Kramers-Heisenberg dispersion formula for scattering of twisted light

An extremely active research topic of modern optics is studying how light can be engineered to possess forms of structure such as a twisting or a helical phase and the ensuing optical orbital angular momentum (OAM) and its interactions with matter. In such circumstances, the plane-wave description no longer suffices and both paraxial and nonparaxial solutions to the wave equation are desired. Within the framework of molecular QED theory, a general formulation is developed for the scattering of twisted light beams by molecular systems through the Kramers-Heisenberg dispersion formula and ensuing scattering cross section, which takes account of the effects of the phase and intensity structure of twisted light, revealing scattering effects not exhibited by unstructured, plane-wave light. The theory is applicable to linear scattering as well as to nonlinear optical effects for both chiral and nonchiral species, and explicit results are derived for Rayleigh and Raman scattering (including second-order contributions), Rayleigh and Raman optical activity, and their circular-vortex differential scattering analogs. These processes necessitate the inclusion of magnetic-dipole and electric-quadrupole coupling terms, as well as the usual leading electric-dipole interaction term. It is seen that the coupling of electric quadrupole moments to structured light affords a unique sensitivity to the phase properties of the beam, most importantly, its optical OAM, and its inclusion permits the contribution to the scattering cross section proportional to the square of the mixed electric dipole-quadrupole polarizability to be evaluated for which interesting features result. These include its discriminatory behavior arising from circularly polarized input radiation and its dependence on the topological charge, which can also serve to enhance scattering. Also presented are results for a contribution of identical order proportional to the pure electric-dipole and quadrupole polarizabilities.

Figure 1

Representative time-ordered Feynman graphs for linear scattering of light. Both time orderings (a) and (b) are required in the total calculation for scattering.

Figure 2

Twisted light handedness (corresponding azimuthal phase inset) for beams of topological charge $\left|\ell \right|=1$. Optical vortices are inherently chiral, screwing to the left for $\ell >0$ and to the right for $\ell <0$ (from the point of view of the source).

Figure 3

Feynman graphs to aid the calculation of single-photon absorption of a photon in mode $\left(k,\eta ,\ell ,p\right)$ and emission of a photon in mode $\left(k^{\prime },{\eta }^{\prime },{\ell }^{\prime },p^{\prime }\right)$ in diagrammatic time-dependent perturbation theory. Time progresses in the upward direction, with space along the horizontal axis.