Birefringence

definition:
The phenomenon of birefringence or the refractive index of a medium is related to polarization.

In literature, birefringence usually has two different meanings. In classical optics, it is referred to as double refraction.
In nonlinear optics and laser technology, birefringence is a property where the refractive index of some non isotropic transparent media depends on the polarization direction (i.e., the direction of the electric field). The latter exhibits birefringence when a non polarized beam is incident on the material.  

The refractive index depends on the polarization state
The refractive index depends on the polarization state and produces the following effects:

• When a light beam undergoes refraction on the surface of a birefringent crystal, the refraction angle is related to the polarization direction. In this way, the non polarized beam is divided into two linearly polarized lights (birefringent) when not vertically incident on the material. When unpolarized light is directed towards an object, if a birefringent crystal is used to observe the object, two images will appear.  
• When a linearly polarized laser beam is transmitted in a birefringent crystal, if the polarization direction does not coincide with the birefringent axis, it will contain two polarization parts with different wave numbers in different directions. Therefore, during transmission, due to the relative phase change between the two polarized components, the polarization state changes.
• This effect can be applied to birefringent tuners as it is wavelength dependent (although the refractive index difference is wavelength independent). This effect is power dependent through self phase modulation and cross phase modulation (see nonlinear polarization rotation), and is sometimes used for passive mode locking in fiber lasers.  
• Similarly, when a laser beam is transmitted in a laser crystal with thermal induced birefringence, the polarization state also changes. This change is related to the position, as the direction of the birefringence axis is changing (usually axial). This change (combined with the polarizing element in the laser resonant cavity) is the source of depolarization loss.  
• The birefringence of nonlinear crystal materials can achieve phase matching of birefringence during nonlinear interactions.  

Example of birefringence
In laser technology and nonlinear optics, birefringence phenomenon usually occurs in non isotropic crystals:

• Some laser crystals, such as vanadate crystals and tungstate crystals, inherently exhibit birefringence. This is very useful when a linearly polarized output without depolarization loss is required.  
• All nonlinear crystals used for nonlinear frequency conversion exhibit birefringence.  
• Birefringent crystals are commonly used to make polarizers.  
• Although optical fibers themselves do not have birefringence, birefringence effects are often encountered in fiber optics: sometimes birefringence comes from fiber bending (causing bending losses) and random disturbances. And there are also polarization maintaining fibers.  

Even isotropic media can experience birefringence due to the presence of non-uniform mechanical stress. This can be observed by placing a piece of organic glass between two crossed polarizers: when stress is applied to the organic glass, a color image can be seen due to the wavelength dependent birefringence effect induced by stress.
Similar effects also exist in bent optical fibers, where depolarization losses occur due to thermal effects in the laser crystal.  
Direct optical fibers have only a small random birefringence, and even so, the polarization state of the light in them will change after a certain distance of transmission. The existence of polarization maintaining fibers utilizes strong birefringence to suppress these effects.  

Quantitative description of birefringence
The following methods can be used to quantitatively describe the magnitude of birefringence:

• For crystals, the refractive index difference in the polarization direction can be considered.  
• In optical fibers and other waveguides, the effective refractive index difference is better described. This is directly related to the difference in the imaginary part of the propagation constant.  

It can also be characterized by polarization beat length, which is the difference between 2 π and the propagation constant. If there are waves with different polarization states simultaneously in the waveguide, their phase relationship remains unchanged after passing through integer multiples of the polarization beat length.