Birefringence

Birefringence is the optical property of a material whose refractive index depends on the polarization direction of the light passing through it. Light polarized along two perpendicular directions travels through such a material at different speeds, so the two components build up a phase difference (the retardance), and an unpolarized beam that enters a birefringent crystal at an angle can be split into two beams. This splitting is called double refraction. Crystals such as calcite, crystalline quartz and sapphire are birefringent by nature, while glasses and polymers that are normally isotropic can become birefringent under mechanical stress or a strong electric field. Liquid crystals are strongly birefringent, and the effective birefringence of a nematic liquid crystal layer can be changed with an applied voltage.

In virtual reality (VR) and augmented reality (AR) hardware, birefringence is both a tool and a source of defects. The quarter-wave plates and multilayer reflective polarizers that fold the light path in pancake lenses are birefringent components, and liquid crystal birefringence is what liquid-crystal displays, liquid crystal on silicon (LCoS) microdisplays, liquid crystal lenses and geometric-phase optics use to control light. Unwanted stress birefringence in injection-molded plastic lenses disturbs the polarization states that a pancake lens depends on, which can lower image contrast and produce ghost images.

Reviewed 11 October 2026. Physics, history and VR/AR claims checked against the cited RP Photonics entries, LCoS and PBOE reviews, 3M SPIE paper, Meta patent US11226482B2, Kopin and Photonic Lattice pages, Half Dome 3 reports and paper metadata. About review dates.

How it works

Ordinary and extraordinary indices

In a uniaxial crystal such as calcite, quartz, sapphire or lithium niobate, the refractive index depends on the angle between the light's electric field and a single direction in the crystal called the optical axis. Light polarized perpendicular to the optical axis sees the ordinary index (n_o). Light polarized along the optical axis sees the extraordinary index (n_e). Light that travels along the optical axis sees n_o for every polarization, so it experiences no birefringence. A crystal is called positive uniaxial when n_e is higher than n_o. Biaxial crystals, such as mica, have three principal axes with different refractive indices, which makes the general case more complicated.

As a quantity, birefringence usually means the difference between the extraordinary and ordinary refractive indices (Δn = n_e - n_o) at a given wavelength. Its size varies widely. Calcite and yttrium vanadate have an index difference of the order of 0.17 to 0.2 in the visible spectrum, while crystalline quartz and magnesium fluoride have values of the order of 0.01. Birefringence also depends on temperature, so the retardance of a waveplate drifts as it warms or cools. Most liquid crystals have a positive birefringence between 0.05 and 0.45.

Retardance and waveplates

For an optical component, birefringence is usually specified as retardance, the difference in phase shift between the two polarization directions. For a layer of thickness d and birefringence Δn, the phase retardation at wavelength λ is 2πΔnd/λ. A waveplate (retarder plate) is a transparent plate with a chosen amount of birefringence and two perpendicular axes, a fast axis and a slow axis. In a quarter-wave plate the phase difference between the two linear polarizations is π/2, which corresponds to a path difference of λ/4; in a half-wave plate it is π. A quarter-wave plate turns linearly polarized light at 45 degrees to its axes into circularly polarized light, and turns circularly polarized light back into linear polarization. A half-wave plate keeps linear polarization linear but rotates its direction; light at 45 degrees to the axes is rotated by 90 degrees.

A waveplate reaches its design retardance only over a limited range of wavelengths and incidence angles. Thin "true zero-order" plates work over a wider range than thicker multiple-order plates, and achromatic waveplates combine materials with different dispersion (for example quartz and magnesium fluoride) to keep the retardance nearly constant over hundreds of nanometers. Common waveplate materials include quartz, calcite, magnesium fluoride, sapphire, mica and some birefringent polymers. Electro-optic crystals and liquid crystal cells can act as electrically controllable waveplates, also called active retarders.

Induced birefringence

Isotropic materials become birefringent when something breaks their symmetry. Mechanical stress is the most familiar case: a piece of acrylic placed between crossed polarizers shows colored patterns when it is stressed. Strong electric fields can induce birefringence in glasses, and in polymers the ordering of molecules during a process such as extrusion can leave the material birefringent. A pancake-lens patent granted to Facebook Technologies (now Meta Platforms Technologies) in 2022 describes how this happens in molded lens parts: plastics with high stress optical coefficients pick up stress during injection molding, and uneven cooling leaves extra stress in the thinner regions of a part. The resulting slow axis runs around the circumference, and the retardance tends to increase from the thicker middle of the part toward its thinner edge.

A related effect, circular birefringence, is a difference in refractive index between left- and right-handed circular polarization. A magnetic field can induce it; this is the Faraday effect.

