Time and Frequency-Resolved Spectroscopy

 In principle, multiple frequency experiment can be preformed in the frequency domain with CW or nanosecond pulses. Frequency resolved techniques provide precise energies of molecular transitions and relative populations of molecular states. However, the implementation of the short femtosecond pulses in coherent Raman experiments can provide many potential advantages: increased signal size, rejection of the nonresonant background, and simultaneous excitation over a broad spectral range and high time resolution.
 The bandwidth of femtosecond pulses (few hundreds of wave numbers) is too broad to give adequate resolution in a frequency-domain approach. On the other hand their temporal width (tens of femtosecond) is too long to resolve the important core vibrations of large molecules in a time-domain approach. Collecting data simultaneously in the combined time and frequency domains can provide an additional control as to the resolution in both, providing better utilization of experimental data, without, of course, violating the imposed by the Fourier limits of resolution.
 Every coherently excited molecular vibration contributes to the signal with a bandwidth equal to the bandwidth of the excitation pulses. Moreover, since the detected coherent signal is a result of homodyne measurement (absolute square of the signal is measured on the square law detector) signal at some vibrational frequency Ω_υcan be referred to as the interference of the emitted electric field at that vibrational frequency with a “DC” field. Therefore, when transform limited Gaussian probe pulse is assumed , the spectral dependence of the observed nuclear response is determined by the multiplication of the probe field spectrum, centered at the carrier frequency ω_c with a field centered at the vibrational frequency ω_c+Ω_υ of width that is determined either by the excitation pulses or by the molecular parameters. According to this model, the spectral distribution of some Raman active mode at Ω_υwill be simply described by a Gaussian centered at Ω_υ/2 (result of the multiplication of two Gaussians). Furthermore when several vibrations are involved we expect an additional contributions to the signal with frequencies that are determined by the difference of the individual vibrations (so called homodyne beats). Once again the beat frequency between two modes with frequencies Ω_A and Ω_B reaches its maximum intensity precisely at the mean frequency (Ω_A + Ω_B)/2.
 In the data presented below for each time delay between the excitation pair of pulses and the probe pulse (τ) the entire spectrum of the signal produced by CHCl₃ is measured by a spectrometer with sufficiently high resolution (0.3 nm)
 Fig. 3a depicts a composite set of measurements, where for each delay (between 500-2600 fs, measured each 20 fs, horizontal axis) the entire spectrum of the DFWM signal was recorded. The measured data was cutoff before the τ = 500 fs in order to remove the intensive non-resonant response (so called coherence peak). The intensity at each detection frequency (at a given detuning form the central frequency at 800nm, vertical axis) is color coded to generate this two dimensional map.
 

                      
Figure 3 a) The full combined time/frequency data. The carrier frequency was 795nm. b) Spectrogram for neat chloroform.

 Taking the one dimensional Fourier transform of the time resolved signal (for each detuning separately), generates a new 2-D map of vibrational mode intensity along the horizontal axis, and detuning frequency along the vertical axis. Fig 3b depicts a 2-D spectrogram showing accurate detuning behavior of the different spectral components.
 In fig. 3b, the most pronounced features are the fundamental vibrational modes of chloroform at 261cm^-1 and 365cm^-1 and the homodyne beats at 104 cm^-1, 315cm^-1 and the additional spectral component at 626cm^-1. In order to extract the accurate detuning dependence for these spectral features, slices along the relevant frequencies are analyzed and shown in fig. 4.
 Fig. 4a and b depicts the amplitude of the signal at the 261cm^-1 and 365cm^-1 vibrational mode as a function of detuning from the carrier frequency. In the case of the 261 cm^-1 vibrational mode the observed behavior exhibits maximum around zero detuning while for the 365 cm^-1 vibrational mode there is a double maximum around 200cm^-1 and -200cm^-1 detuning from a carrier frequency (183cm^-1  and -183cm^-1 expected ) .
 We believe that double maxima behavior originates from a contribution of the two spectroscopic pathways (for more details, see arxiv 0907.3625) . In principle, we expect the 261cm^-1 mode to show similar double maxima (around 130cm^-1 and -130cm^-1 detuning from carrier frequency) behavior as well. But due to the fact that our femtosecond pulses have a broad 400cm^-1 FWHM these two maxima for the 261 cm^-1 line are not resolved.

                                        
 

Figure 4 a) The observed detuning dependence profile for 261cm^-1 vibrational mode of chloroform. b) The observed detuning dependence profile for 365cm^-1 vibrational mode of chloroform. c) The observed detuning dependence profile for 104cm^-1 frequency beat associated with υ₃-υ₆. d) The observed detuning dependence profile for 315cm^-1 frequency beat associated with υ₂-υ₃.
 

 The experimental profile for the homodyne beats at 104cm^-1 (associated with υ₃-υ₆) and 315cm^-1 (associated with υ₂-υ₃) are shown in fig. 4c and d. Both of the components show the double maxima behavior around 320cm^-1 for 104cm^-1 beat and 540 cm^-1 for 315cm^-1 homodyne beat. Those results are in good agreement with the expected values of 313cm^-1 and 523cm^-1 respectively.
The results shown above are in qualitative agreement with the predictions of our model. The distortion of some of the line shapes in the detuning profiles and the differences in the relative intensities might be related to the pulse chirp and require more detailed analysis.