Birefringence

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress. D = ε E {displaystyle mathbf {D} ={oldsymbol {varepsilon }}mathbf {E} }     (1) E = E 0 e i ( k ⋅ r − ω t ) {displaystyle mathbf {E} =mathbf {E} _{0}e^{i(mathbf {k} cdot mathbf {r} -omega t)}}     (2) − ∇ × ∇ × E = μ 0 ∂ 2 ∂ t 2 D {displaystyle - abla imes abla imes mathbf {E} =mu _{0}{frac {partial ^{2}}{partial t^{2}}}mathbf {D} }     (3a) ∇ ⋅ D = 0 {displaystyle abla cdot mathbf {D} =0}     (3b) − ∇ × ∇ × E = ( k ⋅ E ) k − ( k ⋅ k ) E {displaystyle - abla imes abla imes mathbf {E} =(mathbf {k} cdot mathbf {E} )mathbf {k} -(mathbf {k} cdot mathbf {k} )mathbf {E} }     (3c) ( − k ⋅ k ) E + ( k ⋅ E ) k = − μ 0 ω 2 ( ε E ) {displaystyle (-mathbf {k} cdot mathbf {k} )mathbf {E} +(mathbf {k} cdot mathbf {E} )mathbf {k} =-mu _{0}omega ^{2}({oldsymbol {varepsilon }}mathbf {E} )}     (4a) k ⋅ D = 0 {displaystyle mathbf {k} cdot mathbf {D} =0}     (4b) ε = ε 0 [ n x 2 0 0 0 n y 2 0 0 0 n z 2 ] {displaystyle mathbf {varepsilon } =varepsilon _{0}{egin{bmatrix}n_{x}^{2}&0&0\0&n_{y}^{2}&0\0&0&n_{z}^{2}end{bmatrix}}}     (4c) ( − k x 2 − k y 2 − k z 2 ) E x + k x 2 E x + k x k y E y + k x k z E z = − ω 2 n x 2 c 2 E x {displaystyle left(-k_{x}^{2}-k_{y}^{2}-k_{z}^{2} ight)E_{x}+k_{x}^{2}E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=-{frac {omega ^{2}n_{x}^{2}}{c^{2}}}E_{x}}     (5a) ( − k y 2 − k z 2 + ω 2 n x 2 c 2 ) E x + k x k y E y + k x k z E z = 0 {displaystyle left(-k_{y}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{x}^{2}}{c^{2}}} ight)E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=0}     (5b) k x k y E x + ( − k x 2 − k z 2 + ω 2 n y 2 c 2 ) E y + k y k z E z = 0 {displaystyle k_{x}k_{y}E_{x}+left(-k_{x}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{y}^{2}}{c^{2}}} ight)E_{y}+k_{y}k_{z}E_{z}=0}     (5c) k x k z E x + k y k z E y + ( − k x 2 − k y 2 + ω 2 n z 2 c 2 ) E z = 0 {displaystyle k_{x}k_{z}E_{x}+k_{y}k_{z}E_{y}+left(-k_{x}^{2}-k_{y}^{2}+{frac {omega ^{2}n_{z}^{2}}{c^{2}}} ight)E_{z}=0}     (5d) | ( − k y 2 − k z 2 + ω 2 n x 2 c 2 ) k x k y k x k z k x k y ( − k x 2 − k z 2 + ω 2 n y 2 c 2 ) k y k z k x k z k y k z ( − k x 2 − k y 2 + ω 2 n z 2 c 2 ) | = 0 {displaystyle {egin{vmatrix}left(-k_{y}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{x}^{2}}{c^{2}}} ight)&k_{x}k_{y}&k_{x}k_{z}\k_{x}k_{y}&left(-k_{x}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{y}^{2}}{c^{2}}} ight)&k_{y}k_{z}\k_{x}k_{z}&k_{y}k_{z}&left(-k_{x}^{2}-k_{y}^{2}+{frac {omega ^{2}n_{z}^{2}}{c^{2}}} ight)end{vmatrix}}=0}     (6) ω 4 c 4 − ω 2 c 2 ( k x 2 + k y 2 n z 2 + k x 2 + k z 2 n y 2 + k y 2 + k z 2 n x 2 ) + ( k x 2 n y 2 n z 2 + k y 2 n x 2 n z 2 + k z 2 n x 2 n y 2 ) ( k x 2 + k y 2 + k z 2 ) = 0 {displaystyle {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}}} ight)+left({frac {k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}}+{frac {k_{y}^{2}}{n_{x}^{2}n_{z}^{2}}}+{frac {k_{z}^{2}}{n_{x}^{2}n_{y}^{2}}} ight)left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2} ight)=0}     (7) ( k x 2 n o 2 + k y 2 n o 2 + k z 2 n o 2 − ω 2 c 2 ) ( k x 2 n e 2 + k y 2 n e 2 + k z 2 n o 2 − ω 2 c 2 ) = 0 {displaystyle left({frac {k_{x}^{2}}{n_{mathrm {o} }^{2}}}+{frac {k_{y}^{2}}{n_{mathrm {o} }^{2}}}+{frac {k_{z}^{2}}{n_{mathrm {o} }^{2}}}-{frac {omega ^{2}}{c^{2}}} ight)left({frac {k_{x}^{2}}{n_{mathrm {e} }^{2}}}+{frac {k_{y}^{2}}{n_{mathrm {e} }^{2}}}+{frac {k_{z}^{2}}{n_{mathrm {o} }^{2}}}-{frac {omega ^{2}}{c^{2}}} ight)=0}     (8) Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress. Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it in calcite, a crystal having one of the strongest birefringences. However it was not until the 19th century that Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarizations (perpendicular to the direction of the wave vector). A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon. The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behavior. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index no (for 'ordinary') regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that would be perpendicular to the optic axis, and a ray with that polarization is called an ordinary ray and is governed by the same refractive index value no. However, for a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of the optic axis, and this extraordinary ray will be governed by a different, direction-dependent refractive index. Because the index of refraction depends on the polarization, when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray.The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence: The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as if there were no birefringence involved. However the extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between no and ne). However its power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed. When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations are perpendicular to the optic axis and see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to ne) but still has the power flow in the direction of the wave vector. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a waveplate, in which there is no distortion of the image but an intentional modification of the state of polarization of the incident wave. For instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source. The case of so-called biaxial crystals is substantially more complex. These are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, however, the direction of power flow is not identical to the direction of the wave vector in either case. The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution ('spheroid') but is described by three unequal principle refractive indices nα, nβ and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid is a spheroid). Although there is no axis of symmetry, there are two optical axes or binormals which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization. For this reason, birefringent materials with three distinct refractive indices are called biaxial. Additionally, there are two distinct axes known as optical ray axes or biradials along which the group velocity of the light is independent of polarization.