Saturation effects in coherent anti-Stokes Raman scattering

Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\omega }_l-{\omega }_s\simeq {\omega }_{21}$, where ${\omega }_{l,s}$ are the frequencies of the pump fields with powers $P_{l,s}$, and ${\omega }_{21}$ is the frequency of the Raman transition $|1\mathrm{}〉\to |2\mathrm{}〉$) with the possible addition of a one-photon resonance (${\omega }_l\simeq {\omega }_{31}$ is the frequency of the electronic transition $|1\mathrm{}〉\to |3\mathrm{}〉$). For these cases we show that the CARS polarization ${\mathcal{p}}_{\mathrm{CARS}}$ is proportional to the off-diagonal density-matrix element ${\rho }_{21}$. In order to determine ${\rho }_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system $|1\mathrm{}〉$ and $|2\mathrm{}〉$ when ${\omega }_l$ is far from resonance, or for the three-level system $|1\mathrm{}〉,|2\mathrm{}〉$, and $|3\mathrm{}〉$, when ${\omega }_l\simeq {\omega }_{31}$. The steady-state expression for ${\mathcal{p}}_{\mathrm{CARS}}$ in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\propto P_l{P}_s^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\propto P_l^0{P}_s^{\frac{-1\mathrm{}}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\rho }_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When $P_l$ is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\propto P_l^0{P}_s^{\frac{1}{2}}$ for ${\omega }_s\simeq {\omega }_{32}$ and