## 2 Optical Frequency Comb Self-Referencing
### CEO Stabilization Loop
CEO stabilization by pump current modulation has thus been implemented in various DPSSLs that will be described in more details in Sections (3) and (4) [21, 25, 26, 27, 28, 29, 30, 31]. However, the pump current acts onto \(f_{\mathrm{CEO}}\) through the gain medium, and the corresponding modulation bandwidth is limited by the upper-state lifetime of the gain material that induces a roll-off in the modulation response of \(f_{\mathrm{CEO}}\). The upper-state lifetime is in the millisecond range for many important diode-pumped Yb and Er gain materials, which limits the CEO modulation bandwidth to the kilohertz or tens of kilohertz range [21, 26, 27]. Extending the feedback bandwidth beyond the limit of the gain upper-state lifetime can be achieved using a phase-lead filter in the loop [26], or by the use of a derivative component in the servo-controller. However, other approaches to circumvent the gain lifetime limitation and to act on the CEO beat frequency at a higher speed have been proposed based on the use of intra-cavity loss-modulation. As an example, a graphene electro-optic modulator was used in reflection to stabilize the CEO frequency in a Tm-fiber oscillator with a large feedback loop bandwidth of \(>\)1 MHz [16]. Another method to overcome the gain lifetime limitation is opto-optical modulation (OOM) of a semiconductor device inside the cavity. Using a SESAM as a fast actuator for CEO frequency control in a modelocked DPSSL resulted in a feedback loop bandwidth \(>\)100 kHz [44]. The results will be described in more details in Section 3.3. A high SNR of the free-running CEO beat with a low enough frequency noise are prerequisites to achieve the self-referencing of an optical frequency comb as previously discussed in this section. However, these conditions may not be sufficient in some particular cases where the dynamics of \(f_{\mathrm{CEO}}\) may play a crucial role. Such a case was shown and investigated by [PERSON] _et al._ in an Er:Yb:glass DPSSL where \(f_{\mathrm{CEO}}\) is controlled by the pump current [49] and another case will be discussed in Section 4 for a TDL. In the Er:Yb:glass laser, a reversal point occurred in the static response of \(f_{\mathrm{CEO}}\) as a function of the pump current \(I_{\mathrm{pump}}\), where the slope of \(\partial f_{\mathrm{CEO}}/\partial f_{\mathrm{pump}}\) changed its sign (Figure 6a). A tight phase-lock of \(f_{\mathrm{CEO}}\) to an external frequency reference of 20 MHz was straightforwardly achieved at pump currents located below the reversal point, whereas no stabilization at all (even a loose lock) could be achieved at pump currents located on the other side of the reversal point, despite a slightly higher SNR and a very similar frequency noise of the free-running CEO beat. The reason behind this effect was shown to be related to the dynamic response of \(f_{\mathrm{CEO}}\) to pump current modulation, obtained by measuring the transfer function of \(f_{\mathrm{CEO}}\) for a modulation of the pump current. Not only the amplitude, but also the phase of the transfer function is of particular importance for the realization of a stable feedback loop. According to a general rule-of-the-thumb in control systems, a sufficient phase margin is needed at the unity gain frequency to keep a feedback loop stable [50]. As a consequence, the overall phase shift accumulated throughout the loop has to be kept sufficiently higher than \(-180^{\circ}\) to prevent the system to turn into a positive feedback loop leading to an unstable operation. In the considered laser, the high frequency range of the transfer function (above \(\sim\)1 kHz) did not depend on the operating pump current (neither in amplitude nor in phase), whereas a significant difference occurred in the low frequency range. In particular, the phase showed a detrimental change of \(180^{\circ}\) between low and high pump current (Figure 6c).