Physics · Optics · Year 1

Crystal optics

Why does writing seen through Iceland spar appear double?

The situation

In a geology museum a transparent rhombic crystal — Iceland spar — lies on a book: look through it and every letter appears twice. Turn the crystal, and one image stays in place while the other circles round it. In a sugar factory laboratory polarised light is passed through a tube of sugar solution, and the amount of sugar is found from how far the plane of polarisation is turned. Engineers put a plastic model of a part between two polaroids and see from the coloured fringes where the stress builds up. How does a single ray split in two inside a crystal, and why does sugar “twist” light?

Topic

Crystal optics, double refraction. Double image through Iceland spar and rotation of polarisation in sugar solution

Key ideas

Double refraction
Quarter-wave plate
Optical activity

Formulas

Δ n = n_{e} - n_{o}, d = t tg ρ

Birefringence: in calcite $Δ n \approx - 0.172$ (589 nm), and the extraordinary ray is deflected by $ρ \approx 6.2 °$ ; $d$ is the separation of the images and $t$ the thickness of the crystal.

Γ = \frac{2 π | Δ n | d}{λ}, d_{λ / 4} = \frac{λ}{4 | Δ n |}

Wave plate: $Γ = 90 °$ — quarter-wave (circular polarisation), $Γ = 180 °$ — half-wave (rotates the plane).

I = I_{0} \sin^{2} 2 (α - β) \sin^{2} \frac{Γ}{2}, σ_{1} - σ_{2} = \frac{N f_{σ}}{t}

An anisotropic specimen between crossed polaroids; the photoelasticity law: $N$ is the isochromatic fringe order and $f_{σ}$ the material constant.

Δ n = λ B E^{2}

Kerr effect: in an electric field a liquid behaves like a uniaxial crystal; for nitrobenzene $B \approx 2.4 \cdot 10^{- 12}$ m/V².

φ = [α] c l, β = V B l

Optical activity (sucrose: $[α] \approx 66.5$ °·mL/(g·dm)) and the Faraday effect ( $V$ is the Verdet constant).

Theory

Anisotropy: uniaxial and biaxial crystals

In a crystal the speed of light depends on the direction in which $\vec{E}$ oscillates — the medium is anisotropic. Uniaxial crystals (calcite, quartz, ice) have one special direction, the optic axis, along which there is no double refraction. In Huygens' construction the wavefront of the ordinary wave is a sphere, and that of the extraordinary wave is an ellipsoid about the optic axis (in the scene, “Huygens wavefronts”). Biaxial crystals (mica, gypsum) have three principal refractive indices and two optic axes; the refractive indices are described by the optical indicatrix, an ellipsoid.

Ordinary and extraordinary rays

An unpolarised ray entering calcite splits in two. The ordinary (o) ray obeys Snell's law and is polarised perpendicular to the principal section. The extraordinary (e) ray is deflected by $ρ \approx 6.2 °$ even at normal incidence (in a natural rhomb the optic axis makes 44.6° with the surface normal) and is polarised in the principal section. This produces a double image; a polaroid extinguishes them in turn (the brightness of each follows Malus's law: $\cos^{2}$ and $\sin^{2}$ ), and when the crystal is turned the e-image circles round the o-image. In quartz $Δ n$ is 19 times smaller — the double image is hardly noticeable. The Nicol prism works on this phenomenon: the o-ray is totally internally reflected at the cemented layer and goes off to the side, while the e-ray passes through, polarised.

λ/4 and λ/2 plates: circular and elliptical polarisation

In a plate cut parallel to the optic axis the o and e waves travel in the same direction but at different speeds, and acquire a phase difference $Γ = 2 π | Δ n | d / λ$ . Linearly polarised light passing through a λ/4 plate at 45° becomes circularly polarised, and at any other angle elliptically polarised; a λ/2 plate rotates the plane of polarisation by $2 θ$ . A test: a rotating analyser does not change the brightness of circular light, changes it partly for elliptical light and takes it down to zero for linear light. To distinguish circular light from unpolarised light another λ/4 is put in front: circular light turns into linear. For 589 nm a calcite λ/4 plate would be only ≈ 0.86 µm thick — so in practice quartz or mica is used. A ready-made plate works correctly only for its own wavelength — test it by choosing a different laser.

Interference of polarised light and photoelasticity

Two perpendicularly polarised waves leaving an anisotropic plate between crossed polaroids are projected onto one plane in the analyser and interfere: $I = I_{0} \sin^{2} 2 (α - β) \sin^{2} (Γ / 2)$ . In white light each colour acquires its own phase — a coloured picture (chromatic polarisation). Transparent plastic or glass under stress becomes birefringent (D. Brewster, 1816): $Δ n \propto σ_{1} - σ_{2}$ . Fringes of one colour are isochromatic fringes (lines of equal stress difference), and the black isoclinic fringes are where a principal stress is parallel to the polaroid axis; in a circular polariscope the isoclinic fringes disappear. At the centre of a compressed disc $N = 8 P / (π D f_{σ})$ — this formula is used to calibrate the material. The colours in the protractor show stress left over from moulding (a rough model in the scene).

Electro-optics and magneto-optics

The Kerr effect (J. Kerr, 1875): in an electric field polar molecules line up, and the liquid behaves like a uniaxial crystal: $Δ n = λ B E^{2}$ . A Kerr cell between crossed polaroids is a very fast “shutter”: when a voltage is applied, light passes (within nanoseconds). Crystals also show the Pockels effect, which depends linearly on the field — optical communication modulators work this way. The Faraday effect (1845): a magnetic field along the beam rotates the plane of polarisation by $β = V B l$ . The direction of rotation depends not on the direction of the beam but on $\vec{B}$ : a reflected beam is rotated the same way again and the angle doubles — optical isolators are based on this. Radio astronomers use Faraday rotation to measure the magnetic fields of the universe.

Optical activity and saccharimetry

Some substances rotate the plane of polarisation by themselves: quartz along the optic axis (≈ 21.7°/mm at 589 nm), and sugar solutions. The reason is the mirror asymmetry (chirality) of the molecule or crystal lattice: right- and left-circular waves travel at different speeds. In solution $φ = [α] c l$ : sucrose +66.5, glucose +52.7, fructose −92.4 °·mL/(g·dm) (589 nm, 20 °C). The rotation depends on wavelength (roughly $1 / λ^{2}$ ), so in white light the colours change as the analyser is turned. In a half-shade saccharimeter it is much more accurate for the eye to judge the equality of two half-fields than to find the darkest point. The model: the polaroids are ideal; the Verdet constant of water is 3.8 rad/(T·m).

Did you know?

The Danish scientist Erasmus Bartholin described the double image in calcite crystals brought from Iceland in 1669 — this is how birefringence was discovered. This transparent variety of calcite is called “Iceland spar” because it is common in Iceland.
In 1845 Michael Faraday discovered that a magnetic field rotates the plane of polarisation of light in glass — the first experimental evidence of a link between light and electromagnetism.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11
Scattering of light — Year 1
Interferometers and LIGO — Year 1
Resolving power — Year 1
Nonlinear optics: the green laser — Year 2