定义：
双折射现象或者介质折射率与偏振有关。

文献中，双折射通常包含两种不同的含义。经典光学中，就是下面所说的双折射（double refraction）。
而在非线性光学和激光器技术中，双折射则是一些非各向同性透明介质的折射率依赖于偏振方向（即电场方向）的性质。后者的性质时非偏振光束入射到该材料上时产生双折射。 

折射率依赖于偏振态的结果 
折射率依赖于偏振态会产生下面的一些效应： 

• 当光束在双折射晶体表面发生折射是，折射角与偏振方向有关。这样非偏振光束在非垂直入射到材料中的情况下分为两个线性偏振的光（双折射）。当非偏振光射向一个物体，如果采用双折射晶体看该物体，会出现两个像。 
• 当线偏振激光光束在双折射晶体中传输时，如果偏振方向与双折射轴不重合，这时会包含两个方向具有不同波数的偏振部分。因此，在传输过程中，由于两偏振分量之间存在相对相位变化，于是偏振状态发生变化。
• 这一效应可应用于双折射调谐器中，因为它是与波长相关的（尽管折射率差与波长无关）。该效应通过自相位调制和交叉相位调制而与功率相关（参阅非线性偏振态旋转），有时用于光纤激光器中的被动锁模。 
• 类似的，激光光束在存在热效应诱导的双折射效应的激光器晶体中传输时，偏振态也发生变化。这一变化与位置有关，因为双折射轴方向是变化的（例如，通常是轴向变化）。这一变化（与激光器谐振腔中的偏振光元件结合）是去极化损耗的来源。 
• 非线性晶体材料的双折射可以实现非线性作用时的双折射相位匹配。 

双折射举例 
在激光器技术和非线性光学中，双折射现象通常发生在非各向同性晶体中： 

• 一些激光器晶体（例如，钒酸盐晶体和钨酸盐晶体）本身就具有双折射。这在需要无去极化损耗的线偏振输出时非常有用。 
• 所有用于非线性频率转换的非线性晶体都存在双折射。 
• 双折射晶体通常用来制作偏振器。 
• 尽管光纤本身不具有双折射，光纤光学中常常遇到双折射效应：有时双折射来自于光纤弯曲（引起弯曲损耗）和随机扰动。并且还存在保偏光纤。 

即使是各向同性介质，也会由于存在不均匀的机械应力而产生双折射。这可以在两个交叉偏振器间放置一块有机玻璃观察：当施加应力到有机玻璃上，可以看到由于应力诱导的与波长相关的双折射效应而产生的彩色图像。
弯曲光纤中也存在类似的效应，由于激光器晶体中的热效应，会产生去极化损耗。 
直光纤只有很小的随机双折射，即使这样其中的光传输一段距离后偏振状态也会发生变化。存在保偏光纤，是利用了很强的双折射来抑制这些效应。 

定量描述双折射 
可以采用下列方法定量描述双折射的大小： 

• 对于晶体，可以考虑量偏振方向的折射率差值。 
• 光纤和其它波导中，采用有效折射率差值描述更好。这与传播常数虚部的差值直接相关。 

还可以用偏振拍长来表征，是2π 除以传播常数的差值。如果波导中同时存在不同偏振状态的波，经过整数倍的偏振拍长，它们的相位关系不变的。
 

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

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.

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

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 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 bending). In 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). 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_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_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 θ 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_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_e, but a rather a mixture of n_e and n_o. This can be calculated with the following equation:

The equation shows that for θ → 0 (i.e., propagation along the optical axis) we obtain n → n_o, and the observed birefringence vanishes: one obtains n_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, 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:

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 (resulting 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.

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 χ⁽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 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.

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. See the article on optical activity for details.

Bibliography