## 1 Introduction  
Over the past ten years, stimulated Raman scattering (SRS) microscopy has been investigated in nanophotonics [1; 2; 3; 4] as well as in biophotonics as an analytical, label-free, non-invasive technique with unique cellular and tissue imaging capabilities [5; 6; 7; 8]. As almost all the biomolecules contain carbon and hydrogen, the CH-stretching (2800-3100 cm\({}^{-1}\)) region of Raman spectra of biomolecules is the most used in SRS microscopy. The two Raman bands typically investigated are CH\({}_{2}\), near to 2845, and CH\({}_{3}\), near to 2930 cm\({}^{-1}\), corresponding to lipids and proteins, respectively. Due to their large spectral shapes (about 100 cm\({}^{-1}\)) and the difference between peaks of 95 cm\({}^{-1}\), the CH\({}_{2}\) and CH\({}_{3}\) are partially overlapped. Moreover, in the C-H region, the SRS signal level is high because the density of CH bonds is high, while the SRS molecular specificity is assumed to be low. Typically, SRS microscopy is implemented by using two Fourier transform-limited (FTL) tunable picosecond (ps) laser sources with a high spectral resolution (\(\approx\)10 cm\({}^{-1}\)), helpful in the region of interest (i.e., the fingerprint region: 800 \(\div\) 1800 cm\({}^{-1}\)), where Raman peaks are narrow, nearly spaced, and could be packed [8]. However, when ps laser pulses are used to implement an SRS microscope, an equally fruitful imaging in carbon-hydrogen (C-H) stretching is achieved. The drawback is that ps pulses show a low peak intensity, thus needing high laser power for imaging [5; 6; 7; 8]. In the last decade, an improvement of about one order of magnitude of the SRS signal has been demonstrated when ps pulses are replaced with femtosecond (fs) pulses [9]. This improvement is due to a higher peak intensity; thus, higher signal-to-noise ratio (SNR) can be obtained when temporally shorter pulses are used than narrowband, picosecond pulses with an unchanged optical power. However, when ultra-fast sources are used, a low spectral selectivity is obtained and multi-band excitation can occur, not allowing, in principle, the separation of some bands of particular interest in biology, such as those of lipids and proteins as discussed previously. To solve this issue, a number of methods for SRS multicolor imaging, based on broadband femtosecond pulses, have been developed. Among them, frequency tuning [10], multiplexing [11; 12] and spectral focusing [13; 14; 15; 16] implementations have been studied and reported in the literature. It is well known that the light pulse is considered transform-limited when the angular frequency is constant and equals the central angular frequency \(\omega(t)=\omega_{0}\). On the other hand, the chirp of an optical pulse is generally understood as the time dependence of its instantaneous frequency; thus, a chirped pulse having a carrier frequency \(\omega_{0}\) at time \(t\) shows an instantaneous central frequency \(\omega(t)\) that depends on the linear chirp parameter \(\beta\) by the equation: \(\omega(t)=\omega_{0}+2\beta t\)[17]. In detail, a down-chirp (up-chirp) means that the instantaneous frequency decreases (increases) with time. A pulse can gain a chirp, for example, through propagation in a transparent medium due to the effects of chromatic dispersion and nonlinearities. Indeed, due to its wide spectral width and to group velocity dispersion, optical pulse propagating in a transparent medium undergoes a phase distortion inducing an increase in its duration with a different laser frequency distribution in time, as reported in Figure 1. In particular, the time-bandwidth product \(\Delta\omega_{0}\cdot\tau_{0}=4\ln 2\approx 2.77\) corresponds to the area of the ellipse on the left in Figure 1 and it is the bandwidth of an FTL laser pulse.