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 (ω_l-ω_s≃ω₂₁, where ω_l,s are the frequencies of the pump fields with powers P_l,s, and ω₂₁ is the frequency of the Raman transition |1〉→|2〉) with the possible addition of a one-photon resonance (ω_l≃ω₃₁ is the frequency of the electronic transition |1〉→|3〉). For these cases we show that the CARS polarization p_CARS 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 ω_l is far from resonance, or for the three-level system |1〉,|2〉, and |3〉, when ω_l≃ω₃₁. The steady-state expression for p_CARS in the former case gives p_CARS∝P_lP_s^1/2 at low powers and p_CARS∝P_l⁰P_s^-1/2 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 ρ₂₁ 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, p_CARS∝P_l⁰P_s^1/2 for ω_s≃ω₃₂ and p_CARS∝(P_lP_s)^1/2 when ω_s≃ω₃₂±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 p_CARS 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, p_CARS∝P_l^1/2P_s⁰ for ω_l≃ω₃₁ and ω_s≃ω₃₂, and the CARS spectrum is split into five components.