64 5 Symmetric Second Rank Tensors
The index of refractionν(i)for linearly polarized light with the electric field
vector parallel to a principal directione(i), is linked with the principal valueε(i)of
the dielectric tensor by the Maxwell relation
ν(i)=√
ε(i), i= 1 , 2 , 3. (5.29)Birefringence occurs, whenever, at least two of the principal indices of refraction
are different. The differencesδν 12 =ν(^1 )−ν(^2 )orδν 13 =ν(^1 )−ν(^3 )quantify the
size of the birefringence for light propagating in the 3-direction or in the 2-direction,
respectively.
One way to detect birefringence is to measure the transmission of light through
a medium, between crossed polarizer and analyzer. The effect is largest, when the
incident light is linearly polarized with the electric field vector oriented under 45◦
between two principal directions, say betweene(^1 )ande(^3 ). The vector obviously
has components to both these principal directions for which the speed of light is
different, because the indices of refraction are different. After a propagation over
the distanceLthrough the birefringent medium, the two components have a phase
shiftδφ=Lλδν 13 , whereλis the wave length of the light. As a consequence of this
phase shift, the light is elliptically polarized and has a component perpendicular to
the direction of the incident linear polarization. The intensityIof the light which
passes through an analyzer oriented parallel toe(^1 )−e(^3 ), is proportional to
I∼sin^2 (δφ)=sin^2(
L
λδν 13)
. (5.30)
Clearly, this intensity of the light passing through the crossed analyzer can be used
to measure the differenceδν 13 between the indices of refraction which, in turn, is
caused by the symmetric traceless part of the dielectric tensor. Its microscopic origin,
described by an average of the molecular polarizability tensor and the alignment of
optically anisotropic molecules, as well as a non-isotropic arrangement of atoms,
will be discussed later.
5.3.5 Electric and Magnetic Torques
An electric fieldEexerts a torque on an electric dipole momentpel. Similarly, a
magnetic fieldBcauses a torque on a magnetic dipole momentm. These torques,
denoted byTelandTmag,aregivenby
Tμel=εμλνpλEν, Tμmag=εμλνmλBν. (5.31)In general, the electric dipole moment is the sum of a permanent and an induced part,
viz.pel=pperm+pind, cf. Sect.5.3.3. The computation of the permanent moment for
a given charge distribution is presented in Sect.10.3. As discussed above, the induced
dipole moment is proportional to the electric field, when the field strength is small