Ultrafast saturation of electronic-resonance-enhanced coherent anti-Stokes Raman scattering and comparison for pulse durations in the nanosecond to femtosecond regime

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.

Figure 1

Six-level model used for studying ERE-CARS saturation of NO. The rotational manifolds in the ground (excited) vibrational state in the ground electronic state $X{}^2\mathrm{\Pi }$ are represented by $|c\rangle$ and $|c^{\prime }\rangle$ and rovibrational states in the excited electronic state $A{}^2{\mathrm{\Sigma }}^{+}$ are represented by $|a\rangle$ and $|a^{\prime }\rangle$. The pump, Stokes, probe, and generated CARS pulses that are represented by $E_1, E_2, E_3$, and $E_4$, respectively, couple the unprimed states $|a\rangle , |b\rangle$, and $|c\rangle$. The prime states are considered to be the bath levels to account for all of the decays ${\mathrm{\Gamma }}_{ij}$ in the system and the coherence dephasing is represented by ${\gamma }_{ij}$.

Figure 2

(a) Population in the excited electronic state $|a\rangle$, (b) excited vibrational state $|b\rangle$, (c) ground-state (Raman) coherence, and (d) instantaneous polarization in the ERE-CARS transition for different probe intensities. The following parameters are used for the plot: the peak of the pump and Stokes pulse appears at 400 fs and the probe delay is maintained at 200 fs.

Figure 3

Saturation of ERE-CARS signal intensity for different pulse durations in the hybrid ERE-CARS configuration. Pump- and Stokes-pulse durations are maintained at ${\tau }_1={\tau }_2=100$ fs and only the probe-pulse duration ${\tau }_3$ is varied. (a) The ERE-CARS signal $S_{\text{ERE}}$ plotted as a function of peak intensity of the probe $I_{30}$ and (b) pulse area ${\mathrm{\Theta }}_3$ corresponding to $I_{30}$.

Figure 4

Saturation of ERE-CARS signal intensity for different probe delays.

Figure 5

(a) Saturation of integrated ERE-CARS signal intensity for different pulse durations in the longer-pulse regime between 10 ns and 10 ps. (b) Pulse area corresponding to peak intensities of the probe pulse.

Figure 6

(a) Saturation of integrated ERE-CARS signal intensity for different pulse durations in the shorter-pulse regime between 10 ps and 10 fs. (b) Pulse area corresponding to peak intensities of the probe pulse.