produced before, and nobody could say what it was. The prevailing account of light was corpuscular, and polarisation was described as a property of the corpuscles — as “sides” — with no way of saying what a circular one would mean. Fresnel’s rhomb was the demonstration that the phenomenon is a phase relationship between two components, because his device produced it by delaying one component with respect to the other, and no other description of the apparatus is available.

The step required was to accept that light is a transverse wave with two independent components. Fresnel and Young reached that conclusion at almost the same time and both found it uncomfortable, because a transverse wave requires the medium to resist shear, which for the luminiferous ether meant a rigid solid filling all of space. The ether was eventually abandoned and the transversality was not.

What makes the rhomb the good demonstration rather than one of several is that it involves no material with any polarisation-dependent property. A crystal produces circular polarisation and one can always argue that the crystal is doing something obscure. A block of ordinary glass has no preferred direction whatever, and the preferred direction in the experiment is supplied entirely by the plane of incidence.

What is happening in the air outside #

The phases are not arbitrary numbers that fall out of the algebra. They are a statement about something that exists in the medium the light did not enter.

Past the critical angle there is no transmitted ray, and what there is instead is a field decaying into the second medium without carrying energy away. That is the field the phase shift belongs to: the reflection is total in magnitude precisely because nothing leaves, and the depth the evanescent field reaches is what differs between the two polarisations and produces the difference in their phases. Nothing is lost and something is nevertheless different about the two, which is the whole mechanism.

At total internal reflection there is a field in the rarer medium — an evanescent wave, decaying exponentially away from the surface, travelling along it, and carrying no energy away on average. The reflected beam is also displaced sideways along the surface, by a fraction of a wavelength, because it has effectively spent time in that field before returning.

The two polarisations penetrate to slightly different depths and are displaced by slightly different amounts, and the phase difference is the record of that. So the retardation has a physical picture: the two components dip into the forbidden region by different amounts and come back at different times.

The displacement itself is measurable — it is the Goos–Hänchen shift, about a wavelength for a beam near the critical angle — and its polarisation dependence is the same asymmetry expressed as a length instead of as a phase.

The same construction seen as a rotation #

There is a way of reading the two solutions that makes them less arbitrary, and it connects the device to a much larger family.

A polarisation state can be drawn as a point on a sphere, with linear states round the equator and the two circular states at the poles. A retarder is then a rotation of that sphere about an axis: the axis is fixed by the retarder’s orientation and the angle of rotation is the retardation. A quarter-wave plate rotates by ninety degrees, which is exactly what is needed to carry a point on the equator to a pole.

Read that way, the rhomb’s two solutions are two ways of making the same rotation, and the peak between them is the largest rotation one reflection can produce. It also explains why the device is achromatic in the useful sense: what has to be held constant is an angle of rotation, and an angle is a dimensionless quantity that a wavelength cannot enter except through the index.

The same picture makes the failure mode obvious. Any additional retardation — from stress in the glass, from a coating, from a second surface — is another rotation about another axis, and rotations about different axes do not commute or add. So errors in a retarder do not simply add up the way errors in a path length do, which is why two imperfect rhombs in series are not twice as good and not twice as bad but something that has to be computed.

That sphere is the same construction that carries a geometric phase, and the connection is not an analogy: a polarisation taken round a closed loop on it acquires a phase equal to half the area enclosed, regardless of how fast it was taken round. A rhomb is a rotation about one axis and produces no such phase; a sequence of them can.

The instrument, and its awkwardnesses #

A rhomb is used where a wave plate’s chromatic behaviour is intolerable: in polarimeters that must work across a spectrum, in ellipsometry, and wherever a broadband source has to be circularly polarised.

It has three drawbacks and they are all geometric.

The beam is displaced. Light goes in at one place and comes out somewhere else, offset by the length of the block, which is a nuisance in an assembled instrument and impossible in a converging beam.

It is not thin. A wave plate is a disc a fraction of a millimetre thick; a rhomb is a block several centimetres long, because the beam has to travel between two reflections at fifty degrees.

And it only works for a collimated beam. The retardation depends on the angle of incidence, and near the peak that dependence is weak — which is why the peak is a good place to sit — but at the $47.70^\circ$ solution the derivative is not zero. A ray a degree off axis gets a retardation a degree or so wrong. That is much better than a wave plate’s colour error and much worse than a wave plate’s angular tolerance.

The variants in use are attempts on the first two. A Mooney rhomb folds the path to bring the exit beam back in line; a double rhomb cancels the displacement with a second block; and a total internal reflection prism used at the other solution can be made shorter at the cost of angular tolerance.

The threshold has one more thing to say about materials. The condition $n > 1.4966$ is close enough to ordinary glass that the choice matters: crown glasses at $1.52$ clear it comfortably, fused silica at $1.4585$ does not, and water at $1.33$ is nowhere near. So the effect exists at all only for a fairly narrow band of transparent materials, and it is a piece of luck rather than a design that the commonest optical glass sits on the right side of it.

The same phase, doing structural work in a waveguide #

The phases computed at the top of this page are not only the raw material of a device. They appear, with no modification at all, inside the condition that decides which modes a waveguide carries — which is where most working optical engineers meet them without noticing.

