## 1 Introduction
As a result of the chirp, laser pulses can undergo a temporal stretch and the final pulse duration is \(\tau=F\tau_{0}\), where \(\tau_{0}\) is the FTL pulse duration and \(F\) is the stretching factor; at the same time, the instantaneous spectral bandwidth becomes narrower than the FTL spectral bandwidth by a factor of \(1/F\)[18]. With the relation \(\Delta\omega\cdot\tau=2.77\) also being applicable to the chirped pulse width \(\tau\) and to the instantaneous bandwidth \(\Delta\omega\) of the pulse, the duration broadening leads to a decrease in the instantaneous bandwidth by the stretching factor, whereas the whole bandwidth \(\Delta\omega_{0}\) is left unchanged [19], as depicted in Figure 1.  
In order to enhance spectral resolution in SRS microscopy based on fs laser pulses, an option is to force a quadratic spectral phase variation. By equally chirping pump and Stokes beams with an energy spacing corresponding to the Raman line, it is possible to generate a constant instantaneous frequency difference (IFD, \(\Omega=\omega_{\mathrm{p}}-\omega_{\mathrm{s}}\)) that spectrally focuses the excitation energy into a single resonance. We note that the bandwidth \(\delta\Omega\), which ultimately determines the SRS spectral resolution, in the limiting case can simulate the ps SRS system. This method is known as spectral focusing (SF) [13; 14; 15; 16]. Nevertheless, the great disadvantage related to this approach is the large amount of parameters that should be taken into account in the selection and alignment of the optics to obtain the chirp-matching condition. Moreover, since the operative conditions can be altered by  
Figure 1: Time-bandwidth distribution for (**a**) Fourier-transform limited laser pulse and (**b**) the same laser pulse that is linearly chirped. fluctuations in the pump and Stokes wavelengths together with the dispersion in the microscope, perfect chirp-matching is difficult to maintain. For these reasons, the resulting spectral resolution of SF-SRS setups is often worse than theoretically predicted [13; 14; 15; 16]. The comparison among FTL, equally and differently chirped laser sources is shown in Figure 2. In the spectral focusing, when the pump and Stokes pulses are \"chirp-matched\", the bandwidth of IFD is lower than the total bandwidths of the transform-limited pulses, leading to an improvement of the spectral resolution, as reported in Figure 2b. On the other hand, when the chirp-matching condition is not met, the bandwidth of the IFD signal is broader, and, as a consequence, the spectral resolution is poorer with respect to the chirp matched case (see Figure 2c). Interestingly, spectral resolution for differently chirped lasers is better than FTL laser sources [19]. For microscopy, both contrast imaging and spectral tuning are important to assess the good quality of images. However, in coherent Raman scattering, there is a tradeoff between the best spectral resolution, which is reached with ps sources, and the best ratio of image contrast and signal intensity, which is obtained when the spectral resolution and the width of the Raman lines under observation are almost the same [14]. This condition is satisfied in the case of excitation with both ps pulses, since they match the linewidths in the fingerprint region (5-20 cm\({}^{-1}\)), and in the case of broader bandwidth femtosecond (fs) pulses, which are considered the ideal excitation for CH stretching vibrations. We note that the SF approach usually takes advantage by static dispersive elements, such as glass rods, placed into the optical path to stretch both the pump and the Stokes frequencies in time and produce a constant IFD. However, the optical elements embedded in an SRS optical circuitry microscope setup can also change the pulse width.