History

The Danish scientist Erasmus Bartholin (1625-1698) described double refraction in crystals of Iceland spar (calcite) in his 1669 book Experimenta crystalli Islandici disdiaclastici quibus mira & insolita refractio detegitur. He saw that objects viewed through the crystal appeared doubled and that, when the crystal was rotated, one image moved in a circle around the other. He named the two rays solita and insolita and showed that both were produced by refraction. The explanation, that the crystal splits light into two plane-polarized beams, came later. After returning to The Hague in 1676, Christiaan Huygens spent two years there studying the double refraction Bartholin had found in Iceland spar, and his Traité de la lumière argued for a wave theory of light.

Stress-induced birefringence was reported by David Brewster in a letter to Joseph Banks published in the Philosophical Transactions of the Royal Society in 1816. Brewster compressed a piece of plate glass edgewise between two screws and found that it acted on polarized light across its whole breadth. When he bent a narrow strip of glass, the compressed inner edge and the stretched outer edge both produced colored fringes, and the two kinds of fringe matched those of different types of natural crystal.

The term Pancharatnam-Berry phase, used for the geometric-phase optics described below, combines the names of S. Pancharatnam, whose "Generalized theory of interference, and its applications" appeared in the Proceedings of the Indian Academy of Sciences in 1956, and Michael Berry, whose 1984 paper showed that a quantum system carried slowly around a circuit acquires a geometrical phase factor in addition to its dynamical phase. In 2000, Michael F. Weber and co-authors described multilayer mirrors built from polymers with large birefringence in Science. They showed that the index difference in the thickness direction of the film controls the Brewster's angle at the layer interfaces, so the mirrors can keep or even increase their reflectivity at higher angles of incidence. 3M later used polymeric, birefringent, multilayer reflective polarizers to build pancake lens eyepieces for head-mounted displays.

Applications in VR and AR

Pancake lenses

A pancake lens (also called a polarized catadioptric lens or folded optics) passes light through the same optical cavity three times, using polarization to decide when light is reflected and when it is transmitted. The Facebook Technologies patent, filed in 2018, describes the typical light path. Linearly polarized light from the display passes through a quarter-wave retarder oriented at about 45 degrees and becomes circularly polarized. Part of it passes a partial reflector that transmits and reflects roughly 50% each. A second quarter-wave retarder turns it back into linear polarization, and a reflective polarizer sends it back toward the partial reflector. On the return trip the light is reflected by the partial reflector, which reverses its circular handedness, so after the second retarder it is linearly polarized in the perpendicular direction and passes through the reflective polarizer to the eye. Because a half mirror is used to fold the path three times, the theoretical maximum optical efficiency of a conventional pancake lens is 25%.

Both kinds of polarization component in this path rely on birefringence. The same patent notes that quarter-wave plates may be made of birefringent materials such as quartz, stretched organic sheets or liquid crystal. A single quarter-wave plate gives exactly a quarter-wave of retardance only at its design wavelength, so in a color display blue light is over-retarded and red light under-retarded. The resulting elliptical polarization lets some light leak through the reflective polarizer on its first pass, which the patent calls "see-through ghosts". The patent proposes compound retarders made of three quarter-wave plates, arranged so that two act together as a half-wave retarder, to convert a broad visible spectrum to circular polarization. For the reflective polarizer, 3M engineers Timothy Wong, Zhisheng Yun, Gregg Ambur and Jo Etter described polymeric, birefringent, multilayer reflective polarizers in a 2017 SPIE paper. They reported polarized reflectivity of up to 95% across broad wavelengths and at incidence angles from normal to plus or minus 60 degrees, and noted that the films can be formed onto curved spherical, aspheric and freeform lens surfaces.

The 3M paper also demonstrated a dynamically focusing prototype head-mounted display. Its 70-degree design moves a plano-convex beamsplitter lens with a stepper motor to cover a focus range of -8 to +8 diopters, and a gaze-tracking system refocuses the eyepiece to reduce vergence-accommodation conflict. One set of test photographs used a 0.97-inch SXGA transmissive LCoS microdisplay made by Kopin Corporation. Researchers have also proposed replacing the half mirror with a nonreciprocal polarization rotator; in a 2024 feasibility experiment with a commercial Faraday rotator and reflective polarizers, Yuqiang Ding, Zhenyi Luo, Garimagai Borjigin and Shin-Tson Wu reported that the theoretically predicted 100% efficiency could be approximately reached by using two high-extinction-ratio reflective polarizers.

Lens material birefringence, ghosting and contrast

Any birefringence in the lens bodies inside the folded cavity changes the polarization state of light on each pass. The 3M paper states that its plano-convex singlet was designed to be injection molded with minimum birefringence, "which is important for maintaining a high degree of polarization through multiple passes in the folded path". It adds that when second-surface mirrors are used, birefringence in the substrates has a cumulative effect over the three transmissions, while first-surface mirrors minimize it at some cost to eye relief and other design parameters.

