## 2 Analytical Derivation  
### Self-Phase Modulation by a Linearly Chirped Pulse  
We consider that an optical pulse is subject to temporal phase modulation depending on its intensity profile, _i.e._, self-phase modulation (SPM). Assuming that the temporal profile is Gaussian, the Fourier-transform-limited (FTL) pulse is represented in the time and frequency domains, respectively, as,  
\[E_{1}(t)=E_{0}\exp(-at^{2}) \tag{1}\]  
\[\tilde{E}_{1}(\omega)=\frac{E_{0}}{\sqrt{2a}}\exp(-\frac{\omega^{2}}{4a}) \tag{2}\]  
where \(a=2\ln 2/\Delta t^{2}\) and \(\Delta t\) is the pulsewidth (full width at half maximum: FWHM). Applying a group delay dispersion (GDD) \(\phi^{\prime}\)to Equations (1) and (2), the corresponding chirped pulses are expressed in the frequency and time domains, respectively, as follows.  
\[\tilde{E}_{2}(\omega)=\frac{E_{0}}{\sqrt{2a}}\exp\left[-\frac{\omega^{2}}{4a} (1+i\phi^{\prime})\right] \tag{3}\]  
\[E_{2}(t)=\frac{E_{0}}{\sqrt{1+i\phi^{\prime}}}\exp\left(-\frac{at^{2}}{1+i \phi^{\prime}}\right) \tag{4}\]  
Here, \(\phi^{\prime}\)is the normalized GDD given by  
\[\phi^{\prime}=2a\phi^{\prime} \tag{5}\]  
which is a dimensionless quantity roughly equal to a pulse stretching ratio with regard to the FTL pulse having the same spectral width. For example, \(\phi^{\prime}=10\) implies that the pulse is stretched by a factor of approximately 10.  
When a linearly chirped pulse is subject to SPM, the temporal profile of the optical intensity is given by  
\[I_{2}(t)=I_{0}\exp\!\left(-\frac{2 at^{2}}{1+\mathcal{O}^{\prime 2}}\right)=I_{0} \!\left(1-\frac{2 at^{2}}{1+\mathcal{O}^{\prime 2}}\right) \tag{6}\]  
where \(I_{0}\) is the peak intensity. It is noted that in Equation (6) the main part of the pulse can be approximated by a quadratic function. The nonlinear refractive index is expressed as follows.  
\[n(t)\approx n_{0}+n_{2}I_{0}-n_{2}I_{0}\frac{2 at^{2}}{1+\mathcal{O}^{\prime 2 }} \tag{7}\]  
Considering only the time-dependent term, _i.e._, the third term, Equation (4) can be rewritten as,  
\[E_{3}(t)=\frac{E_{0}}{\sqrt{1+i\mathcal{O}^{\prime}}}\exp\!\left[-\frac{at^{2}} {1+\mathcal{O}^{\prime 2}}\{1-i(2B+\mathcal{O}^{\prime})\}\right] \tag{8}\]  
where \(B\) (=\(\frac{2\pi}{\lambda_{0}}n_{2}I_{0}L\)) is the \(B\)-integral, \(\lambda_{0}\) is the center wavelength, and \(L\) is the interaction length. The Fourier transform of Equation (8) yields  
\[\tilde{E}_{3}(\omega)=\frac{E_{0}}{\sqrt{2\,a}}\sqrt{\frac{1-i\mathcal{O}^{ \prime}}{1-i(2B+\mathcal{O}^{\prime})}}\exp\!\left[-\frac{\omega^{2}(1+ \mathcal{O}^{\prime 2})}{4a\{1-i(2B+\mathcal{O}^{\prime})\}}\right] \tag{9}\]  
The GDD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as  
\[\mathcal{O}^{\prime}_{OUT}=\frac{(1+\mathcal{O}^{\prime 2})(2B+\mathcal{O}^{ \prime})}{1+\left(2B+\mathcal{O}^{\prime}\right)^{2}}\mathcal{O}^{\prime} \tag{10}\]  
where as the real part gives the spectral width (FWHM)  
\[\Delta\omega=\frac{4\ln 2}{\Delta t}\sqrt{\frac{1+\left(2B+\mathcal{O}^{ \prime}\right)^{2}}{1+\mathcal{O}^{\prime 2}}} \tag{11}\]  
It is noted that \(\mathcal{O}^{\prime}_{OUT}\) is a dimensionless normalized quantity, being the same as in Equation (5).