Illustrated Physics
physics, drawn
Concept

Birefringence — where it appears

The property of a material whose refractive index depends on the polarisation of the light passing through it. It splits a beam into two that travel at different speeds, which is what a wave plate uses to delay one polarisation with respect to the other — and what a Fresnel rhomb achieves without any such material at all.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

The crystal that answers twice

Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

optics · Polarisation Past the critical angle, all that is left of a reflection is its phase. The phase each polarisation acquires on total internal reflection at an index ratio of 1.518, against angle of incidence, together with the difference between them. Below the critical angle of 41.19° there is a transmitted beam and the reflection coefficients are real; above it the transmitted wavenumber is imaginary, the coefficients have modulus exactly one — every photon comes back — and the only thing that distinguishes one angle from another is the phase. The two polarisations acquire different phases, and their difference peaks at 46.533° at an incidence of 51.05°, which the closed form puts at the same place. That difference is a retardation: a wave plate made out of an angle, with no birefringent material anywhere in it, and — because the expression contains only the index ratio — one that barely changes with colour.

The retarder with no crystal in it

A wave plate turns linear polarisation into circular by making one component travel a little further than the other, which requires a birefringent crystal cut to a thickness and works properly at one wavelength. Total internal reflection does the same job with a phase that comes from the geometry instead — and because a refractive index barely changes across the visible where a wavelength changes by a factor of two, the same block of ordinary glass is a quarter-wave plate for every colour at once.

optics · Polarisation A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

optics · Dispersion How far the vacuum is from being a nonlinear medium. The size of the vacuum's departure from linearity, as a fraction, against the electric field it is subjected to — thirteen decades of field and twenty-eight of correction, both logarithmic. The scale is the Schwinger field, computed here from the electron's mass and the fundamental constants as 1.32e+18 volts per metre: the field at which a pair gains its own rest energy over a Compton wavelength, and therefore the field at which the vacuum stops being a passive backdrop. The marks are the strongest fields that exist. a laboratory magnet, 10 T is 2.3e-9 of it; a hydrogen atom's own field is 3.9e-7 of it; a 10²² W/cm² laser focus is 2.1e-4 of it; the Schwinger field is 1.0e+0 of it; a magnetar, 10¹¹ T is 2.3e+1 of it. So a laboratory is twenty-eight decades from making the effect large, and a magnetar's field is above the critical one — which is why the only places the vacuum's nonlinearity has been seen are the two where the fields are not human: the ultraperipheral collision of two heavy nuclei, and the surface of a neutron star.

The one medium that was supposed to add exactly

Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

waves · Superposition A squeezed disc between crossed polarisers. The dark fringes seen in a 50 mm disc of polycarbonate, 6 mm thick, squeezed across a vertical diameter by 500 N and viewed between crossed circular polarisers in light of 550 nm: each dark line is a contour on which the principal stresses differ by a whole number of fringe values, from the stress-optic law with a coefficient of 78 Brewsters. At the centre the fringe order is 3.61; it climbs steeply towards the two loading points, where the fringes crowd together and the stress grows without limit, and falls to zero at the free edges level with the centre. The pattern is the exact stress field of an elastic disc under diametral load; every fringe can be counted from a dark point, and the count times the fringe value is the difference of the principal stresses at that place.

The stress that polarised light can see

Squeeze a disc of clear plastic and look at it between two crossed polarising filters, and it fills with dark lines and coloured bands that crowd towards the points where it is pressed. Each line is a contour of stress. Glass, plastic and most transparent solids become doubly refracting when they are loaded, in proportion to the difference between their largest and smallest stresses, and polarised light turns that difference into a pattern that can be counted. David Brewster found it in 1816, the same Brewster whose angle polarises reflected light, and for a century the way to learn the stresses in a bridge or a dam was to cut its shape from plastic, load it and photograph the fringes.

optics · Polarisation Light that turns with the molecules it passes. Through a layer of liquid crystal whose molecules turn steadily through 90° from the entrance face to the exit (dashed line), the direction of the light's polarisation against depth, for linearly polarised light entering along the molecules, in a thick cell and a thin one; u = 2dΔn/λ measures the layer's birefringence against the wavelength. In the thick cell the polarisation turns with the molecules, never more than a few degrees behind, and leaves rotated by 90°: 100.0 per cent of the light passes an analyser set along the exit molecules. In the thin cell the light barely notices the twist; its polarisation lags further and further behind and leaves turned only part of the way, with 31 per cent passing. Mauguin saw the first case in 1911: a birefringent medium that twists slowly enough drags the light's polarisation round with it.

The twist that carries light round a corner

Between two crossed polarisers, a sandwich of liquid crystal a few micrometres thick turns black to white. Its long molecules lie flat, but turned a quarter-turn from one face of the layer to the other, and polarised light entering along the molecules at one face comes out along them at the other — rotated by ninety degrees and able to pass the second polariser. Charles Mauguin found in 1911 that a slowly twisting birefringent layer drags light's polarisation round with it. Sixty years later the effect became the liquid-crystal display. The layers in those displays, though, are too thin to drag anything: the light inside them wanders off and comes back, at exactly the right thickness.

optics · Polarisation

Named alongside it

The objects these essays reach for when they reach for this one.

PolarisationWave plateCircular polarisationRefractive indexRetardationSuperpositionAchromaticAdiabatic followingAnisotropyComplex amplitudeCross-sectionCrystal

All concepts