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Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals or boron nitride, depending on the polarization of the light. This effect can occur only if the structure of the material is anisotropic (directionally dependent). If the material has a single axis of anisotropy or optical axis, (i.e. it is uniaxial) birefringence can be formalized by assigning two different refractive indices to the material for different polarizations. The birefringence magnitude is then defined by

- $Delta\; n=n\_e-n\_o,$

The reason for birefringence is the fact that in anisotropic media the electric field vector $vec\; E$ and the dielectric displacement $vec\; D$ can be nonparallel (namely for the extraordinary polarisation), although being linearly related.

Birefringence can also arise in magnetic, not dielectric, materials, but substantial variations in magnetic permeability of materials are rare at optical frequencies.

- Birefringence results when isotropic materials are deformed such that the isotropy is lost in one direction (ie, stretched or bent). Example
- Applying an electric field can induce molecules to line up or behave asymmetrically, introducing anisotropy and resulting in birefringence. (see Pockels effect)
- Applying a magnetic field can cause a material to be circularly birefringent, with different indices of refraction for oppositely-handed circular polarizations (see Faraday effect).

Material | n_{o}
| n_{e}
| Δn |
---|---|---|---|

beryl Be_{3}Al_{2}(SiO_{3})_{6}
| 1.602 | 1.557 | -0.045 |

calcite CaCO_{3}
| 1.658 | 1.486 | -0.172 |

calomel Hg_{2}Cl_{2}
| 1.973 | 2.656 | +0.683 |

ice H_{2}O
| 1.309 | 1.313 | +0.004 |

lithium niobate LiNbO_{3}
| 2.272 | 2.187 | -0.085 |

magnesium fluoride MgF_{2}
| 1.380 | 1.385 | +0.006 |

quartz SiO_{2}
| 1.544 | 1.553 | +0.009 |

ruby Al_{2}O_{3}
| 1.770 | 1.762 | -0.008 |

rutile TiO_{2}
| 2.616 | 2.903 | +0.287 |

peridot (Mg, Fe)_{2}SiO_{4}
| 1.690 | 1.654 | -0.036 |

sapphire Al_{2}O_{3}
| 1.768 | 1.760 | -0.008 |

sodium nitrate NaNO_{3}
| 1.587 | 1.336 | -0.251 |

tourmaline (complex silicate ) | 1.669 | 1.638 | -0.031 |

zircon, high ZrSiO_{4}
| 1.960 | 2.015 | +0.055 |

zircon, low ZrSiO_{4}
| 1.920 | 1.967 | +0.047 |

Many plastics are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is moulded or extruded. For example, cellophane is a cheap birefringent material, and Polaroid sheets are commonly used to examine for orientation in birefringent plastics like polystyrene and polycarbonate. Birefringent materials are used in many devices which manipulate the polarization of light, such as wave plates, polarizing prisms, and Lyot filters.

There are many birefringent crystals: birefringence was first described in calcite crystals by the Danish scientist Rasmus Bartholin in 1669.

Birefringence can be observed in amyloid plaque deposits such as are found in the brains of Alzheimer's patients. Modified proteins such as immunoglobulin light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet conformation. Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.

Cotton (Gossypium hirsutum) fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.

Slight imperfections in optical fiber can cause birefringence, which can cause distortion in fiber-optic communication; see polarization mode dispersion.

Silicon carbide, also known as Moissanite, is strongly birefringent.

The refractive indices of several (uniaxial) birefringent materials are listed below (at wavelength ~ 590 nm)

Material | n_{α}
| n_{β}
| n_{γ} |
---|---|---|---|

borax | 1.447 | 1.469 | 1.472 |

epsom salt MgSO_{4}·7(H_{2}O)
| 1.433 | 1.455 | 1.461 |

mica, biotite | 1.595 | 1.640 | 1.640 |

mica, muscovite | 1.563 | 1.596 | 1.601 |

olivine (Mg, Fe)_{2}SiO_{4}
| 1.640 | 1.660 | 1.680 |

perovskite CaTiO_{3}
| 2.300 | 2.340 | 2.380 |

topaz | 1.618 | 1.620 | 1.627 |

ulexite | 1.490 | 1.510 | 1.520 |

A common feature of optical microscopes is a pair of crossed polarizing filters. Between the crossed polarizers, a birefringent sample will appear bright against a dark (isotropic) background.

It is also used as a spatial low-pass filter in electronic cameras, where the thickness of the crystal is controlled to spread the image in one direction, thus having the effect of a spatial low-pass filter (by increasing the spot-size). This is essential to the proper working of all television and electronic film cameras, to avoid spatial aliasing, the folding back of frequencies higher than can be sustained by the pixel matrix of the camera.

In ophthalmology, scanning laser polarimetry utilises the birefringence of the retinal nerve fibre layer to quantitate its thickness indirectly, which is of use in the assessment and monitoring of glaucoma.

Propagation direction | Ordinary ray | Extraordinary ray | ||
---|---|---|---|---|

Polarization | n_{eff}
| Polarization | n_{eff} | |

z | xy-plane | $n\_o$ | n/a | n/a |

xy-plane | xy-plane | $n\_o$ | z | $n\_e$ |

xz-plane | y | $n\_o$ | xz-plane | $n\_e\; <\; n\; <\; n\_o$ |

other | analogous to xz-plane |

For a uniaxial material with the z axis defined to be the optical axis, the effective refractive indices are as in the table on the right. For rays propagating in the xz plane, the effective refractive index of the e polarization varies continuously between $n\_o$ and $n\_e$, depending on the angle with the z axis. The effective refractive index can be constructed from the Index ellipsoid.

