A ray entering a uniaxial crystal splits into ordinary (o) and extraordinary (e) rays. Watch the double image and crossed-polariser colours.
Anisotropic crystals such as calcite have a refractive index that depends on the polarisation direction of light. An incoming ray is split into an ordinary ray (o), which obeys Snell's law, and an extraordinary ray (e), which refracts differently. The two rays travel at different speeds, build up a phase difference, and recombine between crossed polarisers to produce vivid interference colours.
Γ = 2π·(n_e − n_o)·d / λ
n_e(θ) = 1 / √(cos²θ/n_o² + sin²θ/n_e²)
I = I₀·sin²(2α)·sin²(Γ/2) (crossed polarisers)
The Vikings may have used a calcite "sunstone" to locate the sun on overcast days: rotating the crystal until the two refracted images matched in brightness pointed toward the hidden sun, thanks to the polarisation of the sky.
What is birefringence?
Birefringence is the optical property of a material having a refractive index that depends on the polarisation and direction of light. A single incident ray is split into two — the ordinary and extraordinary rays — that travel at different speeds.
What is the difference between the ordinary and extraordinary rays?
The ordinary ray (o-ray) obeys Snell's law and sees a fixed refractive index n_o regardless of direction. The extraordinary ray (e-ray) sees an index n_e that varies with direction and generally does not obey Snell's law, so it bends differently and is offset.
Why does calcite show a double image?
Calcite is strongly birefringent. The ordinary and extraordinary rays refract at different angles inside the crystal and emerge as two parallel beams, so any object viewed through it appears doubled.
The two rays accumulate a phase difference (retardation) that depends on wavelength. Between crossed polarisers, some wavelengths interfere constructively and others destructively, so white light is split into bright interference colours.
Retardation is the phase difference between the ordinary and extraordinary rays after passing through the crystal, given by Γ = 2π(n_e − n_o)d / λ, where d is thickness and λ the wavelength.
A uniaxial crystal has a single optic axis along which both polarisations travel at the same speed. Calcite and quartz are uniaxial. Light travelling along any other direction splits into ordinary and extraordinary rays.
Thicker crystals produce larger retardation, so the interference colours cycle through higher orders. Very thin crystals show grey or first-order colours, while thicker ones show repeating pastel sequences.
Rotating the crystal changes the angle between its optic axis and the polariser. Brightness is maximum at 45° and drops to zero (extinction) every 90°, when the axis aligns with a polariser.
Yes. It is the basis of liquid-crystal displays (LCDs), wave plates, polarising microscopes for mineralogy, stress analysis in transparent materials, and optical isolators.
If n_e > n_o the crystal is optically positive (e.g. quartz). If n_e < n_o it is optically negative (e.g. calcite). The sign affects which ray is faster and the appearance of interference figures.
This simulation models the phenomenon of birefringence, in which an anisotropic crystal splits a single incident light ray into two beams — the ordinary (o) ray and the extraordinary (e) ray — that travel at different speeds and refract at different angles. The underlying physics is that the refractive index of the crystal depends on both the polarisation direction of the light and the angle between the ray and the crystal’s optic axis, leading to measurable separation of the two beams and an accumulation of optical retardation. Users can adjust the crystal material, its thickness, the optic-axis orientation, and the angle of incidence to observe how each parameter changes the separation of the rays, the retardation, and the vivid interference colours produced between crossed polarisers.
Birefringence was first described systematically in calcite (Iceland spar) by Erasmus Bartholin in 1669 and has since become central to optics, mineralogy, and photonics. It underlies the operation of liquid-crystal displays, wave plates, polarising microscopes used to identify minerals, photoelastic stress analysers, and optical isolators in fibre-optic networks.
Birefringence is an optical property of anisotropic materials in which the refractive index differs depending on the polarisation direction of the light passing through. When an unpolarised ray enters such a material it is split into two rays — the ordinary and extraordinary — that travel at different speeds. The difference between the two principal refractive indices, Δn = ne − no, is called the birefringence of the material.
Use the Crystal dropdown to select calcite, quartz, rutile, or a custom material. The Thickness slider sets the crystal depth in micrometres and directly controls the retardation. The Optic-axis angle slider (or click-drag on the canvas) rotates the optic axis; set it to 45° for maximum intensity between crossed polarisers. The Incidence angle slider tilts the incoming ray, while the ne (custom) slider lets you explore arbitrary extraordinary indices. The colour swatch in the top-right corner shows the interference colour expected between crossed polarisers for the current settings.
