Birefringences and dichroisms#

Light experienced in everyday life is mostly non-polarized - that is, the tip of the electric or magnetic field vectors moves randomly in the plane perpendicular to the propagation direction. With the use of appropriate tools, unpolarized light can be turned into a wide variety of polarization states, becoming linearly, elliptically or circularly polarized. In these cases, the tip of the field vector(s) oscillates in the plane containing the direction of propagation (linear polarization) or rotates along a circle (circular polarization) or an ellipse (elliptical polarization) in the plane perpendicular to it. The polarization of light plays an important role in the interactions of light with a molecule.

Birefringences and dichroisms are two aspects of this interaction which are closely related to the polarization status of light. They arise through a variety of different mechanisms and interactions between the electromagnetic field and the molecule, but they are all characterized by the simultaneous measurements of the refractive (birefringences) or absorptive (dichroisms) index of light along two directions, different with respect to some predefined laboratory or molecular frame. The difference in the dispersion or absorption at these two directions is described by a quantity of a specific sign and magnitude. All these properties give detailed information about the molecular structure, and in particular some of them can distinguish between a sample and its mirror image. Distinguishable mirror images can occur naturally, in {chiral} molecules where the mirror image of the molecule cannot be superimposed on the original molecule itself - that is, for so-called enantiomers - or they can occur by introducing an asymmetry in the experimental set-up that makes the mirror image of the experimental design impossible to superimpose on the original set-up. In the latter case, since the experimental design is itself chiral, the observed effect does not vanish even if the molecule is not chiral. Note that only molecules possessing no improper or rotation-reflection axes may be chiral.

Let us consider first an isotropic liquid sample subject to a strong external static electric field. The refractive index \(n_\|\) experienced by the component of the polarization vector aligned parallel to the direction of the external field \(\mathbf{F}\) in this case differs from the index of refraction \(n_\bot\) for the component aligned perpendicular to the direction of the vector \(\mathbf{F}\). As a consequence, the initially linearly polarized light beam will exit the region of the sample with an ellipticity. This is an example of a {\em{linear birefringence}},

\[\begin{equation*} \Delta n^{\rm lin} = n_\| - n_\bot, \end{equation*}\]

and the effect is known as {{electric-field-induced optical birefringence}} or more often as the {{Kerr (electro-optical) effect}}. It exhibits a quadratic dependence on the strength of the applied electric field. Its existence is due mainly to the fact that the external electric field, in a system possessing an anisotropic electric dipole polarizability tensor, tends to align the molecules preferentially in its direction.

Other mechanisms in general also contribute to the emergence of an anisotropy, the most important being the effect of electronic rearrangements. They play a role through the higher-order polarizabilities of the system, in particular by the second electric-dipole hyperpolarizability \(\gamma\) and, in the presence of permanent electric dipoles, by the first electric-dipole hyperpolarizability \(\beta\). Other examples of linear birefringences are the Cotton-Mouton and the Buckingham effects, and the Jones and magneto-electric birefringences, the latter occurring when linearly polarized light goes through an isotropic fluid perpendicularly to both electric and magnetic static fields.

{\em {Circular birefringence}} is observed when the two circular components of a linearly polarized beam propagate with different circular velocities, and therefore an anisotropy arises between the two refractive indices \(n_+\) and \(n_-\):

\[\begin{equation*} \Delta n^{\rm circ} = n_+ - n_-. \end{equation*}\]

The net result is a rotation of the plane of polarization. An example of a circular birefringence is {\em{optical activity}}, which is observed when a medium composed of chiral molecules (with an excess of one enantiomer) is subject to linearly polarized electromagnetic radiation. The dependence of the corresponding anisotropy on the wavelength is described by the {{optical rotatory dispersion}} (ORD). The Faraday effect is another well-known example of a circular birefringence.

An {\em{axial birefringence}} occurs with {{unpolarized}} light, and an example is the so-called {{magnetochiral birefringence}}. It can be seen in isotropic samples composed of chiral molecules, and it is observed when a static magnetic induction field is switched on parallel to the direction of propagation of the unpolarized probe beam. The refractive index experienced by the beam propagating parallel to the external magnetic field, \(n_{\uparrow\uparrow}\), becomes different from that experienced by a beam propagating antiparallel to the field, \(n_{\uparrow\downarrow}\)

\[\begin{equation*} \Delta n^{\rm ax} = n_{\uparrow\uparrow} - n_{\uparrow\downarrow}. \end{equation*}\]

The corresponding anisotropy has a different sign for the different enantiomers.

An external field will have an orientational effect on the molecules of the sample, and this is a mechanism responsible for the emergence of birefringences. In the presence of external fields the sample becomes anisotropic, and the extent to which this happens depends on the temperature. Indeed, the general form of the optical anisotropy (for measurements taken at a fixed pressure) is usually of the form

\[\begin{equation*} \Delta n \propto A_0 + \frac{A_1}{T} + \frac{A_2}{T^2} + \cdots, \end{equation*}\]

where the term \(A_0\) includes all contributions arising from the reorganization of the electrons due to the action of the external field(s), whereas the terms \(A_1\), \(A_2\), \(\ldots\) are connected to different mechanisms of reorientation of the molecules, involving the interaction of the field(s) with permanent electric or magnetic multipole moments. The temperature-dependent terms vanish for systems of spherical symmetry; the contributions exhibiting a nonlinear dependence are usually connected to the presence of permanent electric or magnetic dipole moments, or higher-order processes involving more complicated interactions between fields and multipoles.

When the inducing field is time-dependent, the birefringence is said to be ``optically induced’’. In general, the information provided by birefringences induced by static fields can differ from that obtained observing the corresponding optically induced phenomena. For example, in the static Kerr effect of dipolar molecules, an important role is played by the permanent dipoles, which tend to be aligned by the static field. This contribution vanishes for dynamic inducing fields, where the birefringence arises only from the rearrangement of the anisotropy of the electric dipole polarizability. The contribution resulting from the interaction with permanent multipoles averages over time to zero.