Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes

1 A tensor susceptibility?

The electrons bound within, and binding, the atoms of a dielectric crystal are not uniformly distributed, but are restricted in their motion by the potentials which confine them. In response to an applied electric field, they may therefore move a greater or lesser distance, depending upon the strength of their confinement in the field direction. As a result, the induced polarization varies not only with the strength of the applied field, but also with its direction. The susceptibility– and properties which depend upon it, such as the refractive index– are therefore anisotropic, and cannot be characterized by a single value.

The scalar electric susceptibility, χe, is defined to be the coefficient which relates the value of the ~ ~ local electric field, Eloc, to the local value of the polarization, P : ~ ~ P = χe0Elocal. (1)

As we discussed in the last seminar we can ‘promote’ this relation to a tensor relation with χe → (χe)ij. In this case the dielectric constant is also a tensor and takes the form

ij = [δij + (χe)ij] 0. (2)

Figure 1: A cartoon showing how the electron is held by anisotropic springs– causing an electric susceptibility which is different when the electric field is pointing in different directions.

How does this happen in practice? In Fig. 1 we can see that in a crystal, for instance, the electrons will in general be held in bonds which are not spherically symmetric– i.e., they are anisotropic.

1 Therefore, it will be easier to polarize the material in certain directions than it is in others. In par- ticular, the bonds in the x, y, and z directions may all be different leading to a electric susceptibility which is different when the local electric field points in the x, y, or z directions:     1 0 0 (χe)xx 0 0 ij = 0  0 1 0  +  0 (χe)yy 0  . (3) 0 0 1 0 0 (χe)zz

Now, when electromagnetic waves propagate through an anisotropic linear dielectric each component of the electric field follows a different wave equation– with a different propagation speed. Everything is the same as in Griffiths Sec. 9.3.1 (on the propagation of light in linear media) except now the speed of propagation (and hence the index of refraction) depends on the direction of the electric field. Such a material causes an effect called birefringence. To see how this can lead to interesting effects, consider a slab of a material which is birefringent. Consider an electromagnetic wave which is traveling in thez ˆ-direction. If the wave has an electric p field which oscillates along thex ˆ-axis the wave will travel at speed c/ 1 + (χe)xx; if the wave has an p electric field which oscillates along they ˆ-axis the wave will travel at different speed c/ 1 + (χe)yy. If the light is linearly polarized at 45◦ relative to the x-y axes then the electromagnetic waves which make up the light are equally distributed between being polarized in thex ˆ andy ˆ directions. If we consider a wave E E~ = √0 ei(kz−ωt)(ˆx +y ˆ), (4) 2 entering the material, what will the wave look like after it travels a distance D within the material? The speed of thex ˆ-component of the wave has a wave vector with a magnitude k0: 0 p 0 ω/k = c/ 1 + (χe)xx = c/nx → k = nxω/c = nxk so that

E0 i(nxkz−ωt) Ex = √ e , (5) 2 and the same happens to the y-component:

E0 i(nykz−ωt) Ey = √ e . (6) 2 After traveling a distance D there is now a phase difference between the x and y-components of the electric fields: ∆φ = (nx − ny)kD. (7) As you can easily confirm, when thex ˆ andy ˆ components of the electric field are 90-degrees out of phase we have circularly polarized light (defined by the fact that the electric field is always non-zero): π 2πD λ 1 = (nx − ny)kD = (nx − ny) → D = . (8) 2 λ 4 nx − ny This is known as a quarter wave plate and is able to take linearly polarized light and produce circularly polarized light.