# Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter
## 2 Materials and Methods  
Third-order nonlinear polarization \(\left(P^{(3)}\right)\) is the origin of the scattered signal at shifted frequency in coherent Raman spectroscopy. Here, we will consider a case of coherent anti-Stokes Raman scattering (CARS) with three incident fields involved. The nonlinear polarization is then expressed in terms of the third-order susceptibility (\(\chi^{(3)}\) (\(\omega_{as}\); \(\omega_{1}\), \(-\omega_{2}\), \(\omega_{3}\))) and is a nonlinear response of the medium to incident optical fields \(E_{1}\), \(E_{2}\), \(E_{3}\):  
\[P^{(3)}(\omega_{as})=6\chi^{(3)}(\omega_{as};\omega_{1},-\omega_{2},\omega_{3} )E_{1}E_{2}^{*}E_{3}, \tag{1}\]  
If one chooses to represent the polarization through the molecular susceptibility tensor (\(\alpha\)), the nonlinear polarization expression can be written as following:  
\[P^{(3)}(\omega_{as})=\frac{1}{2}N\frac{\partial\alpha}{\partial q}Q(\omega_{1 }-\omega_{2})E_{3}, \tag{2}\]  
\(N\)--density of molecules, \(Q\)--coherent amplitude of molecular or lattice vibrations, \(\omega_{as}\)--anti-Stokes frequency. The macroscopic coherence is the statistical average of molecular displacements (\(q_{i}\)) with Raman frequency (\(\omega_{R}\)) and independent phases (\(\varphi_{i}\)). Upon the action of the driving pair of pulses (\(E_{1}\), \(E_{2}\)), the displacements are forced to accept common phase (\(\varphi_{1}-\varphi_{2}\)). The difference in frequency of the pair (\(\omega_{1}-\omega_{2}\)) is tuned to match the Raman active vibration (\(\omega_{R}\)) in order to achieve the maximum value of \(Q\). After the action of the two fields, the coherence decays freely in time as a result of inter- and intra-atomic/molecular interactions that lead to changes in the individual phases (pure dephasing) [25; 26]. Third wave/pulse \(E_{3}\), which can be delayed in time with respect to the driving fields, is scattered on the time-dependent coherence \(Q(t)\), producing a signal at the anti-Stokes frequency (\(\omega_{as}\)). The coherent amplitude can be represented in terms of an instantaneous response function \(G(t)\), which is the characteristic of the ensemble oscillating at Raman frequency (\(\omega_{R}=\omega_{1}-\omega_{2}\)), coupled to the finite width driving pulses [26]:  
\[Q(t)=\kappa_{12}\int\limits_{-\infty}^{\infty}G(t^{\prime}-t)E_{1}(t^{\prime} )E_{2}(t^{\prime})dt^{\prime} \tag{3}\]  
Time-domain CARS signal (\(S_{as}(t_{d})\)) can be expressed as the following:  
\[S_{as}(t_{d})=\kappa\int\limits_{-\infty}^{\infty}|Q(t)|^{2}|E_{3}(t-t_{d})|^ {2}dt, \tag{4}\]  
The \(\kappa\) constant represents combined field coupling coefficients to the molecular/atomic system (\(\kappa_{12}\)) for \(E_{1}\), \(E_{2}\) fields and the \(E_{3}\) field to the coherent amplitude. The above equations for the coherent amplitude and CARS signal are Fredholm type-I integral equations and can be solved numerically or using the Fourier transform method [27]. The spectra and autocorrelation functions of the \(E_{1}\), \(E_{2}\), and \(E_{3}\) pulses are known and can be measured. Provided the condition that \(Q(t)\) is a real function, a Fourier transform of the response function \(G(t)\) can be obtained. This is true for many types of vibrational systems for which diffusional phase shifting events are minimized or non-existent. Therefore, fine features in the spectra can be effectively resolved in the vicinity of Raman active vibrations driven by fairly broadband ultrashort pulses. In the case of extremely short optical pulses, or when \(G(t)\) does not change much on a timescale of the pulsewidths,the integrations are not necessary to calculate the signal (\(S_{as}(t)\)).