^{1,a)}, Paul J. Kinnius

^{1}, Sukesh Roy

^{2}and James R. Gord

^{3}

### Abstract

A theoretical analysis of coherent anti-Stokes Raman scattering (CARS) spectroscopy of gas-phase resonances using femtosecond lasers is performed. The time-dependent density matrix equations for the femtosecondCARS process are formulated and manipulated into a form suitable for solution by direct numerical integration (DNI). The temporal shapes of the pump, Stokes, and probe laser pulses are specified as an input to the DNI calculations. It is assumed that the laser pulse shapes are Gaussians and that the pulses are Fourier-transform limited. A single excited electronic level is defined as an effective intermediate level in the Raman process, and transition strengths are adjusted to match the experimental Raman polarizability. The excitation of the Raman coherence is investigated for different -branch rotational transitions in the fundamental band of diatomic nitrogen, assuming that the pump and Stokes pulses are temporally overlapped. The excitation process is shown to be virtually identical for transitions ranging from to . The excitation of the Raman coherences is also very efficient; for laser irradiances of , corresponding approximately to a , pulse focused to , approximately 10% of the population of the ground Raman level is pumped to the excited Raman level during the impulsive pump-Stokes excitation, and the magnitude of the induced Raman coherence reaches 40% of its maximum possible value. The theoretical results are compared with the results of experiments where the femtosecondCARS signal is recorded as a function of probe delay with respect to the impulsive pump-Stokes excitation.

Funding for this research was provided by the National Science Foundation, Combustion and Plasmas Program, under Award No. 0413623-CTS, by the U.S. Department of Energy, Division of Chemical Sciences, Geosciences and Biosciences, under Grant No. DE-FG02-03ER15391, by the Air Force Office of Scientific Research, and by the Air Force Research Laboratory, Propulsion Directorate, Wright-Patterson Air Force Base, under Contract No. F33615-03-D-2329. Stimulating discussions with Professor Marlan O. Scully are gratefully acknowledged.

I. INTRODUCTION AND MOTIVATION

II. DENSITY MATRIX ANALYSIS FOR THE INTERACTION OF A TWO-STATE RESONANCE WITH MONOCHROMATIC LASER RADIATION

A. Density matrix elements for transition between states in ground level and states in excited level

B. Summary of the equations for the density matrix elements for the CARS interaction

C. Solution of the density matrix equations for the CARS interaction

D. Calculation of the CARS signal at frequency

III. IMPULSIVE PUMP-STOKES EXCITATION OF -BRANCH RESONANCES IN NITROGEN

IV. FEMTOSECONDCARS SIGNAL GENERATION: INTERACTION OF THE PROBE LASER WITH THE INDUCED RAMAN COHERENCE

V. CONCLUSIONS AND FUTURE WORK

### Key Topics

- Stimulated Raman scattering
- 67.0
- Coherence
- 44.0
- Polarization
- 21.0
- Matrix equations
- 14.0
- Irradiance
- 13.0

## Figures

Schematic diagram of the energy level structure for the femtosecond CARS calculations for the transition in the fundamental (1,0) Raman band of the nitrogen molecule. The (, 1, 2, 3, and 4) transition strengths were varied to match the Raman cross section of .

Schematic diagram of the energy level structure for the femtosecond CARS calculations for the transition in the fundamental (1,0) Raman band of the nitrogen molecule. The (, 1, 2, 3, and 4) transition strengths were varied to match the Raman cross section of .

Temporal dependence of the real and imaginary components and the magnitude of the induced Raman coherence for the (a) , (b) , (c) , (d) , and (e) transitions in the fundamental (1,0) band of . The coherence matrix elements are normalized by dividing by the population of level prior to laser excitation. The difference between the central frequencies of the pump and Stokes lasers is . The Raman frequencies for the , , , , and transitions are 2329.8, 2329.4, 2328.0, 2325.8, and , respectively. The collisional dephasing rate for each Raman transition is , corresponding to a Raman linewidth of . The pump and Stokes laser temporal pulse shapes are both Gaussian with widths of (full width at half maximum). The pump and Stokes pulses are overlapped exactly in time. The peak irradiance for both the pump and Stokes pulses is .