Take a slab of high-index material between two lower-index regions, and a ray bouncing along it past the critical angle. For the ray to be a mode rather than an arbitrary zig-zag, the field has to reproduce itself after a round trip across the slab, so the total phase accumulated must be a whole number of cycles. That total has two parts: the ordinary $2k_0 n_1 d\cos\theta$ from crossing the slab twice, and twice the reflection phase $\delta(\theta)$ from the two bounces.

$$2 k_0 n_1 d \cos\theta - 2\delta(\theta) = 2\pi m.$$

Without the second term the condition would be an elementary one giving evenly spaced angles. With it the equation is transcendental and has to be solved numerically, and two consequences follow that are worth having.

The first is that the two polarisations do not share a solution. $\delta_s$ and $\delta_p$ differ by exactly the amount this essay has been computing, so the guided modes of an entirely isotropic waveguide split into two sets with different propagation constants. That is modal birefringence, it exists with no birefringent material anywhere, and it is why a length of ordinary fibre does not preserve a polarisation state.

The second is about the lowest mode. At the critical angle the reflection phase goes to zero, so the condition for $m = 0$ is satisfied at some angle however thin the slab and however weak the index contrast. A symmetric slab therefore always guides at least one mode — the statement the fibre ladder makes without deriving — and the reason is the behaviour of $\delta$ at its lower endpoint, which is visible in the first figure on this page as the point where both curves leave zero.

The mirror that is a retarder nobody asked for #

A metal reflects by a mechanism that looks quite different and produces the same asymmetry, and it is a practical nuisance rather than an opportunity.

A metal’s refractive index is complex, so the Fresnel coefficients are complex at every angle rather than only past a critical one. There is therefore a phase difference between the two polarisations on any non-normal reflection from any mirror — a few tens of degrees at forty-five degrees of incidence for aluminium in the visible, varying with wavelength.

The consequence is that a fold mirror is a wave plate. Send light linearly polarised at forty-five degrees to the plane of incidence into a periscope and it comes out elliptical, with an ellipticity nobody designed and that changes with colour. Instruments that care — polarimeters, ellipsometers, anything measuring the polarisation of astronomical sources — either work at near-normal incidence, where the effect vanishes with the angle, or use mirrors in pairs oriented at right angles to one another, so that what was $s$ at the first surface is $p$ at the second and the two retardations subtract. That is the rhomb’s argument run backwards: two reflections chosen to cancel rather than to add.

The same quantity, measured deliberately, is an instrument of its own. Ellipsometry reflects polarised light off a surface and measures the amplitude ratio and the phase difference between the two components; from those two numbers a film’s thickness and index can be extracted, and because a phase is being measured rather than an intensity, the sensitivity reaches a fraction of a nanometre. It is the standard thickness measurement in semiconductor manufacture, and its whole signal is the difference between $\delta_s$ and $\delta_p$.

Where the model runs out #

Everything assumes a clean, uncoated, undamaged surface. The reflecting faces must not be coated and must not be touched — a fingerprint changes the index in contact with the glass, which changes the critical angle and therefore the retardation, and a rhomb is one of the few optical components ruined rather than merely dirtied by being handled.

The glass must be good enough, and there is a threshold. The maximum retardation from one reflection is $2\arctan\!\big((n^2-1)/2n\big)$, which reaches $45^\circ$ only for $n > 1.4966$. Below that no angle gives a quarter of the required amount and a two-reflection rhomb is impossible. Fused silica at $1.4585$ is below the threshold, so a rhomb cannot be made of it — which matters, because fused silica is what one would otherwise choose for ultraviolet work.

Stress birefringence competes with the effect being used. The block is solid glass and any residual strain from annealing or from mounting adds a retardation of its own, which is not achromatic and which the design has no way to distinguish from the intended one. Rhombs are made from annealed glass and mounted without clamping for that reason.

And the analysis is for a plane wave at one angle. A real beam has an angular spread, each ray gets its own retardation, and the emerging polarisation is a mixture rather than a state. For a beam of a few milliradians the effect is small; for a focused one it is not, which is the same restriction the collimation requirement above states in different words.

A travelling wave has two fields at right angles to each other and to the direction of travel, and the polarisation is the direction of one of them. Everything in this essay is about arranging a quarter-cycle delay between two such directions — which is why a device with no crystal in it can do what a crystal does, and why the property being exploited is a phase and not an absorption.

The ladder from here #

Later rungs on this anchor: the Goos–Hänchen and Imbert–Fedorov shifts, where the same phase asymmetry is measured as a displacement rather than as a retardation; achromatic wave plates built from two crystals of opposing dispersion, which are the competing solution; the Berry phase in a coiled optical fibre, where the retardation comes from the path’s geometry in a still more literal sense; and Pancharatnam’s phase, which is the same idea for polarisation states rather than for directions in space.

The neighbouring ladders are the reflection that happens where the glass is not, which is the evanescent field the phase records, the crystal that answers twice, which is the material way of doing the same job, and the angle at which reflection picks a side, where the same coefficients are real and the polarisation dependence shows up as an amplitude.