## Appendix 4 Derivation of the Equation 22  
Equation 22 gives the spectral phase in the case of minimum deviation and at central frequency, \(\omega_{0}\). This relation is obtained using Equation 21, which is valid for any frequency. At central frequency, as represented on Figure 6, the following relations take place:  
\[ \begin{split}\theta_{4}\bigl{(}\omega_{0}\bigr{)}& =\theta_{1}\bigl{(}\omega_{0}\bigr{)}\\ \theta_{2}\bigl{(}\omega_{0}\bigr{)}&=\beta\,/\,2 \end{split}  \tag{12}\]  
Taking into account Equation 12 in Equation 21 at \(\omega=\omega_{0}\) and using the central wavelength, \(\lambda_{0}\), instead of the central frequency, \(\omega_{0}\), one obtains immediately Equation 22.  
Appendix 5 Discussion on the Possibility to Stabilize the CEP of a ML Laser Oscillator with an EO System, Outside the Oscillator Cavity  
We discuss here the ability of an EO prism pair system to stabilize the CEP of a mode-locked (ML) laser oscillator. The electric field at the output of a mode locked system can be written:  
\[E(t)=\sum_{n}E_{p}\left(t-n\tau\right)e^{i\left(\omega_{0}-n\omega_{\tau}+n \omega_{\tau}+\omega_{0}\right)} \tag{13}\]  
In this formula, \(\tau=1/\,f_{rup}\) is the period between two pulses, \(\omega_{0}\) is the circular carrier wave frequency and \(\Delta\varphi_{CE}\) is the phase slippage (modulo \(2\pi\)) between two pulses. Applying Fourier transform and Fourier series theory, one obtains the following expression [28]:  
\[E\bigl{(}\omega\bigr{)}=e^{i\omega_{0}}\tilde{E}_{p}\left(\omega-\omega_{c} \right)\sum_{m}\delta\bigl{(}\Delta\varphi_{ce}-\omega\tau-2\pi m\bigr{)} \tag{14}\]  
where \(\delta\) is the Dirac function. This corresponds, as is well known, to a frequency comb, with frequencies given by:  
\[\omega_{m}=-\Delta\varphi_{ce}\bigl{/}\tau+2\pi m\bigr{/}\tau=f_{0}+mf_{rep} \tag{15}\]A method was recently proposed and demonstrated to stabilize the CEP of the optical pulse outside the oscillator cavity [29]. This method is based on an acousto-optic device in order to shift the frequency comb. As EO systems are currently used in the ns domain as frequency shifters through a linear temporal phase sweep, it is natural to wonder whether an EO phase CEP shifter could be used to stabilize the CEP of the pulse train outside the ML oscillator by shifting the comb frequencies. To our knowledge, the only way to apply a linear temporal phase shift on every pulse is to use a sinusoidal phase modulator at frequency f\({}_{\text{rep}}\) (or an integer multiple of f\({}_{\text{rep}}\)), as shown on Figure 11. The electric field at the output of the modulator can then be written:  
\[E(t)= \left(\sum_{n}E_{p}\left(t-n\tau\right)e^{i\left(\omega_{p}-n\omega_{p}+n \omega_{p}+\omega_{0}\right)}\right)e^{i\tau\sin\left(\omega_{up}\tau\right)} \tag{100}\]  
where \(\Gamma\) is the amplitude modulation index, which depends linearly on the radio frequency (RF) electric field applied to the crystal, and \(\omega_{rep}\) is the angular frequency corresponding to \(f_{rep}\). Applying the Anger-Jacobi development [30] to the last term of Equation 100 leads to:  
\[E(t)= \left(\sum_{n}E_{p}\left(t-n\tau\right)e^{i\left(\omega_{p}-n\omega_{p}+n \omega_{p}+\omega_{0}\right)}\right)\left(\sum_{-\infty}^{\infty}J_{k}\left( \Gamma\right)e^{i\omega_{up}\tau}\right) \tag{101}\]  
In the spectral domain, the electric field has the following form:  
\[\tilde{E}\left(\omega\right)\propto e^{i\omega_{0}}\left(\tilde{E}_{p}\left( \omega-\omega_{0}\right)\sum_{m}\delta\left(\omega-m\omega_{rep}+\omega_{0} \right)\right)\ast\left(\sum_{-\infty}^{\infty}J_{k}\left(\Gamma\right)\delta \left(\omega-k\omega_{rep}\right)\right) \tag{102}\]  
where the symbol * stands for the convolution product and \(J_{k}\) are Bessel functions.