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Definition: the polarization dependence of the refractive index of a medium

German: Doppelbrechung

Category: general optics

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Birefringence is the property of optically non-isotropic transparent materials that the refractive index depends on the polarization direction (direction of the electric field). For example, it is observed for crystalline quartz, calcite, sapphire and ruby, also in nonlinear crystal materials like LiNbO3, LBO and KTP. Figure 1 shows that for α-quartz.

Often, birefringence results from non-cubic crystal structures. In other cases, originally isotropic materials (e.g. crystals with cubic structure and glasses) can become anisotropic due to the application of mechanical stress, or sometimes by application of a strong electric field; both can break their original symmetry. In optical fibers, birefringence can result from an elliptical shape of the fiber core, from other asymmetries of the fiber design (particularly for photonic crystal fibers), or from mechanical stress (e.g. caused by by bending). In case of polymers (plastics), birefringence can result from the ordering of molecules which is caused by an extrusion processes, for example.

The term birefringence is sometimes also used as a quantity (see below), usually defined as the difference between extraordinary and ordinary refractive index at some optical wavelength.

refractive indices of quartz
Figure 1: Refractive indices of α-quartz vs. wavelength. For this positive uniaxial material, the extraordinary index is higher.

Uniaxial and Biaxial Optical Materials

Depending on the symmetry of the crystal structure, a crystalline optical material can be uniaxial or biaxial.

The simpler case is that of uniaxial crystals (examples: calcite, quartz, sapphire, LiNbO3). Those have a so-called optical axis, and the refractive index for given wavelength depends on the relative orientation of electric field director and optical axis:

extraordinary index

The equation shows that for θ → 0 (i.e., propagation along the optical axis) we obtain n → no, and the observed birefringence vanishes: one obtains no for any polarization direction.

One distinguishes positive and negative uniaxial crystals; in the former case, the extraordinary index is higher than the ordinary index.

For biaxial crystals (examples: mica, CaTiO3 = perovskite, LiB3O5 = LBO, β-BaB2O4 = BBO), such calculations are substantially more complicated, at least for arbitrary propagation directions. There are three mutually orthogonal principal axes associated with different refractive indices. Frequently, however, one deals with cases where the propagation direction is in one of the planes spanned by the principal axes of index ellipsoid, and in such cases the calculation is again reasonably simple. This is usually the case in calculations for phase matching of nonlinear frequency conversion processes.

For optical fibers and other waveguides, the distinction between uniaxial and biaxial does not apply, since the propagation direction is essentially determined by the waveguide.

Consequences of a Polarization-dependent Refractive Index

The polarization dependence of the refractive index can have a variety of effects, some of which are highly important in nonlinear optics and laser technology:

Examples of Birefringence

In laser technology and nonlinear optics, the phenomenon of birefringence occurs mainly in the context of non-isotropic crystals:

Even in a naturally isotropic medium, birefringence can be induced e.g. by inhomogeneous mechanical stress. This can be observed e.g. by placing a piece of acrylic between two crossed polarizers: when stress is applied to the acrylic, one observes colored patterns resulting from the wavelength-dependent effect of stress-induced birefringence. Similar effects occur in bent optical fibers, and also due to thermal effects in laser crystals, which can lead to depolarization loss.

Straight optical fibers usually exhibit only a small degree of random birefringence, which can however scramble the polarization state of guided light over some propagation distance, e.g. 1 m. There are polarization-maintaining fibers, where a strong artificial birefringence can be used for suppressing such effects.

Quantifying Birefringence

The magnitude of birefringence can be specified in different ways:


[1]R. Ulrich et al., “Bending-induced birefringence in single-mode fibers”, Opt. Lett. 5 (6), 273 (1980)
[2]S. J. Garth, “Birefringence in bent single-mode fibers”, J. Lightwave Technol. 6 (3), 445 (1988)

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See also: refraction, polarization beat length, birefringent tuners, birefringent phase matching, polarization-maintaining fibers, polarization of laser emission, spatial walk-off, fiber polarization controllers, Lyot filters, Spotlight article 2007-05-26
and other articles in the category general optics

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