Temporal dependence of the real and imaginary components and the magnitude of the induced Raman coherence for the (a) , (b) , (c) , (d) , and (e) transitions in the fundamental (1,0) band of . The coherence matrix elements are normalized by dividing by the population of level prior to laser excitation. The difference between the central frequencies of the pump and Stokes lasers is . The Raman frequencies for the , , , , and transitions are 2329.8, 2329.4, 2328.0, 2325.8, and , respectively. The collisional dephasing rate for each Raman transition is , corresponding to a Raman linewidth of . The pump and Stokes laser temporal pulse shapes are both Gaussian with widths of (full width at half maximum). The pump and Stokes pulses are overlapped exactly in time. The peak irradiance for both the pump and Stokes pulses is .

Temporal dependence of the phase of the induced Raman coherence for the , , and transitions in the fundamental (1,0) band of . The collisional dephasing rate and the pump and Stokes pulse parameters are the same as given in the caption of Fig. 2.

Temporal dependence of the phase of the induced Raman coherence for the , , and transitions in the fundamental (1,0) band of . The collisional dephasing rate and the pump and Stokes pulse parameters are the same as given in the caption of Fig. 2.

Temporal dependence of the excited state populations and the normalized excited populations for the , , , , and transitions in the fundamental (1,0) Raman band of . The collisional dephasing rate and the pump and Stokes pulse parameters are the same as given in the caption of Fig. 2.

Temporal dependence of the excited state populations and the normalized excited populations for the , , , , and transitions in the fundamental (1,0) Raman band of . The collisional dephasing rate and the pump and Stokes pulse parameters are the same as given in the caption of Fig. 2.

Response of the Raman transition to excitation at two different irradiance levels. The real and imaginary components and the magnitude of the induced Raman coherence are plotted in (a) and (b) for peak pump and Stokes irradiances of and , respectively. The normalized excited level population is plotted in (c) and (d) for peak pump and Stokes irradiances of and , respectively. The transition frequency is , and the difference between the central frequencies of the pump and Stokes lasers is . All resonance and laser parameters are the same as listed in the caption of Fig. 2 except for the peak irradiances for the pump and Stokes pulses.

Response of the Raman transition to excitation at two different irradiance levels. The real and imaginary components and the magnitude of the induced Raman coherence are plotted in (a) and (b) for peak pump and Stokes irradiances of and , respectively. The normalized excited level population is plotted in (c) and (d) for peak pump and Stokes irradiances of and , respectively. The transition frequency is , and the difference between the central frequencies of the pump and Stokes lasers is . All resonance and laser parameters are the same as listed in the caption of Fig. 2 except for the peak irradiances for the pump and Stokes pulses.

Comparison of the results from the measurement of the FWM signal as a function of probe delay and DNI theoretical calculations for room air. For the experiment, the pump, Stokes, and probe pulse energies were 10, 100, and , respectively. The measured pulse widths were approximately . For the calculations, the peak irradiances for the pump, Stokes, and probe beams were , , and , respectively, corresponding to an estimated focal diameter for each beam. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

Comparison of the results from the measurement of the FWM signal as a function of probe delay and DNI theoretical calculations for room air. For the experiment, the pump, Stokes, and probe pulse energies were 10, 100, and , respectively. The measured pulse widths were approximately . For the calculations, the peak irradiances for the pump, Stokes, and probe beams were , , and , respectively, corresponding to an estimated focal diameter for each beam. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

Calculation of the CARS signal from and CARS plus nonresonant background signal from room air. The peak irradiances for the pump, Stokes, and probe beams were (a) , , and , respectively; (b) , , and , respectively; (c) , , and , respectively; (d) , , and , respectively; and (e) , , and , respectively. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

Calculation of the CARS signal from and CARS plus nonresonant background signal from room air. The peak irradiances for the pump, Stokes, and probe beams were (a) , , and , respectively; (b) , , and , respectively; (c) , , and , respectively; (d) , , and , respectively; and (e) , , and , respectively. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

Normalized CARS signal from room air. The CARS signal for peak irradiances of is the base case. For each curve, the calculated CARS signal is divided by the peak CARS signal for the base case, and by the normalized peak irradiance product . The peak irradiance products for each curve are indicated in the legend. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

Normalized CARS signal from room air. The CARS signal for peak irradiances of is the base case. For each curve, the calculated CARS signal is divided by the peak CARS signal for the base case, and by the normalized peak irradiance product . The peak irradiance products for each curve are indicated in the legend. The collisional dephasing rate and the laser pulse parameters for the calculations are the same as given in the caption of Fig. 2.

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