# Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter
## 3 Results and Discussion
This is governed by the exponential decay trend with a time constant that is clearly longer than the one observed for \(\
u_{1}\)(\(A_{1g}\)) and \(\
u_{2}\)(\(E_{g}\)) modes, as well as narrower linewidths. Multiple Raman active modes can be driven by \(E_{1}\), \(E_{2}\)-pulses at the same time, given fairly broad bandwidth pulses. When KTP phonon modes centered at \(\sim\)830 cm\({}^{-1}\) are targeted, a complex nature of Raman active vibrations in the crystal is clearly seen in the detected CARS transients. Figure 3a shows the time-domain CARS signal obtained for the case of coherently driven vibrations with the center wavelengths for \(E_{1}\), \(E_{2}\)-pulses adjusted to 967 nm and 1052 nm, respectively. The signal carries a well pronounced modulation pattern that represents a coherent beat signal due to interference of at least two spectral components. This is governed by the exponential decay trend with a time constant that is clearly longer than the one observed for \(\
u_{1}(A_{1g})\) and \(\
u_{2}(E_{g})\) modes, as well as narrower linewidths. Raman spectra retrieved from the time-domain signal are represented by the imaginary part of \(\chi^{(3)}(\omega)\) and shown in Figure 3b. Figure 3c shows the corresponding real part of \(\chi^{(3)}(\omega)\) in the frequency domain. Three separate components can be clearly seen in the retrieved spectra. The corresponding separations between the components are 18.5 cm\({}^{-1}\) and 45.9 cm\({}^{-1}\). This is in fairly good agreement with reported spontaneous Raman scattering data that provided the range of 20.4-29.0 cm\({}^{-1}\) and 34.6-39.2 cm\({}^{-1}\) for the two [34]. Linewidths (FWHM) for the three components are 9.1 \(\pm\) 0.4 cm\({}^{-1}\), 7.5 \(\pm\) 0.6 cm\({}^{-1}\), and 11.2 \(\pm\) 0.5 cm\({}^{-1}\), respectively. This compares fairly well to ranges of 10.2-12.6 cm\({}^{-1}\), 9.2-10.8 cm\({}^{-1}\), and 14.0-16.4 cm\({}^{-1}\), respectively, that have been reported for the spontaneous Raman spectroscopy measurements. A component amplitude ratio of 23:4:15 can also  
Figure 2: (**a**) Time-domain CARS signal (open circles) obtained from KTP crystal when the OPO wavelengths were set to 1019 nm and 1098 nm (\(\sim\)706 cm\({}^{-1}\)). The red solid line represents the theoretical fit that assumes single exponential decay for \(G(t)\) with the dephasing time of 495 fs; **(b**) Time-domain CARS signal (open circles) obtained from KTP crystal when the OPO wavelengths were set to 967 nm and 1041 nm. The solid line represents theoretical fit that assumes a Raman doublet with 62 cm\({}^{-1}\) frequency difference and decaying with the dephasing times of 495 fs and 515 fs; (**c,d**) Corresponding spectra of the resonant third-order nonlinearity obtained with the help of the equation presented in the text. The real part of the third-order susceptibility (\(\chi^{(3)}\)) is shown in black open circles, while the imaginary part is shown in blue circles. Solid lines show the results of simulations for both parts of the nonlinearity, assuming a Lorentzian profile for the two components with parameters outlined in the text. be retrieved and represents the best fit to the imaginary part of the \(\chi^{(3)}\) spectra. As was mentioned above, due to a slower decay, phonon linewidths are more than two times narrower when compared to the high frequency modes (\(\
u_{1}(A_{1g})\) and \(\
u_{2}(E_{g})\)), as shown in the results presented in Figure 2. This is explained by the fact that the \(\
u_{1,2}\) modes have overtone decay channels with a higher phonon density of states for resulting lower energy phonons in parametric decay processes. Thus, lack of the efficient decay channels for the \(\sim\)830 cm\({}^{-1}\) mode leads to a significantly lower damping rate for the driven vibrations. We believe that the mode belongs to Ti-O-P intergroup vibrations opposite to the former case (i.e., \(\sim\)700 cm\({}^{-1}\)), representing vibrations within the TiO\({}_{6}\) atomic group.