## 2 Optical Frequency Comb Self-Referencing
### CEO Stabilization Loop  
The detection of a CEO beat with a sufficient SNR by _f_-to-2\(f\) interferometry after SC generation constitutes a first mandatory step for the self-referencing stabilization of a frequency comb. A modelocked laser is sometimes said to be self-referenceable once this CEO detection has been achieved. However, the subsequent stabilization of \(f_{\rm CEO}\) is not a straightforward step, especially to achieve a tight CEO lock. The most common method for CEO stabilization consists of phase-locking \(f_{\rm CEO}\) to an external reference oscillator, or to a sub-harmonic of the repetition rate \(f_{\rm rep}\) in some cases, using a phase-locked loop (PLL). A first requirement for a proper operation of such a PLL is to have a sufficient SNR of the CEO beat. A rule-of-thumb typically requires a SNR larger than \(\sim\)25 dB. Figure 3: Octave-spanning SC spectrum generated in a PCF from the output of an Yb:CALGO DPSSL with gigahertz repetition rate: comparison between the measured (blue) and simulated (red) spectra. The Raman soliton at 1360 nm and the dispersive wave at 680 nm (shown by the yellow areas) are distant of one octave and have a high degree of coherence \(g^{1}\) shown by the black curve displayed with respect to the right vertical axis; they are used to generate the CEO beat in an _f_-to-2\(f\) interferometer. (Reproduced with permission from [21]. © The Optical Society of America, 2014). A PLL is typically made of three distinct components (Figure 4): (i) a phase detector that measures the phase fluctuations between _f_CEO and the reference signal to generate an error signal; (ii) a proportional-integral-derivative (PID) servo-controller that amplifies the error signal to produce a correction voltage; and (iii) an actuator that enables controlling_f_CEO from the correction signal. A large phase noise generally affects the CEO beat, whose free-running linewidth is typically in the range from a few kilohertz [27] to a few megahertz [43]. Consequently, a double-balanced mixer (DBM) cannot be directly used as a phase detector as a result of its narrow operation range. The CEO beat is thus either frequency-divided before being compared to a reference signal in a DBM, or a digital phase detector with a much larger linear phase response is used to measure the phase fluctuations of the CEO beat. For instance, in our comb CEO stabilization experiments [21, 27, 28, 44], we typically used the digital phase detector DXD200 [45] that is also implemented in the frequency combs of the market leader MenloSystems (Martinsried/Munich, Germany) and provides a large linear phase response of \(\pm 64\pi\). The availability of a suitable actuator to control _f_CEO is another key component for the successful implementation of a PLL to phase-stabilize the CEO beat. The actuator should provide a linear control of _f_CEO over a sufficiently large range to keep a stable long-term lock. The bandwidth of the actuator, _i.e.,_ the speed at which it can act on _f_CEO to correct for fast fluctuations, is another key parameter to achieve a proper stabilization. A broad linewidth of the free-running CEO beat mainly arises from high frequency noise components. Therefore, a fast actuator is required to correct for these components if a significant linewidth reduction is aimed for, for instance to achieve a tight phase lock. A tight lock is characterized by the suppression of the full width at half maximum (FWHM) linewidth of the signal and the appearance of a coherent peak (delta function) in the center of the RF spectrum, surrounded by a noise pedestal that corresponds to the residual phase noise of the signal (Figure 5b). A tight lock is typically achieved when the integrated phase noise \(\Delta\Phi_{rms}=\left[\int S_{\phi}(f)df\right]^{1/2}\) is smaller than 1 [46], where \(S_{\phi}(f)\) is the phase noise power spectral density (PSD). In such a case, a fraction \(\eta=\exp[-\int S_{\phi}(f)df]\) of the signal power is in the coherent peak.