Birefringence

Birefringence is the property of some transparent optical materials that the refractive index depends on the polarization direction – which is defined as the 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.

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

Intrinsic and Induced Birefringence

Birefringence can occur only in optically non-isotropic materials. Often, it results from non-cubic lattice structures of optical crystals, i.e., an intrinsic anisotropy.

In other cases, birefringence can be induced in originally isotropic optical materials (e.g. crystals with cubic structure, glasses or polymers) can become anisotropic due to the application of some external influence which breaks the symmetry:

• In some cases, mechanical stress has that effect. That can easily be observed with 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.
• In other cases, application of a strong electric field has similar effects, e.g. in glasses. The temporary application of such a field can even cause a frozen-in polarization, which means that the induced birefringence remains even after removing the external field.

In optical fibers, birefringence can be caused by an elliptical shape of the fiber core, by other asymmetries of the fiber design (particularly for photonic crystal fibers), or by mechanical stress (e.g. caused by bending). In the case of polymers (plastics), birefringence can result from the ordering of molecules which is caused by an extrusion processes, for example.

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, β-BaB₂O₄ = BBO). 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:

• If the electric field has the direction of the optical axis, one obtains the extraordinary index <$n_\textrm{e}$>. This is possible only if the propagation direction (more precisely, the direction of the <$k$> vector) is perpendicular to the optical axis. For the other polarization direction, one then obtains the ordinary index <$n_\textrm{o}$>.
• For propagation along the optical axis, the electric field can only be perpendicular to that axis, so that one obtains the ordinary index for any polarization direction. In that situation, no birefringence is experienced.
• For an arbitrary angle <$\theta$> between propagation direction and optical axis, one can find two linear polarization directions exhibiting different refractive indices. The first one is perpendicular to the <$k$> vector and the optical axis; here, we have the ordinary index <$n_\textrm{o}$>, and such a wave is called an ordinary wave. The other polarization direction is perpendicular to that and to the <$k$> vector. The latter has a refractive index which is generally not the extraordinary index <$n_\textrm{e}$>, but a rather a mixture of <$n_\textrm{e}$> and <$n_\textrm{o}$>. This can be calculated with the following equation:

The equation shows that for <$\theta \rightarrow 0$> (i.e., propagation along the optical axis) we obtain <$n \rightarrow n_\textrm{o}$>, and the observed birefringence vanishes: one obtains <$n_\textrm{o}$> 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), 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:

Double Refraction

When a beam is refracted at the surface of a birefringent crystal, the refraction angle depends on the polarization direction. An unpolarized light beam can then be split into two linearly polarized beams when hitting surfaces of the material with non-normal incidence (double refraction). When some object, which is illuminated with unpolarized light, is viewed through a birefringent crystal (e.g. made of calcite), two images occur which are slightly displaced.

Changes of Polarization States

If a linearly polarized laser beam propagates through a birefringent medium, there are generally two polarization components with different wavenumbers. Therefore, the optical phases of the two linear polarization components evolve differently, and consequently the resulting polarization state (from the superposition of the two components) changes during propagation. This effect can be applied, for example, in birefringent tuners because it is wavelength-dependent (even if the difference in refractive indices is not wavelength-dependent). It can also be power-dependent (→ nonlinear polarization rotation) through self- and cross-phase modulation, e.g. in an optical fiber, and this effect is sometimes used for passive mode locking of fiber lasers.

Similarly, the polarization state of a laser beam in a laser crystal with thermally induced birefringence is distorted. The kind of distortion depends on the position, since the birefringent axis has a varying (e.g. always radial) orientation. This effect (combined with a polarizing optical element in the laser resonator) is the origin of depolarization loss.

Birefringent Phase Matching

The birefringence of nonlinear crystal materials allows for birefringent phase matching of nonlinear interactions. Essentially, this means that birefringence compensates the wavelength dependence of the refractive index. This is the most common method of phase matching for various types of nonlinear frequency conversion such as frequency doubling and optical parametric oscillation.

Spatial Walk-off

For extraordinary waves, where the refractive index depends on the angular orientation, there is a spatial walk-off: the direction of power propagation is slightly tilted against that of the <$k$> vector. This effect can severely limit the efficiency of nonlinear frequency conversion processes, particularly when using tightly focused laser beams.

Different Group Velocities

Not only the refractive index, but also the group index becomes polarization-dependent. This matters e.g. for the propagation of ultrashort pulses: components with different polarization propagate with different group velocities. If a femtosecond pulse propagates through a piece of birefringent material, the polarization components experience different group delays (time delays); the pulse may effectively be split into two pulses. Than can be exploited in divided-pulse amplification, for example.

For a numerical example, we again consider α-quartz and assume a wavelength of 1060 nm. The group indices for ordinary and extraordinary polarization, respectively, are 1.549353 and 1.558489. Within a crystal length of <$L$> = 10 mm, this causes a differential time delay of <$L (n_g - n_o) / c$> = 305 fs.

Examples of Birefringence in Laser Technology and Nonlinear Optics

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

• Some laser crystals (e.g. vanadate or tungstate crystals) are naturally birefringent. This is often helpful for obtaining a linearly polarized output without depolarization loss.
• All nonlinear crystals for nonlinear frequency conversion are birefringent. This is because they can have their <$\chi^{(3)}$> nonlinearity only by being non-isotropic, and that also causes birefringence.
• Birefringent crystals are also used for making polarizers of various types.
• Straight optical fibers are often nominally symmetric, but nevertheless exhibit some small degree of random birefringence because of tiny deviations from perfect symmetry – for example due to bending, other mechanical stress or small microscopic irregularities. There are polarization-maintaining fibers, which have and intentionally introduced stronger birefringence.

Quantifying Birefringence

The magnitude of birefringence can be specified in different ways:

• For an optical component with some birefringence, one can specify the retardance, which is the difference in phase shifts for the two polarization directions.
• For bulk optical materials, it is also common to consider the difference of refractive indices for the two polarization directions. The larger that difference, the larger the obtained retardance per millimeter of propagation length.
• For optical fibers and other waveguides, it is more appropriate to consider the difference of effective refractive indices. This is directly related to the difference in imaginary values of the propagation constants.
• Alternatively, one may specify the polarization beat length, which is <$2\pi$> divided by the difference of the propagation constants. If waves with different polarization directions propagate together in the waveguide, their phase relation is restored after integer multiples of the propagation beat length.

Circular Birefringence

Beyond the ordinary kind of (linear) birefringence, there is also the phenomenon of circular birefringence. Here, the refractive index difference between left-hand and right-hand circular polarization. Circular birefringence can be induced by a magnetic field; this is called the Faraday effect. See the article on optical activity for details.

More to Learn

Encyclopedia articles:

• retardance
• refraction
• polarization of light
• polarization beat length
• birefringent tuners

Blog articles:

• The Photonics Spotlight 2007-05-26