Light propagation through crystalline media is often associated with in- teresting physical effects. One of them is double refraction or birefringence.
60 Wave Optics: Basic Concepts and Contemporary Trends
Birefringence occurs when light entering a uniaxial (with one optic axis) crys- tal splits into two beams. These two beams have mutually orthogonal po- larization, usually referred to as the ordinary and the extraordinary waves.
The ordinary wave behaves just like a wave in an isotropic medium, while the extraordinary wave has some peculiar characteristics. Most important is the dependence of the magnitude of the refractive index on the direction of prop- agation of light. Note that the ordinary wave has the same refractive index in all directions. Along one direction both the waves have the same refractive index. This particular direction is identified as the optic axis of the crystal.
There can be uniaxial or biaxial crystals depending on whether there exists one or two optic axes.
It is clear that the dependence of the refractive index increases due to the anisotropic nature of the crystal through which the light propagates. It was mentioned earlier that all the information about the optical properties of the material is carried by a pair of physical quantities, namely, the dielectric permittivityεand the magnetic permeabilityµ. For a nonmagnetic medium, the anisotropic nature of the dielectric functionε reflects the directional de- pendence. In particularε is a second-rank tensor (denoted by ¯ε below) with 3×3 elements, while it is a scalar for an isotropic medium. In general for an anisotropic nonmagnetic medium, the relation between D and E can be written as
D= ¯εE. (4.43)
In terms of components
Dx=εxxEx+εxyEy+εxzEz, (4.44) Dy=εyxEx+εyyEy+εyzEz, (4.45) Dz=εzxEx+εzyEy+εzzEz. (4.46) Thus fields along y can cause induction along thex direction, provided εxy
is nonzero. Note that in an isotropic medium,Ey can lead to induction only along theydirection. By means of coordinate transformation, the 3×3 matrix εij can be diagonalized.
4.3.1 Birefringence in crystals like calcite
Calcite (CaCO3) can be found in nature as large crystals, which can be made into a cube, pressed slightly along one great diagonal. It can also be pol- ished along the crystal faces. Objects seen through this cube form two images, which explains the origin of the name: double refraction or birefringence. If a narrow beam is directed along the normal to a natural face of the crystal, two beams exit from the opposite side parallel to the incident beam. The ordinary beam passes along the direction of the incident beam while the extraordinary beam is shifted with respect to the direction of the original beam. To put it differently, the angle of refraction of the extraordinary beam is not zero. A rotation of the crystal about the axis of the incident beam, the spot due to
the extraordinary beam rotates in a circular path about the same axis. For unpolarized light the two spots from ordinary and extraordinary beams have the same intensity. Since the two rays have mutually orthogonal polarization, an analyzer placed after the crystal can filter them out. The rotation of the analyzer results in a periodic decrease in the ordinary ray intensity and an increase of the same for the extraordinary wave. This can be understood from a different angle. Let a plane polarized light be incident on the crystal having optic axis alongOO′ perpendicular tok. Let the incident field vector oscillate along AA′, which makes an angleα with the normal to the optic axisBB′. The ordinary wave E oscillates along BB′, while that of the extraordinary wave oscillates alongOO′. Thus amplitudes of oscillations in the ordinary and extraordinary waves are given by
(E0)o= (E0)inccosα, (4.47) (E0)e= (E0)incsinα. (4.48) For intensities we have
Io=Iinccos2α, (4.49)
Ie=Iincsin2α. (4.50)
Hence
Ie
Io
= tan2α, (4.51)
Iinc=Io+Ie. (4.52)
It is clear from the above equations that the ratio of intensities vary in a periodic fashion, and in the region of overlap of the spots, the intensity is constant. In the literature this is often referred to as Malus’ law.
4.3.2 Polarizers based on birefringence: Nicol and Wollaston prisms
The Nicol prism is cut from a calcite crystal as shown in Fig. 4.2. This crys- tal is split into two and joined using glue with refractive indexnglue= 1.549,
FIGURE 4.2: Schematics of a Nicol prism.
62 Wave Optics: Basic Concepts and Contemporary Trends
O1’ O1
O2’ O2
FIGURE 4.3: Schematics of a Wollaston prism.
which is in between the ordinary and extraordinary refractive indicesno= 1.66 and ne = 1.49. For the specified geometry, the ordinary wave undergoes to- tal internal reflection while the extraordinary ray passes through. The two outgoing waves are linearly polarized with mutually orthogonal polarization.
The Wollaston prism gives rise to two orthogonally polarized light beams as shown in Fig. 4.3. It is made of two triangular prisms glued along the hypotenuse in such a way that its optic axesO1O′1andO2O′2are orthogonal to each other. Both ordinary and extraordinary rays travel in the same direction in the first prism. An extraordinary wave leaving the first prism travels in the second as an ordinary wave and vice versa. The refractive index of the initially extraordinary beam in the second prism is higher and hence it bends upward, closer to the normal to the interface, while the second beam (initially ordinary) sees a higher to lower refractive index and bends downward, away from the normal.
4.3.3 Retardation plates
Consider a thin plate of a uniaxial material. Assume the material to be cut in such a way that the optic axis is parallel to the surfaces of the plate. Let a plane polarized monochromatic wave be incident on the plate normally. Let also the direction of oscillation of the field vectorEmake some angle with the optic axis. ThenEx(the projection onto the normal to the optic axis) andEy
(the projection along the optic axis) determine the ordinary and extraordinary components within the plate. Both propagate in the same direction albeit with
different phase velocities sincene6=no. At the input face both are in phase, while the phase difference at the output face of a crystal with widthhis given by
δ=k0(ne−no)h, (4.53)
wherek0=ω/c= 2π/λis the vacuum wave-vector magnitude. In general this will lead to elliptic polarization. In particular, forEx=Ey (when the angle is 45◦) andδ=π/2, the superposition of the orthogonal components will lead to circular polarization. For example, for a negative crystal, when ne < no, the phase of the ordinary wave is retarded by δ with respect to that of the extraordinary ray. The field at the output face can be written as
E(t) = ˆxE0
√2cos(ωt) + ˆyE0
√2cos(ωt+δ). (4.54) For
δ= (2m+ 1)π/2 or h(ne−no) = (2m+ 1)λ/4, (4.55) the tip of the vector E undergoes a rotation in a circular path and for this case, the total field can be written as
E(t) = ˆxE0
√2cos(ωt)−yˆE0
√2sin(ωt). (4.56)
It is clear that a retardation plate, also called a λ/4 plate, converts linear polarization to circular polarization under certain conditions (Eq. (4.55)). We can devise a matrix tool based on Jones vectors in order to calculate the conversion between various states of polarization.