# Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter
## 3 Results and Discussion  
Figure 2a shows the time-domain CARS signal obtained in potassium titanyl phosphate (KTiOPO4) crystal at a condition where one of the strongest fundamental modes has been driven by a pair of pulses at 1019 nm and 1098 nm. The crystal is flux-grown and was angle cut at \(\varphi\) = 40\({}^{\circ}\) and \(\theta\) = 0\({}^{\circ}\). This ensures that the \(A_{1}\) and \(A_{2}\) symmetry tensor components are involved [32] when \(E_{1}\), \(E_{2}\), and \(E_{3}\) beam polarizations are set to parallel. The targeted mode is assigned to an asymmetric stretching mode vibration (\(\
u_{1}\)(\(A_{1g}\))) within the TiO6 octahedron [33]. The resonant contribution is dominant, as evidenced by an absence of a resolution limited peak at zero time delay. The experimental CARS signal can be fitted very well by a single exponential decay function for \(G(t)\) suggesting that, as expected for high quality crystals, the Raman line is homogenously broadened and corresponds to the loss of vibrational energy via a phonon parametric process (rather than pure dephasing) that primarily includes an overtone decay channel. The best fit yields a phonon decay time constant \(T_{2}\) = 495 \(\pm\) 10 fs that translates to a linewidth (\(\Delta\
u\) = 1/(\(\pi c\)\(T_{2}\))) of 21.4 \(\pm\) 0.5 cm\({}^{-1}\). In the case of 967 nm and 1041 nm combination for \(E_{1}\), \(E_{2}\)-pulses, the time domain signal shows clear modulation corresponding to quantum beats of two components separated by \(\sim\) 62 cm\({}^{-1}\). The experimental signal shown in Figure 2b can be fit well when the two line components (\(\omega_{R1}\) = 706 cm\({}^{-1}\) and \(\omega_{R2}\) = 768 cm\({}^{-1}\)) are involved. The corresponding response function is then \(G(t)=h(t)^{*}\left(A_{1}^{*}\exp\left(-t/T_{21}-i\omega_{R1}^{*}t\right)+A_{2} ^{*}\exp\left(-t/T_{22}-i\omega_{R2}t\right)\right)\). The best fit is achieved when the amplitude ratio is 9:5 and the corresponding \(T_{21}\), \(T_{22}\) times are 495 \(\pm\) 10 fs and 515 \(\pm\) 10 fs. It is obvious that in this case, due to the broad bandwidths of \(E_{1}\), \(E_{2}\)-pulses (\(\sim\)85 cm\({}^{-1}\)), the two components can be driven with nearly equal strength opposite to the result shown in Figure 2a. In other words, the result shown in Figure 2a corresponds to a phonon mode targeting condition where primarily only one phonon mode centered at (i.e., \(\omega_{1}-\omega_{2}\)\(\sim\)706 cm\({}^{-1}\)) is driven and probed. This results in absence of the beat pattern and decay of the signal that is best fit with the single exponential decay response function presented above. Figure 2c,d show the corresponding spectra for \(\chi^{(3)}\), both for real and imaginary parts, obtained using the formalism described in Section 2 that required solutions for the integral Equations (3) and (4). Solid lines are simulated spectra assuming Lorentzian lineshape and line parameters that have been mentioned above. This highlights the asymmetry seen in the spectra obtained using the experimental data. The best fit to the imaginary part is obtained assuming three components with ratios of 26:15:11, linewidths of 21 cm\({}^{-1}\), 24 cm\({}^{-1}\), 17 cm\({}^{-1}\) and line separations of 65 cm\({}^{-1}\) and 16 cm\({}^{-1}\), respectively. 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.