# Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter
## 2 Materials and Methods
Taking into account Equations (1) and (2), a final expression for \(\chi^{(3)}\) can be straightforwardly derived:  
\[\chi^{(3)}(\omega_{as};\omega_{1},-\omega_{2},\omega_{3})=\frac{1}{12}N\frac{ \partial\alpha}{\partial q}\kappa\int\limits_{0}^{\infty}\left(\frac{S_{as}(t )}{S_{as}(0)}\right)^{\frac{1}{2}}e^{-i\omega t}dt, \tag{5}\]  
Thus, the experimental CARS transient signal (\(S_{as}(t_{d})\)) can be used as a source function to obtain the imaginary and real parts of \(\chi^{(3)}\) with the help of Equation (5). The density matrix approach [28] will also bring the same result as the time evolution of the non-diagonal elements and is identical to the one of the coherent amplitude. Coherently driven phonons have been the subject of detailed studies in the past [29]. The experimental design exploits a multicolor femtosecond pulse laser set up that is used here to characterize solid state and biological media. The design also ensures imaging capability with sub-micron spatial resolution. The detection part of the experimental set up is shown in Figure 1. As was mentioned above, three colors are needed to enable time-domain CARS. This is ensured by having time-synchronized optical parametric oscillator (OPO) pulses at two independently tunable wavelengths [30; 31]. The OPOs are driven by the same pump source, a high-power mode-locked Ti:sapphire oscillator (Coherent Mira HP, 3.5 W, 76 MHz, \(t_{p}\) = 110 fs, 720-960 nm). The third color is a small fraction that is split from the Ti:sapphire oscillator output. The OPO gain material is periodically poled lithium tantalate (PPSLT) with a grating period that can be selected for a particular parametric process. Typical power outputs from the OPOs are, depending on wavelength, between 200-375 mW, their pulsewidths are 115-150 fs, and their pulse bandwidths are in the range of 95-126 cm\({}^{-1}\). Continuous tuning and wavelength adjustment at a fixed grating period is accomplished by using intracavity Lyott filters and by varying OPO cavity lengths. The three beams are directed to a galvo scanner and imaged onto the entrance pupil of an apochromat objective lens (OBJ) with high numerical aperture (NA = 1.2). A photomultiplier tube (PMT) with high gain and quantum efficiency was used to detect signal photons at wavelengths selected by a combination of bandpass filters and diffraction gratings. The output from the PMT was digitized by a high-speed data acquisition card (DAQ). Figure 1: _Cont._  
Software code (written in Labview) was used for synchronized control of the galvo scanner, acquisition of the signal, and for the formation and display of images in 2D. With the available full angle scan provided by galvo drivers and with the DAQ card, an area of about 180 \(\times\) 180 \(\upmu\)m\({}^{2}\) could be scanned up to a maximum rate of 4 frames/s. Another software code was used to provide control of the optical delay line and acquire a time-domain CARS signal. Typical point signal average times for sample images and time-domain CARS data were in the range of 1-200 ms, depending on the type of measurement and desired signal-to-noise ratio. Three-color pulse CARS images at different time-delays can be generated with a spatial resolution of 380 nm. We were able to trace transient non-resonant CARS signals, generated in the forward direction, within higher than five orders of magnitude in solid glass material while the average power of the combined beams was kept under 30 mW (\(\sim\)0.15 nJ/pulse per beam) at the sample. The non-resonant time-domain signal in glass (see Figure 1c) represents instrument function showing our capability to detect decay in coherence with a characteristic dephasing time (\(T_{2}\)) shorter than 150 fs. This is shown by comparing the experimental signal (blue ovals) with simulated CARS transients assuming \(G(t-t^{\prime})=\langle q_{i}(t)q_{i}(t^{\prime})\rangle=h(t-t^{\prime})\exp( -|t-t^{\prime}|/T_{2})\), where \(h(t)\)--Heavyside function and \(T_{2}\) were set to 150 fs (black curve) and 250 fs (red curve).