## 2 Optical Frequency Comb Self-Referencing
### CEO Stabilization Loop
When a tight lock cannot be achieved, the stabilized signal keeps a finite linewidth and one generally refers to this situation as a loose lock (Figure 5a). Figure 4: Scheme of principle of a PLL for coherent stabilization of _f_CEO to an external reference frequency, with its three main components: (i) a phase detector \(D\) to measure the phase fluctuation between the signal and the reference to produce an error signal; (ii) a PID servo-controller \(G\) that amplifies the error signal; and (iii) an actuator \(C\) that tunes _f_CEO from a control voltage. A simple approximation to determine the linewidth of a laser or of a beat signal from an arbitrary frequency noise spectrum was proposed by [PERSON] _et al._, based on the concept of the \(\beta\)-separation line [47]. This line shown in Figure 5 and defined as \(S_{\delta\
u}(f)\ =(8\ln(2)/\pi^{2})\cdot f\) splits the frequency noise PSD \(S_{\delta\
u}(f)\) into two areas with a strongly different impact on the lineshape. Only the noise components for which \(S_{\delta\
u}(f)\ >(8\ln(2)/\pi^{2})\cdot f\), corresponding to the slow modulation area, contribute to the FWHM linewidth of the signal. The other components for which \(S_{\delta\
u}(f)\ <(8\ln(2)/\pi^{2})\cdot f\) (fast modulation area) only affect the wings of the lineshape, without contributing to the FWHM linewidth. A simple approximation of the FWHM linewidth is obtained from the surface \(A\) of the slow modulation area corresponding to the geometrical area under the frequency noise PSD obtained for all Fourier frequencies for which \(S_{\delta\
u}(f)\) exceeds the \(\beta\)-separation line [47]:  
\[FWHM=\sqrt{8\ln(2)A} \tag{4}\]  
\[A=\int H(S_{\delta\
u}(f)-8\ln(2)f/\pi^{2})\ S_{\delta\
u}(f)df \tag{5}\]  
with \(H(x)\) being the Heaviside step unit function (\(H(x)=1\) if \(x\geq 0\) and \(H(x)=0\) if \(x<0\)). An experimental validation of the approximated linewidth given by Equation (5) performed with CEO  
Figure 5: **(a)** Loose lock and **(b)** tight lock achieved for the CEO beat in an Er:Yb:glass DPSSL comb [27, 28]. The upper graphs show the RF spectrum with the presence of a coherent carrier in the case of the tight lock. The lower graphs show the corresponding frequency noise PSD for the free-running (grey curves) and stabilized (blue curves) CEO beat. The linewidth of the locked CEO beat can be approximated using the \(\beta\)-separation line introduced by [PERSON] _et al._[47] (dashed red line). In the case of a loose lock **(a)**, a finite linewidth is obtained that is proportional to the colored area for which the frequency noise PSD exceeds the \(\beta\)-separation line. For a tight lock **(b)**, the frequency noise PSD is completely reduced below the \(\beta\)-separation line, which corresponds to a FWHM linewidth reduced to zero that is representative of the presence of a coherent carrier. beat signals of fiber and DPSSL frequency combs proved its accuracy (within the experimental uncertainties) over more than three decades of linewidth values [48]. The concept of the \(\beta\)-separation line offers a simple tool to evaluate the feedback bandwidth that is required to achieve a tight lock in a stabilization loop [47]. As a tight lock is characterized by the suppression of the signal linewidth, the stabilization loop must be able to reduce all noise components that contribute to the linewidth below the \(\beta\)-separation line. The highest frequency contributing to the signal linewidth corresponds to the crossing point between the frequency noise PSD and the \(\beta\)-separation line. Therefore, this crossing point constitutes a useful metric for the required loop bandwidth. The most commonly used actuator to tune and stabilize \(f_{\mathrm{CEO}}\) in DPSSLs and fiber lasers is the current of the pump diode. The pump current has a direct influence on the pulse parameters of the femtosecond laser, such as pulse duration and energy, which directly translates into a change of \(f_{\mathrm{CEO}}\).