The stress that polarised light can see

The direction of the shaking introduced polarisation as the one property of a transverse wave that a filter can select, and the angle at which reflection picks a side found David Brewster’s angle, at which water reflects no light of one polarisation at all. The crystal that answers twice found calcite doubling the print beneath it, because its structure gives light two speeds depending on how it is polarised, and the retarder with no crystal in it found a way to delay one polarisation behind the other without any crystal at all. The later arguments followed light with no direction of shaking and the pattern scattered sunlight writes across the sky.

All of those materials were doubly refracting by nature, or not at all. Glass, water and most plastics are isotropic: their atoms or molecules are arranged with no preferred direction, and light travels through them at one speed whatever its polarisation. In 1816 Brewster found that this is true only as long as nobody squeezes them. A block of glass under load becomes doubly refracting, with its two axes along the directions of the stress and its strength proportional to how unequal the stress is. Put the loaded glass between crossed polarisers and the stress appears as a pattern of light and dark. This essay is about that pattern, how to read it and what it cannot say.

Squeezing makes a crystal #

The reason a load makes glass birefringent is simple to state. Light is slowed in glass because its field polarises the electron clouds of the atoms it passes, and the clouds re-radiate. In unloaded glass the clouds respond the same way whatever the direction of the field. Squeeze the glass in one direction and the atoms are pushed closer along that direction and slightly apart across it; the clouds are distorted, and their response to a field along the squeeze differs from their response to a field across it. The material now has two indices of refraction, one for light polarised along each principal direction of stress, and to a very good approximation their difference is proportional to the difference between the two principal stresses:

$$\Delta n = C\,(\sigma_1 - \sigma_2).$$

That is the stress-optic law, and $C$ is the material’s stress-optic coefficient, measured in brewsters — $10^{-12}$ per pascal. Optical glass has about three; epoxy resin, about fifty-five; polycarbonate, about eighty. A plate of thickness $t$ delays the light polarised along one principal direction behind the light polarised along the other by a path difference $\Gamma = C t (\sigma_1 - \sigma_2)$, which is exactly what a wave plate does. A loaded plate is a wave plate whose retardation varies from point to point with the stress.

A discovery made twice, and a law Maxwell wrote down #

Brewster was not alone. Seebeck in Germany had noticed coloured patterns in glass that had been heated and cooled unevenly a few years earlier, and Brewster himself found the effect in glass, in gums and in animal jelly, squeezed, bent or warmed. He saw at once that it could make stress visible, and proposed using glass models to study the forces in the arches of bridges — the idea that became a whole branch of engineering a century later, when transparent plastics made models practical.

The law itself was written down by Maxwell, as a young man, in 1853. Studying the patterns in unannealed glass and in strained jelly, he showed that the optical effect depends only on the difference of the two principal stresses in the plane of the plate, and that the directions of the optical axes are the directions of those stresses. It was one of his first papers, and it used the light to work out the elastic equilibrium of the solid — a calculation he could check against the fringes, which is the earliest instance of the method being used as a measurement rather than a curiosity.

Fringes that are contours #

Place the plate between two crossed polarisers, with a quarter-wave plate after the first and another before the second, so that light enters the specimen circularly polarised and the second filter blocks it unless the specimen has changed it. Where the specimen is unstressed it changes nothing, and the field is dark. Where it is stressed it delays one component of the light behind the other, and the light that comes out is no longer circularly polarised and is partly passed by the second filter. The transmitted intensity is

$$I = I_0 \sin^2\!\left(\frac{\pi\Gamma}{\lambda}\right),$$

dark whenever the retardation is a whole number of wavelengths. The dark lines are the isochromatics, contours on which the principal stresses differ by the same amount, and successive lines differ by one fringe: a stress difference of $\lambda/Ct$.

The figure computes the fringes for the classic demonstration: a disc squeezed between two points on a diameter. The stress field inside such a disc was found in closed form by Hertz and Michell at the end of the nineteenth century, and it is drawn here exactly. The fringes form two families of loops crowding into the loading points, where the load is concentrated and the stress grows without limit, and long curves running down the sides of the disc. At the centre the fringe order is 3.61, and at the two free edges level with the centre it falls to zero, because there the disc is barely stressed. Counting fringes from a place known to be unstressed, a corner or a free edge, gives the order at any point, and the order times the fringe value gives the stress difference there.

