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Definition: the phenomenon of double refraction, or the polarization dependence of the refractive index in a medium
In the literature, the term birefringence occurs with two different meanings. In classical optics, it is normally considered to have the same meaning as double refraction, as explained below. In nonlinear optics and laser technology, however, birefringence is usually meant to be the property of some non-isotropic transparent material that the refractive index depends on the polarization direction (direction of the electric field). The latter property makes this material (then called birefringent) capable of exhibiting double refraction, when being hit by an unpolarized light beam.
Consequences of a Polarization-dependent Refractive Index
The polarization dependence of the refractive index can have a variety of effects:
- 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.
- If a linearly polarized laser beam propagates through a birefringent medium, with the polarization direction not being aligned with one of the birefringent axes, there are two polarization components with different wavenumbers. Therefore, the polarization state changes during propagation due to the change in relative phase of the two linearly polarized components. 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.
- The birefringence of nonlinear crystal materials allows for birefringent phase matching of nonlinear interactions.
Examples of Birefringence
- 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.
- Birefringent crystals are also used for making polarizers.
- Although optical fibers are in most cases not birefringent by nature, birefringence is frequently encountered in fiber optics: some birefringence can result from bending (which also causes bend losses) and from random perturbations. Also, there are polarization-maintaining fibers.
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.
The magnitude of birefringence can be specified in different ways:
- For bulk crystals, it is natural to consider the difference of refractive indices for the two polarization directions.
- 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 2π 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.
|||R. Ulrich et al., “Bending-induced birefringence in single-mode fibers”, Opt. Lett. 5 (6), 273 (1980)|
|||S. J. Garth, “Birefringence in bent single-mode fibers”, J. Lightwave Technol. 6, 445 (1988)|
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
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