The same patent, granted in January 2022 with Gary Dean Sharp as inventor, describes the problem in molded parts. It states that plastics with high stress optical coefficients may give a low contrast, for example 3:1, which may be raised to 10:1 or even 100:1 by adding an azimuthal compensator. The compensator is a spatially patterned film of uniaxial birefringent material, which can be made from a reactive mesogen by non-uniform spin coating, that cancels the circumferential retardance of the molded lens. In an Optics Express paper published in 2024, Zhenyi Luo, Yuqiang Ding, Qian Yang and Shin-Tson Wu traced ghost images in pancake VR optics to stray light from multiple surface reflections and to imperfect polarization control inside the optical system.

Material birefringence has shaped lens choices. When Kopin announced its P95 all-plastic pancake optics on 15 June 2021, it stated that earlier pancake optics "needed at least one spherical glass lens to avoid image artifacts caused by birefringence" of the plastic, and that plastics "usually have large birefringence that disturbs the polarization state". Kopin said its two-element plastic design has "virtually no birefringence". According to Kopin, the P95 gives a 95-degree field of view with its 1.3-inch 2560 x 2560 OLED microdisplay, at 16 mm thickness and 15 g per lens set. Optical metrology makers sell instruments for this check. Photonic Lattice markets the VRG-100 as a VR lens evaluation system that measures the retardation and axis orientation of single lenses and of lenses with quarter-wave plates at 466 nm, 543 nm and 650 nm, and states that in pancake optics the retardation of a single lens must be kept as low as possible, and that it tends to increase near the injection gate.

Liquid crystal displays and LCoS

In liquid crystal devices, an applied voltage reorients the liquid crystal molecules. Because the material is birefringent, its effective refractive index changes, and so does the phase retardation of light passing through it. Electro-optic modes investigated for LCoS devices include twisted nematic, electrically controlled birefringence (ECB), vertically aligned nematic and optically compensated birefringence. In the twisted nematic mode, for example, polarized light leaves the unpowered cell with a changed, elliptical polarization. Amplitude-modulating LCoS devices, used mainly for video and image projection, rely on this polarization change to form an image.

Phase-only LCoS spatial light modulators use the ECB effect to change only the phase of light at each pixel. In a review in Light: Science & Applications, Zichen Zhang, Zheng You and Daping Chu explain that a reflective phase-only device needs twice the liquid crystal thickness of a reflective amplitude device to reach 2π of phase, which makes it about four times slower, since response time depends on the square of the thickness. The thickness can only be reduced by using a liquid crystal with higher birefringence. The authors list head-up displays and future holographic 3D displays among the applications of phase-only LCoS holography.

Geometric-phase optics and varifocal displays

Liquid-crystal Pancharatnam-Berry phase optical elements (PBOEs), also known as diffractive waveplates, geometric phase optics or geometric phase holograms, are flat films in which the orientation of the birefringent liquid crystal varies across the surface. Each point acts as a half-wave plate, so the layer thickness is chosen to satisfy Δnd = λ/2 at the design wavelength. The local orientation of the liquid crystal sets the phase given to the light, which allows lenses and gratings to be written into a thin layer.

A Pancharatnam-Berry lens has positive optical power for one circular polarization and the same negative power for the other, and it flips the handedness of the light that passes through it. Its power can therefore be switched either by a polarization rotator in front of it, such as a twisted nematic cell with a quarter-wave plate, or by applying voltage across the element itself, which reorients the liquid crystal and switches off its lens effect. Zhan and colleagues describe how such lenses can build multiplane displays that address vergence-accommodation conflict, and how reflective and transmissive polarization volume gratings can serve as input couplers for AR waveguide displays with nearly 100% first-order diffraction efficiency. If the half-wave condition is not met across the visible spectrum, light leaks into the zero order and appears as ghost images.

Facebook Reality Labs took a related polarization-switching approach in its Half Dome 3 varifocal prototype, one of two newer Half Dome versions that Michael Abrash discussed at Oculus Connect 6 in September 2019. Half Dome 3 replaced the moving mechanical parts of earlier varifocal prototypes with a stack of liquid crystal lenses made from alternating polarization-dependent lenses, whose focal length changes with the polarization state of the light, and switchable half-wave plates. Each additional pair doubles the number of focal planes; in the example Meta described, six liquid crystal lenses sweep through 64 focal planes.

See also

• Pancake lenses
• Liquid crystal on silicon
• Liquid crystal lens
• Liquid-crystal display
• Varifocal display
• Waveguide display
• Spatial light modulator
• Beam splitter
• Optical aberration
• Polarized 3D system