The picture is useful exactly because it is a map of the whole field at once. A strain gauge measures one point; photoelasticity measures every point of a model in one photograph, and the places where fringes crowd together are the places where stress is concentrated — at holes, notches, re-entrant corners and the points where loads are applied — which are exactly the places a designer most needs to see.

Dark bands that move #

Without the quarter-wave plates, between crossed plane polarisers, a second kind of dark line appears. Where one of the principal stresses lies along the polariser’s axis, the light entering is polarised along a principal direction and travels through as a single wave with no delay between components; it emerges unchanged and is blocked. The transmitted intensity picks up a second factor, $\sin^2 2\alpha$, with $\alpha$ the angle between the principal stress and the polariser. These dark lines are the isoclinics, and unlike the isochromatics they depend on how the polarisers are turned. At zero degrees the isoclinics in the disc lie along its two diameters, where by symmetry the principal stresses are vertical and horizontal. Turn the polarisers together by 22.5 degrees and the isoclinics swing into curves running from the loading points to the rim; at 45 degrees they bow out further.

Recording the isoclinics at a series of angles maps the directions of the principal stresses everywhere, which the isochromatics alone do not. The two families together give the stress difference and its orientation; the circular polariscope, by removing the isoclinics, leaves the fringes clean for counting. The quarter-wave plates that do it are the device the retarder with no crystal in it built from total internal reflection, put here to the use for which they are most commonly made.

The tension that splits a disc #

What the fringes measure is the difference of the principal stresses, not the stresses themselves, and the distinction matters most at the disc’s centre. Along the horizontal diameter the exact solution says the stress across the load is a tension, $2P/\pi t D$ at the centre — 1.06 megapascals for this disc — and the stress along the load is a compression three times as large. The fringes see only their difference, 4.24 megapascals, order 3.61. To separate them takes one more piece of information: at a free edge one principal stress is zero, so the fringe order there gives the other directly; inside, a second measurement or a calculation is needed.

The tension is the surprising part, and it is not an artefact of the drawing. Squeezing a disc pulls its middle apart sideways, with a tension proportional to the load and uniform along most of the loaded diameter. A brittle material, strong in compression and weak in tension, fails by splitting along that diameter rather than by crushing. Engineers use exactly that to measure the tensile strength of concrete and rock, which are hard to grip and pull: they squeeze a cylinder across its diameter until it splits, and compute the strength from $2P/\pi t D$. The method is called the Brazilian test, after the engineer who proposed it for testing concrete in 1943, and the photoelastic disc shows why it works.

Why the fringes are coloured #

In light of one colour the fringes are black lines. In white light they are colours, and the reason is in the formula: each wavelength goes dark at its own multiples. Blue light is extinguished where the retardation is 450 nanometres, green at 550, red at 650. Near zero retardation all three are transmitted weakly and the field is grey, brightening to white as the retardation grows; near 550 nanometres green is removed and red and blue together make a vivid purple, the tint of passage, which marks the first order precisely; beyond it come blue, yellow and red bands of the second order, and then paler and paler ones as the colours drift out of step.

The sequence is the same as the colours of a soap film or an oil slick, which why two lamps never interfere traced to interference between two paths of different length, and for the same reason: two waves separated by a path difference, recombined. Here the two waves are the two polarisations, separated by the stress; in the film they are two reflections, separated by the film’s thickness. Mineralogists use the same chart, in a polarising microscope, to identify minerals by the colours their thin sections show, and in photoelasticity it lets an experienced eye read the fringe order to a tenth without counting from a dark point.

A measurement thickness does not change #

A model twice as thick carries the same load at half the stress, and the light crosses twice as much of it, so the retardation is the same. For any two-dimensional model loaded in its own plane the thickness cancels exactly, and the fringe order at the disc’s centre is $8CP/\pi D\lambda$: proportional to the load and the material’s coefficient, and to nothing else but the diameter and the wavelength. At 500 newtons a polycarbonate disc shows 3.6 fringes at its centre, an epoxy one 2.5 and a glass one barely an eighth of a fringe.

That is why photoelastic models are made of plastics with large coefficients, and why the stresses in glass are invisible to the eye in ordinary light but are routinely measured by instruments in glass factories. Glass that has cooled unevenly keeps stresses locked inside it, which make it fragile, and the manufacturing check for proper annealing is a polariscope. Toughened glass is the exception that proves the point: its surface is deliberately quenched into compression, and through polarising sunglasses the rear window of a car often shows a grid of dark and light patches — the pattern of the air jets that cooled it, frozen into the stress.

