Birefringence (double refraction) occurs in anisotropic crystals such as calcite, quartz, and mica, where the refractive index depends on both the direction of propagation and the polarisation of light. A uniaxial crystal has a single special direction called the optic axis; light polarised perpendicular to it travels as the ordinary ray (o-ray) with a constant index n₀, while light polarised in the plane containing the optic axis forms the extraordinary ray (e-ray) whose effective index n₁(θ) varies smoothly between n₀ and n₁ as the angle to the axis changes. This simulation lets you choose the optic axis orientation, the ordinary and extraordinary indices (defaults are calcite: n₀ = 1.658, n₁ = 1.486), and the wavelength of light, then watches both rays refract at different angles and accumulate a phase difference inside the crystal slab.
The wave-view mode reveals the accumulated phase retardance Δφ = 2π d Δn / λ between the two polarisation components. When Δφ = π/2 the crystal acts as a quarter-wave plate converting linear to circular polarisation; when Δφ = π it is a half-wave plate that rotates the polarisation plane. The interactive index ellipsoid diagram in the corner shows how n₁(θ) is read geometrically: it is the radius of the intersection of the ellipsoid with the plane perpendicular to the wave vector. Birefringence underlies LCD screens, optical isolators, waveplates, polarising microscopes, and gemological identification of minerals.
What is birefringence?
Birefringence is the property of certain transparent materials ā notably crystals ā to split an incident light beam into two beams with different speeds and therefore different refractive indices, depending on the polarisation direction. The two resulting beams are called the ordinary ray and the extraordinary ray.
What is the difference between the ordinary and extraordinary ray?
The ordinary ray (o-ray) is polarised perpendicular to the plane containing the optic axis and the wave direction; it obeys the standard Snell's law with a fixed index n₀. The extraordinary ray (e-ray) is polarised in that plane and experiences an effective index n₁(θ) that varies with the propagation angle θ relative to the optic axis, between n₀ and the principal extraordinary index n₁.
What is the optic axis of a crystal?
The optic axis is the unique direction in a uniaxial crystal along which both polarisation modes travel with the same speed (n₀). Light propagating along the optic axis sees no birefringence and is not split. Perpendicular to the optic axis the splitting is maximum, with the largest difference between n₀ and n₁.
The extraordinary ray index follows the index ellipsoid: 1/n₁(θ)² = cos²(θ)/n₀² + sin²(θ)/n₁², where θ is the angle between the wave vector and the optic axis. At θ = 0 it equals n₀ (no splitting); at θ = 90° it equals the principal extraordinary index n₁, giving the maximum birefringence Δn = |n₀ − n₁|.
A wave plate is a thin birefringent slab cut so that the optic axis lies in the crystal face. Linearly polarised light enters at 45° to the axis, splits into o- and e-components that travel at different speeds, and exits with a controlled phase difference. A quarter-wave plate (Δφ = π/2) converts linear to circular polarisation; a half-wave plate (Δφ = π) rotates the polarisation direction by 90°.
Calcite (CaCO₃) is a trigonal crystal with a highly anisotropic electron distribution. The carbonate groups lie in planes perpendicular to the optic axis, so the polarisability ā and hence the refractive index ā differs strongly between directions parallel and perpendicular to those planes. Its indices n₀ = 1.658 and n₁ = 1.486 give Δn ≈ 0.172, one of the largest values among naturally occurring minerals.
Both n₀ and n₁ vary with wavelength due to dispersion, and they do not vary at the same rate, so the birefringence Δn itself is wavelength-dependent. This means the retardance Δφ of a waveplate changes with colour, which is why white light passed through a birefringent crystal between crossed polarisers produces vivid interference colours ā a phenomenon exploited in polarising microscopy.
Birefringent components are essential in liquid-crystal displays (the LC layer is a switchable birefringent medium), optical isolators, Wollaston prisms (used to separate polarisations in laser systems), waveplates for quarter- and half-wave retardation, polarising microscopes for mineral and biological identification, and coherence-domain optical fibre sensors.
The index ellipsoid (optical indicatrix) is an ellipsoid whose semi-axes are the principal refractive indices of the crystal. For a uniaxial crystal the two axes perpendicular to the optic axis equal n₀, and the axis along the optic axis equals n₁. The refractive index for any direction of polarisation is found by cutting the ellipsoid with the plane perpendicular to the wave vector and reading the relevant semi-axis of the resulting ellipse.