Birefringence or double refraction

Birefringence usually occurs in anisotropic crystals. In these crystals the arrangement of atoms is different in different directions. As a consequence the crystal properties are direction dependent. As an example consider a crystal where the refractive index is $n_e$ when the $\vec{E}$ vector is along the $x$ axis and it is $n_0$ when $\vec{E}$ is perpendicular to the $x$ axis. Here the $x$ axis is referred to as the optic axis. The optical properties are the same in all directions perpendicular to the optic axis and it is different if $\vec{E}$ is along the optic axis. Crystals may have more than one optic axis. Here we only consider a situation where there is a single optic axis. Such birefringent crystals are referred to as uniaxial crystals. Through birefringent crystals one sees two images of the object unless the direction of viewing is particularly well chosen (in which case there is only one image) as shown in the Figure 16.10. So there are two refracted rays for each incident beam and hence the name. Optic axis of the crystal is in the vertical direction in the Figure. We find that the second image is slightly displaced from the original one. The image which is not displaced (refractive index $n_0$) is having a polarization perpendicular to the optic axis whereas the displaced image (with refractive index $n_e$) has a polarization along the optic axis of the crystal.

Figure 16.10: Birefringence in Calcite

The difference $\Delta n=n_e-n_o$ is a measure of the birefringence and is often called the birefringence. A material is referred to as a positive material if $\Delta n >0$ and a negative material if $\Delta n <0$. We tabulate below the refractive indices for some birefringent materials.

Crystal	$n_o$	$n_e$
$\lambda_0=589.3 \, {\rm nm}$
Tourmaline	1.669	1.638
Calcite	1.6584	1.4864
Quartz	1.5443	1.5534
Sodium Nitrate	1.5854	1.3369

Figure 16.11: a) Nicol and 16.11: b) Wollaston prisms

Nicol prism is a smart device often used in the laboratories to produce linearly polarized beam. Here two calcite pieces cut in a special way (SQP and PQR) are cemented in a manner shown in the left of Figure 16.11. The two prisms are glued with a material called Canada balsam, a transparent material having a refractive index, , which is between the $n_o=1.6584$ and $n_e=1.4864$ of calcite. Once the unpolarized light is incident on the surface QS of the prism as shown in the middle of the Figure  16.11, it is divided into two rays due to birefringence. One of the rays is totally internally reflected by the layer of Canada balsam, QP, and is absorbed by a black paint on the wall, SP, of the crystal. The other ray transmits through the Canada balsam to produce a plane polarized light.

The Wollaston prism is a device that uses birefringence to separate the unpolarized incident light into two linearly polarized components. Two triangular prisms are glued together as shown in Figure 16.11. Both prisms are made of the same birefringent material. The optic axis of the two prisms are mutually perpendicular as shown in the Figure. We decompose the unpolarized incident light into $\vec{E}_{\parallel}$ and $\vec{E}_{\perp}$ respectively parallel and perpendicular to the plane of the paper. In the first prism $\vec{E}_{\parallel}$ has refractive index $n_e$ and $\vec{E}_{\perp}$ has $n_o$. The situation reverses when the light enters the second prism where $\vec{E}_{\parallel}$ has refractive index $n_o$ and has $n_e$. Figure 16.11 shows the paths of the two polarizations through the Wollaston prism. The two polarizations part ways at the interface of the two prisms and they emerge in different directions as shown in Figure  16.11.