Calcite has one of the largest birefringences of any natural mineral (Δn ≈ −0.172). The ordinary and extraordinary rays refract at distinctly different angles inside the crystal and emerge as two parallel but laterally offset beams. Any object viewed through the crystal therefore appears doubled. The separation between the two images grows with crystal thickness and is proportional to |ne − no|, so thicker or more strongly birefringent crystals produce a wider double image.
Optical retardation Γ is the phase difference accumulated between the ordinary and extraordinary rays after travelling through a crystal of thickness d: Γ = 2π(ne − no)d / λ, where λ is the wavelength of light. Equivalently, the optical path difference (OPD) is simply (ne − no)×d, expressed in nanometres. When this value equals half a wavelength (λ/2) the two recombined beams interfere destructively for that colour between crossed polarisers; when it equals a full wavelength (λ) the interference is constructive. The colourful Michel-Lévy chart used by geologists maps OPD versus crystal thickness to identify minerals by their interference colour.
Liquid-crystal displays rely on the electrically controllable birefringence of liquid-crystal molecules. In the off state the molecules are twisted, rotating the polarisation of the backlight by 90° so it passes through a crossed analyser (bright pixel). When a voltage is applied the molecules align with the electric field, birefringence is reduced, polarisation rotation stops, and the analyser blocks the light (dark pixel). Colour is added by RGB sub-pixel filters. Every pixel in a TFT-LCD is essentially a voltage-controlled retardation plate exploiting precisely the physics demonstrated in this simulation.
Yes. Visible ray separation (double image) requires that the crystal be thick enough and Δn large enough to shift the beams by at least a fraction of a millimetre at the exit face. Many birefringent samples — such as optical fibres or thin polymer films — have very small Δn or are very thin, so the two rays emerge almost coincidentally and no double image is seen. Retardation and interference colours between polarisers can still be detected even for sub-nanometre path differences, making polarimetry far more sensitive than direct image separation.
Erasmus Bartholin, a Danish scientist, first published a systematic account of double refraction in calcite (then called Iceland spar) in 1669. Christiaan Huygens explained it in 1690 using his wave theory of light, introducing the concept of the ordinary and extraordinary wave surfaces. Augustin-Jean Fresnel later gave a complete mathematical treatment in the early 19th century using the theory of transverse waves, correctly predicting both the ray directions and the polarisation states of the two refracted beams.
Birefringence is closely related to optical activity (circular birefringence), in which the refractive indices for left- and right-handed circularly polarised light differ, causing rotation of the polarisation plane. Photoelasticity is stress-induced birefringence used in engineering to map stress distributions in transparent models. Form birefringence arises in nanostructured composites whose feature size is smaller than the wavelength. In fibre optics, unwanted birefringence from core ellipticity or mechanical stress causes polarisation-mode dispersion, limiting data rates. All of these can be explored with the polarimetry concepts shown in this simulation.
A wave plate (retardation plate) is a precisely cut birefringent crystal whose thickness is chosen so that the OPD equals exactly λ/4 (quarter-wave plate) or λ/2 (half-wave plate) for a target wavelength. A quarter-wave plate converts linear to circular polarisation; a half-wave plate rotates the polarisation plane. These components are essential in laser systems, optical coherence tomography, ellipsometry, and quantum optics. An optical isolator combines a Faraday rotator (magneto-optic birefringence) with wave plates to allow light to pass in only one direction, protecting laser sources from back-reflections in fibre-optic networks.
Active research areas include giant birefringence in van der Waals two-dimensional materials such as black phosphorus (Δn exceeding 1.5 in the mid-infrared), which could enable atomically thin wave plates. Birefringent metasurfaces composed of subwavelength antenna arrays offer engineered retardation at arbitrary spatial positions, enabling flat-optic lenses and holograms. In quantum information science, controlled birefringence is used to manipulate polarisation-encoded qubits. Non-linear birefringence and phase-matched second-harmonic generation in crystals like lithium niobate (LiNbO3) remain central to optical frequency conversion and quantum light sources.