# Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter
## 3 Results and Discussion
The presence of the triplet line for the cluster centered at 830 cm\({}^{-1}\) can be explained by shifted frequencies for vibrations of different symmetry within the Ti-O-P intergroup. Raman studies of oil have recently attracted a strong interest in different fields of research. Unlike in crystal lattice, molecules in oil can move freely. Therefore, CARS signal decay is caused by pure dephasing due to collisions and translational motion of molecules. Collisions themselves result in random phase shifts in q\({}_{i}\). The contribution of the translational motion results in phase changes that are linear in time, due to the Doppler effect in between successive collisions. The dephasing rate due to collisions is uniform and does not depend on velocity distribution. The dephasing due to the Doppler effect (translational motion) is accounted for by averaged phase drift in time due to thermal velocity distribution. For a Gaussian distribution, the ensemble averaging results in a real function for \(Q(t)\)  
Figure 3: (**a**) CARS transients (open circles) obtained from KTP crystal when the OPO wavelengths were set to 967 nm and 1052 nm (\(\sim\)830 cm\({}^{-1}\) vibration). The solid line represents the theoretical fit that assumes three spectral components and is obtained using a software algorithm that varies multiple parameters for each component (amplitude, spectral position, dephasing time), assuming a Raman doublet with 62 cm\({}^{-1}\) frequency difference and decaying with the dephasing times of 950 fs and 1160 fs; (**b**) Corresponding spectrum of the resonant third-order nonlinearity (\(\chi^{(3)}(\omega)\)) obtained using experimental CARS transient and applying Equation (5). The imaginary part of \(\chi^{(3)}\) is shown in blue open circles. The solid black line is the best fit to the spectrum assuming three components shown in separate colors and with parameters provided in the text; (**c**) The real part of \(\chi^{(3)}(\omega)\). Namely, \(\mathcal{Q}(t)=q_{0}e^{-(\Gamma t-f(t))}\) where \(f(t)=\alpha t^{2}\) for \(t\ll t_{v}\) and \(f(t)=\beta t\) for \(t>t_{v}\)[35]. Constants \(\alpha\) and \(\beta\) are molecular parameters (e.g., mean velocity, diffusion rate coefficient, etc.). Constant \(t_{v}\) is velocity correlation time. Figure 4a shows the time-domain CARS signal for olive oil. The driving fields (\(E_{1}\) and \(E_{2}\)) have been wavelength tuned to match the Raman active mode frequency of \(\sim\)1072 cm\({}^{-1}\). Based on spontaneous Raman data, this is a relatively weak C-C stretching vibration. The CARS transient (Figure 4a) shows a characteristic decay time of about 1.4 ps. The estimated diffusional dephasing time in oil is \(\mathrm{T}^{*}_{\mathrm{D}}=2/\beta\)\(\sim\)20 ms. Thus, the obvious conclusion is that the dephasing is entirely due to collisions. Also, the CARS transient exhibits clear modulation (beat pattern), suggesting the presence of a Raman doublet. The corresponding parameters for dephasing times of 2.8 ps and 1.9 ps provided the best fit to the data. By applying a solution provided by Equation (5) to the CARS transient, we can find that the strongest line component on Figure 4b fits fairly well into a Lorentzian profile with linewidth \(\Delta\
u_{L1}\) = 3.8 \(\pm\) 0.1 cm\({}^{-1}\), while the second component is noticeably broader (\(\Delta\
u_{L2}\) = 5.6 cm\({}^{-1}\)). The most recent data available in the literature for the vibrational mode shows a single peak with a \(\sim\)7 cm\({}^{-1}\) linewidth [36]. The discrepancy is explained by a lack of sensitivity and resolution (\(\sim\)5 cm\({}^{-1}\)) to provide more precise data. We have applied our method and experimental capabilities to enable quantitative microscopy where, as an example, we characterized biological tissue. We demonstrate imaging of tissue constituents, and then perform dephasing time measurements in targeted areas of the tissue by driving and probing in time characteristic Raman active vibrations. Figure 5a shows a CARS image of mouse tissue (dermis area).