When the stress is unwanted, and when it is put there on purpose #

The same effect is a nuisance wherever light must keep its polarisation. A lens held too tightly in its mount, or a window bonded to a metal frame that expands differently, becomes slightly birefringent where it is squeezed, and blurs or depolarises the light passing through it; designers of polarisation-sensitive instruments and of the lenses that print computer chips specify mounts to keep the stress, and so the retardation, below a few nanometres. An optical fibre bent round a drum or squeezed by its cable becomes birefringent too, and a long fibre with randomly varying stress along its length spreads a pulse’s two polarisations apart by a random amount, the delay that is a random variable.

And the effect is used deliberately to cure that. A polarisation-maintaining fibre has two rods of a glass with a different thermal expansion embedded in its cladding on either side of the core. As the fibre cools from the drawing furnace the rods shrink more than the surrounding silica and squeeze the core along one direction, locking in a large, uniform birefringence. Light launched polarised along one of the stress axes stays polarised along it for kilometres, because the built-in difference in index is so much larger than anything the random bends can add. The fibre is a stressed photoelastic model drawn out to the length of a cable, and its job is exactly the one Brewster’s glass did in reverse: to make the stress control the light.

A century of models, and what replaced them #

From the 1930s to the 1970s photoelasticity was a standard tool of engineering design. Coker and Filon’s treatise of 1931 set out the method, and models of gear teeth, aircraft frames, turbine blades, dams and the joints of bridges were cut from celluloid, Bakelite and later epoxy, loaded, and photographed. Three-dimensional problems were handled by “stress freezing”: an epoxy model was loaded while warm, cooled under load so that the birefringence was locked in, and then sliced, each slice read like a two-dimensional model. The stress concentrations of most ordinary engineering shapes were first mapped this way.

Computation took over, because a finite-element calculation can be run on any shape without making a model, and gives the stresses themselves rather than their difference. Photoelasticity survives where its unique strength matters: in seeing a whole field at once in a real object, as in the inspection of glass and transparent plastic parts, and in materials too complicated to compute. The most striking modern use is in granular matter. A box of photoelastic discs under load shows the force passing through it not evenly but along branching chains of strongly loaded grains, with most of the grains between them carrying almost nothing — the force chains that the silo that does not weigh what it holds found carrying the weight of the grain to the walls. No calculation predicted their pattern before polarised light showed it.

What the pictures cannot show #

The fringes and stresses are computed for an ideal elastic disc loaded at two points, with a constant stress-optic coefficient and light of one wavelength. Real loads are spread over a small contact patch, which removes the infinite stress at the loading points and the crowd of fringes into them; real plastics creep under load, so their fringes drift over minutes, and their coefficients depend on temperature and on the wavelength. The coefficients quoted are typical values for each material. The colour figure draws three wavelengths, not the continuous spectrum the eye integrates, and the colours the eye sees also depend on the light source and the eye’s own sensitivity. And no figure here shows the residual stresses that a real moulded or machined specimen carries before it is loaded, which add their own fringes and must be subtracted.

Still open: measuring all three dimensions without slicing #

A two-dimensional photoelastic image integrates the birefringence along the light’s path, which is exact for a flat model loaded in its plane and an average for anything else. For a genuinely three-dimensional object the stress varies along the path, and reconstructing it from images taken in many directions — integrated photoelasticity, or photoelastic tomography — is an inverse problem with the same difficulty as other tomographies plus one more, because the birefringence’s axes rotate along the path and the effect on the light is not a simple sum. Methods exist for special cases, and full reconstruction of an arbitrary three-dimensional stress field from polarised light alone remains an active problem, of interest for glass and transparent polymer parts that cannot be cut open.

The habit worth carrying away is to ask what a pattern is a contour of. A loaded transparent solid becomes a wave plate whose retardation is $Ct(\sigma_1 - \sigma_2)$, so between crossed circular polarisers its dark fringes are contours of the stress difference, one fringe per λ/Ct, and between plane polarisers a second family marks the stress directions. A squeezed disc shows 3.6 fringes at its centre and a tension there that splits brittle discs, and the thickness of the model never enters.