## 2 Analytical Derivation
### Self-Phase Modulation by a Nonlinearly Chirped Pulse  
Applying both GDD and third-order dispersion (TOD) \(\mathcal{O}^{\prime\prime}\) to a FTL pulse, the chirped pulse is expressed in the frequency and time domains, respectively, as follows.  
\[\tilde{E}_{4}(\omega)=\frac{E_{0}}{\sqrt{2\,a}}\exp\!\left[-\frac{\omega^{2} }{4a}(1+i\mathcal{O}^{\prime})-i\frac{\omega^{3}\mathcal{O}^{\prime\prime}}{1 2a\omega_{0}}\right] \tag{12}\]  
\[E_{4}(t)=\frac{E_{0}}{\sqrt{1+i\mathcal{O}^{\prime}}}\exp\!\left[-\frac{at^{2 }}{1+i\mathcal{O}^{\prime}}-\frac{2\mathcal{O}^{\prime\prime}\left\{1-3 \mathcal{O}^{\prime 2}-i\mathcal{O}^{\prime\prime}(3-\mathcal{O}^{\prime 2}) \right\}}{3\omega_{0}\left(1+\mathcal{O}^{\prime 2}\right)^{3}}a^{2}t^{3}\right] \tag{13}\]Here, \(\mathbf{\phi^{\prime\pi}}\) is the normalized TOD given by  
\[\mathbf{\phi^{\prime\pi}}=2a\omega_{0}\phi^{\prime\pi} \tag{14}\]  
where \(\omega_{0}\) is the center wavelength. It should be noted that this is also a dimensionless quantity. Although the Fourier transform of the exponential function including the cubic function is the Airy function, we use following approximations assuming that the contribution of the cubic term is relatively small, _i.e._, \(\left|b\dot{t}^{2}\right|>\left|ct^{3}\right|\) and \(\left|b\dot{\omega}^{2}\right|>\left|c\dot{\omega}^{3}\right|\) in the following equations, respectively.  
\[\mathcal{F}\left[\exp(-bt^{2}-ct^{3})\right]\approx\frac{1}{\sqrt{2b}}\exp \left[-\frac{\omega^{2}}{4b}-ic\left(\frac{\omega}{2b}\right)^{3}\right] \tag{15}\]  
\[\mathcal{F}^{-1}\left[\exp(-b\omega^{2}-ic\omega^{3})\right]\approx\frac{1}{ \sqrt{2b}}\exp\left[-\frac{t^{2}}{4b}-c\left(\frac{t}{2b}\right)^{3}\right] \tag{16}\]  
The same treatment given in the previous section is applied to Equation (13), including the time-dependent refractive index.  
\[\begin{split} E_{s}(t)=&\frac{E_{0}}{\sqrt{1+i\mathbf{ \phi^{\prime\pi}}}}\exp\left[-\frac{at^{2}}{1+\mathbf{\phi^{\prime\pi}}^{2}}\left\{ 1-i\left(2B+\mathbf{\phi^{\prime\pi}}\right)\right\}\right.\\ &\left.-\frac{2\mathbf{\phi^{\prime\pi}}\left\{1-3\mathbf{\phi^{\prime 2}}-i \left(2B(1-3\mathbf{\phi^{\prime\pi}}^{2})+\mathbf{\phi^{\prime\pi}}(3-\mathbf{\phi^{ \prime\pi}}^{2})\right)\right\}}{3\omega_{0}\left(1+\mathbf{\phi^{\prime\pi}}^{2} \right)^{3}}a^{2}t^{3}\right]\end{split} \tag{17}\]  
The Fourier transform of Equation (17) based on Equation (16) yields  
\[\begin{split}\tilde{E}_{s}(\omega)=&\frac{E_{0}}{ \sqrt{2a}}\sqrt{\frac{1-i\mathbf{\phi^{\prime\pi}}}{1-i\left(2B+\mathbf{\phi^{\prime \pi}}\right)}}\exp\left[-\frac{\omega^{2}}{4a\left\{1-i\left(2B+\mathbf{\phi^{ \prime\pi}}\right)\right\}}\left(1+\mathbf{\phi^{\prime\pi}}^{2}\right)\right.\\ &\left.-i\frac{\mathbf{\phi^{\prime\pi}}\left\{1-3\mathbf{\phi^{\prime 2}}-i \left(2B(1-3\mathbf{\phi^{\prime\pi}}^{2})+\mathbf{\phi^{\prime}}(3-\mathbf{\phi^{\prime \pi}}^{2})\right)\right\}}{12a\omega_{0}\left[1-i\left(2B+\mathbf{\phi^{\prime \pi}}\right)\right]^{3}}\omega^{3}\right]\end{split} \tag{18}\]  
The TOD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as  
\[\mathbf{\phi^{\prime\pi}_{OUT}}=\frac{\left(1-3\mathbf{\phi^{\prime\pi}}^{2}\right) \left\{1-\left(2B+\mathbf{\phi^{\prime\pi}}\right)^{4}\right\}+2\mathbf{\phi^{\prime \pi}}\left(1+\mathbf{\phi^{\prime\pi}}^{2}\right)\left(2B+\mathbf{\phi^{\prime\pi}} \right)\left\{3-\left(2B+\mathbf{\phi^{\prime\pi}}\right)^{2}\right\}}{\left\{1+ \left(2B+\mathbf{\phi^{\prime\pi}}\right)^{2}\right\}^{3}}\mathbf{\phi^{\prime\pi}} \tag{19}\]  
It should be noted again that this is a dimensionless quantity. Both the GDD and spectral width of the output pulse are independent of the initial TOD, being the same as shown in Equations (10) and (11), respectively. This is because the contribution of TOD is assumed to be small. Strictly, the GDD and spectral width would be weak functions of the initial TOD.