More generally, birefringence can be defined by considering a dielectric permittivity and a refractive index that are tensors. Consider a plane wave propagating in an anisotropic medium, with a relative permittivity tensor ε, where the refractive index n, is defined by $ncdot\; n\; =\; epsilon$. If the wave has an electric vector of the form:

{{Equation|1=mathbf{E=E_0}exp i(mathbf{k cdot r}-omega t) ,|2=2}}

where r is the position vector and t is time, then the wave vector k and the angular frequency ω must satisfy Maxwell's equations in the medium, leading to the equations:

{{Equation|1=-nabla times nabla times mathbf{E}=frac{1}{c^2}(mathbf{epsilon} cdot frac{part^2 mathbf{E} }{partial t^2})|2=3a}}

{{Equation|1= nabla cdot (mathbf{epsilon} cdot mathbf{E}) =0 |2=3b}}

where c is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:

{{Equation|1=mathbf{k}^2mathbf{E_0}-mathbf{(k cdot E_0) k}= frac{omega^2}{c^2} (mathbf{epsilon} cdot mathbf{E_0}) |2=4a}}

{{Equation|1=mathbf{k} cdot (mathbf{epsilon} cdot mathbf{E_0}) =0 |2=4b}}

For the matrix product $(epsiloncdotmathbf\; E)$ often a separate name is used, the dielectric displacement vector $mathbf\; D$. So essentially birefringence concerns the general theory of linear relationships between these two vectors in anisotropic media.

To find the allowed values of k, E_{0} can be eliminated from eq 4a. One way to do this is to write eqn 4a in Cartesian coordinates, where the x, y and z axes are chosen in the directions of the eigenvectors of ε, so that

{{Equation|1=mathbf{epsilon}=begin{bmatrix} n_x^2 & 0 & 0 0& n_y^2 & 0 0& 0& n_z^2 end{bmatrix} ,|2=4c}}

Hence eqn 4a becomes

{{Equation|1=(-k_y^2-k_z^2+frac{omega^2n_x^2}{c^2})E_x + k_xk_yE_y + k_xk_zE_z =0|2=5a}} {{Equation|1=k_xk_yE_x + (-k_x^2-k_z^2+frac{omega^2n_y^2}{c^2})E_y + k_yk_zE_z =0|2=5b}} {{Equation|1=k_xk_zE_x + k_yk_zE_y + (-k_x^2-k_y^2+frac{omega^2n_z^2}{c^2})E_z =0|2=5c}}

where E_{x}, E_{y}, E_{z}, k_{x}, k_{y} and k_{z} are the components of E_{0} and k. This is a set of linear equations in E_{x}, E_{y}, E_{z}, and they have a non-trivial solution if their determinant is zero:

{{Equation|1=detbegin{bmatrix} (-k_y^2-k_z^2+frac{omega^2n_x^2}{c^2}) & k_xk_y & k_xk_z k_xk_y & (-k_x^2-k_z^2+frac{omega^2n_y^2}{c^2}) & k_yk_z k_xk_z & k_yk_z & (-k_x^2-k_y^2+frac{omega^2n_z^2}{c^2}) end{bmatrix} =0,|2=6}}

Multiplying out eqn (6), and rearranging the terms, we obtain

{{Equation|1=frac{omega^4}{c^4} - frac{omega^2}{c^2}left(frac{k_x^2+k_y^2}{n_z^2}+frac{k_x^2+k_z^2}{n_y^2}+frac{k_y^2+k_z^2}{n_x^2}right) + left(frac{k_x^2}{n_y^2n_z^2}+frac{k_y^2}{n_x^2n_z^2}+frac{k_z^2}{n_x^2n_y^2}right)(k_x^2+k_y^2+k_z^2)=0, |2=7}}

In the case of a uniaxial material, where n_{x}=n_{y}=n_{o} and n_{z}=n_{e} say, eqn 7 can be factorised into

{{Equation|1=left(frac{k_x^2}{n_o^2}+frac{k_y^2}{n_o^2}+frac{k_z^2}{n_o^2} -frac{omega^2}{c^2}right)left(frac{k_x^2}{n_e^2}+frac{k_y^2}{n_e^2}+frac{k_z^2}{n_o^2} -frac{omega^2}{c^2}right)=0,.|2=8}}

Each of the factors in eqn 8 defines a surface in the space of vectors k — the surface of wave normals. The first factor defines a sphere and the second defines an ellipsoid. Therefore, for each direction of the wave normal, two wavevectors k are allowed. Values of k on the sphere correspond to the ordinary rays while values on the ellipsoid correspond to the extraordinary rays.

For a biaxial material, eqn (7) cannot be factorized in the same way, and describes a more complicated pair of wave-normal surfaces.

Birefringence is often measured for rays propagating along one of the optical axes (or measured in a two-dimensional material). In this case, n has two eigenvalues which can be labeled n_{1} and n_{2}. n can be diagonalized by:

{{Equation|1=mathbf{n} = mathbf{R(chi)} cdot begin{bmatrix} n_1 & 0 0 & n_2 end{bmatrix} cdot mathbf{R(chi)}^textrm{T} |2=9}}

where R(χ) is the rotation matrix through an angle χ. Rather than specifying the complete tensor n, we may now simply specify the magnitude of the birefringence Δn, and extinction angle χ, where Δn = n_{1} − n_{2}.

- http://www.olympusmicro.com/primer/lightandcolor/birefringence.html
- Video of stress birefringence in Polymethylmethacrylate (PMMA or Plexiglas).
- Application note on the theory of birefringence (see no.14)
- Austine Wood Comarow: Paintings in Polarized Light, James Mann, Wasabi Publishing, 2005, ISBN 0-9768198-0-5.

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Last updated on Saturday October 04, 2008 at 12:09:52 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 04, 2008 at 12:09:52 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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