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 (ωₗ-ωₛω₂₁, where ωl,s are the frequencies of the pump fields with powers Pl,s, and ω₂₁ is the frequency of the Raman transition |1〉→|2〉) with the possible addition of a one-photon resonance (ωₗω₃₁ is the frequency of the electronic transition |1〉→|3〉). For these cases we show that the CARS polarization pCARS is proportional to the off-diagonal density-matrix element ρ₂₁. In order to determine ρ₂₁, we use Laplace transforms to solve the Bloch equations for the effective two-level system $|1〉$ and $|2〉$ when ωₗ is far from resonance, or for the three-level system $|1〉,|2〉$, and $|3〉$, when ωₗω₃₁. The steady-state expression for pCARS in the former case gives pCARS∝PₗPₛ1/2 at low powers and pCARS∝Pₗ⁰Pₛ^-12 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 ρ₂₁ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When Pₗ is high, for example, pCARS∝Pₗ⁰Pₛ1/2 for ωₛω₃₂ and pCARS∝(PₗPₛ)1/2 when ωₛω₃₂±V₁₃ where V₁₃ is the one-photon Rabi frequency for the |1〉→|3〉 transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for pCARS is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, pCARS∝Pₗ1/2Pₛ⁰ for ωₗω₃₁ and ωₛω₃₂, and the CARS spectrum is split into five components.