## 2 Optical Frequency Comb Self-Referencing
### CEO vs. CEP Stabilization  
The previously discussed frequency comb self-referencing where \(f_{\mathrm{CEO}}\) is phase-locked to an external RF reference signal consists of stabilizing the CEO frequency \(f_{\mathrm{CEO}}\). In such a case, the carrier slips by a constant phase difference \(\Delta\phi_{\mathrm{CEO}}\) relative to the pulse envelope between two consecutive pulses according to Equation (2), but this phase slip is not zero. As a consequence, the phase difference between the carrier and the envelope (the so-called carrier-envelope phase (CEP)) varies from pulse to pulse and consecutive pulses do not exactly reproduce themselves. An identical pulse is retrieved only after a certain number of pulses given by the ratio \(f_{\mathrm{rep}}/f_{\mathrm{CEO}}\) (Figure 7a). Most of the combs applications in frequency metrology or spectroscopy rely on such CEO-stabilized combs where the absolute phase of the carrier within the pulse is irrelevant. It is only in the particular case where \(f_{\mathrm{CEO}}=0\) that a constant CEP is achieved and successive pulses are identical (Figure 7b). In such a case, the comb is said to be CEP-stabilized. This condition is stronger than CEO-stabilized, as it requires a constant  
Figure 6: **(a)** Static tuning curve of \(f_{\mathrm{CEO}}\) with respect to the pump current showing a reversal point at \(I_{\mathrm{pump}}\approx 860\) mA. The CEO frequency is stabilized to 20 MHz in this DPSSL comb, which can be realized in principle at two pump currents located on each side of the reversal point. **(b)** Amplitude and **(c)** phase of the dynamic response of \(f_{\mathrm{CEO}}\) to pump current modulation measured at different pump currents below the reversal point (green curves), at the reversal point (yellow curve) and above the reversal point (orange to red curves). carrier phase to be achieved with respect to the pulse envelope in all pulses and not only a constant change of the phase difference between the carrier and pulse envelope from one pulse to the next. Therefore, a CEP-stabilized laser is always also CEO-stabilized, whereas the reverse is generally not true (excepted in the particular case of \(f_{\mathrm{CEO}}\) = 0). However, even if all pulses are identical when \(f_{\mathrm{CEO}}\) = 0, their absolute CEP remains arbitrary and different CEP values lead to the same comb frequencies \(\
u_{N}\) = \(Nf_{\mathrm{rep}}\). CEP stabilization is of particular importance in some time domain experiments where a physical effect under study occurs for a single pulse. For instance, extreme nonlinear optical experiments are directly sensitive to the electric field and not to the intensity of a light pulse. Therefore, they directly depend on the \"absolute\" CEP [51]. A typical example is high harmonic generation (HHG) where a gas target illuminated by an intense laser pulse emits high harmonics of the incident light frequency, which is a means to produce extreme ultra-violet frequency combs. Currently it is possible to determine and stabilize the change of the carrier-envelope phase between successive pulses via the CEO frequency as previously described. However, extending this to the direct measurement of the carrier-envelope phase itself is much more challenging. Stabilizing the CEO frequency to \(f_{\mathrm{CEO}}\) = 0 is a first step in this direction, as it enables the production of a train of identical pulses (zero carrier-envelope phase shift \(\Delta\phi_{\mathrm{CEO}}\)). However, the absolute CEP is still not controlled in  
Figure 7: Illustrative examples of the correspondence between frequency (left) and time (right) representations of a pulse train emitted by a modelocked laser [51]. The red oscillation shows the evolution of the carrier and the blue line represents the pulse envelope.