Third Order Polarization for Multifrequency Coherent Anti-Stokes Raman Scattering.

A set of complete and unambiguous third order polarizations were derived for multifrequency (broadband) Coherent Anti-Stokes Raman Scattering (CARS) in molecular gases. The final macroscopic equations were expressed in a laboratory reference frame to facilitate the interpretation of experimental results. These polarizations were made applicable for a general vector polarization and any nonoverlapping frequency distribution of the electric field. The derivation started with the dipole moment approximation and two laser fields as inputs. These two fields, the pump and Stokes, had their central frequencies separated by a Raman rotational -vibrational frequency. The generated fields considered were also limited to two; an anti-Stokes and second Stokes. To more accurately determine the field amplitudes, polarizations were also developed for five related third order nonlinear processes; two parametric (Coherent Stokes Raman Scattering and Coherent Mixed Raman Scattering) and three nonparametric (Stimulated Raman Scattering at the Stokes, pump and second Stokes frequencies). Coherence properties of the fields were not included. Simplification of the general results obtained to the accepted monochromatic equations was noted from the final results. A semiclassical approach was used in the polarization derivation. Both classical and fully quantized excursions were made for understanding and to clarify the results. Specifically, the induced third order molecular polarizability was obtained from quantum mechanical perturbation theory. This polarizability was then combined with the electric field present to define the dipole moment of the molecule. By applying classical orientational averaging to a random distribution of molecules, the gas polarization was derived. This derivation was performed using laboratory coordinates and clearly established the role of each polarization vector component. The gas polarization was defined by the set of equations for each of the six nonlinear processes and four central frequencies. From the equations for the polarization, macroscopic susceptibilities were defined that depend on pseudo polarizability invariants. These invariants were shown to have the same form as the Raman polarizability invariants. By carefully deriving the Raman differential scattering cross section and establishing its dependence on the Raman polarizability invariants, the relationship between the susceptibilities and the correct Raman scattering cross section was determined.