Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers
Pl,s, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization
pCARS is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system
∣1〉 and
∣2〉 when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system
∣1〉,∣2〉, and
∣3〉, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for
pCARS in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ 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 ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When
Pl is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where
V13 is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ 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, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.
The saturation threshold of a probe pulse in an ultrafast electronic-resonance-enhanced (ERE) coherent anti-Stokes Raman spectroscopy (CARS) configuration is calculated. We demonstrate that while the underdamping condition is a sufficient condition for saturation of ERE-CARS with the long-pulse excitations, a transient gain must be achieved to saturate the ERE-CARS signal for the ultrafast probe regime. We identify that the area under the probe pulse can be used as a definitive parameter to determine the criterion for a saturation threshold for ultrafast ERE-CARS. From a simplified analytical solution and a detailed numerical calculation based on density-matrix equations, the saturation threshold of ERE-CARS is compared for a wide range of probe-pulse durations from the 10-ns to the 10-fs regime. The theory explains both qualitatively and quantitatively the saturation thresholds of resonant transitions and also gives a predictive capability for other pulse duration regimes. The presented criterion for the saturation threshold will be useful in establishing the design parameters for ultrafast ERE-CARS.
A generalized formulation is presented for determining the saturation thresholds for optical processes excited by ultrafast pulses based on the pulse area of the excitation pulse. It is demonstrated that the threshold of driving-pulse intensity for absorption and fluorescence saturation in a two-level system is inversely proportional to the square of the duration of the excitation pulse. These results are obtained from both a simplified analytical solution assuming a Gaussian excitation pulse shape and a detailed numerical calculation based on density-matrix equations. The calculation is generalized further to obtain the saturation condition for a two-photon Raman process by defining a two-photon pulse area both analytically and numerically. These results not only provide predictive capabilities for determining thresholds of signal saturation using ultrashort durations with arbitrary pulse shapes and durations but also open up possibilities for predetermining the threshold intensities of various resonant nonlinear processes.
The role of collisional decay on the evolution of molecular coherence and excited-state population in an electronic-resonance-enhanced coherent anti-Stokes Raman scattering (ERE-CARS) configuration is studied. A four-level model scheme is proposed and a density-matrix equation is derived to determine the system evolution. It is shown that even for significantly large collisional decays, a suitable (rather strong) probe-laser intensity prevents significant depletion of the excited-state population and enhances the ground-state coherence. A physical understanding is developed for the reported insensitivity [Roy et al. Appl. Phys. Lett. 2006, 89, 104105] of the ERE-CARS signal to the rate of collisional decay at the excited electronic level.