Theoretical Discussion of the Inverse Faraday Effect, Raman Scattering, and Related Phenomena

The inverse Faraday effect (IFE) occurs when circularly polarized light incident on a nonabsorbing crystal induces a magnetic moment proportional to the Verdet constant of that material, the proportionality constant being independent of the material. Some experimental details relating to the observation and proof of the IFE are presented. A microscopic, or quantum-mechanical, explanation of the IFE which also demonstrates its connection with the Faraday effect is given in terms of an optically induced "effective Hamiltonian" ${\mathcal{H}}_{\mathrm{eff}}$. Conventional calculations of the optical constants like the Verdet constant compute optical-frequency polarizabilities for a given electronic configuration. It is shown here that these same constants are obtained by computing the average effects that optical fields have on the atomic configuration. The optically induced ${\mathcal{H}}_{\mathrm{eff}}$ is linearly dependent on the low-frequency components of $E\left(t\right)E\left(t\right)\mathrm{}$ and one can show the thermal average ${〈{\mathcal{H}}_{\mathrm{eff}}〉}_{\mathrm{thermal}}$ is equal to the temporl average ${〈\mathbf{E}\mathrm{}\left(t\right)\mathrm{}·\mathbf{\varepsilon }·\left(\mathbf{E}t\right)〉}_{\mathrm{time}}$. Both the Verdet constant and the IFE coefficient arise from a thermal average of the same term in ${\mathcal{H}}_{\mathrm{eff}}$. Similarly the Cotton-Mouton effect and its inverse arise from one other term in ${\mathcal{H}}_{\mathrm{eff}}$. Definition of ${\mathcal{H}}_{\mathrm{eff}}$ is one technique for linearizing some nonlinear optical problems. Following linearization one obtains Kramers-Kronig-type dispersion relations between constants of seemingly different phenomena. For example, a Kramers-Kronig relation is obtained between the Verdet constant and the gain coefficient for stimulated Raman scattering by paramagnetic ions in a large dc magnetic field. The effective Hamiltonian is explicitly evaluated from some illustrative examples and its general phenomenological form for a paramagnetic ion in cubic symmetry is written down and discussed.