## 2 Optical Frequency Comb Self-Referencing
### Nonlinear Interferometry for _f_CEO Detection  
The physical origin of _f_CEO arises from the difference between group and phase velocities inside the laser resonator, which induces a slippage \(\Delta\phi_{\text{CEO}}\) between the carrier phase and the pulse envelope from one pulse to the next in the time domain (see Figure 1a). From the properties of the Fourier transform that links the time and frequency domains, this corresponds to a frequency shift _f_CEO in the spectral domain (Figure 1b):  
\[f_{\text{CEO}}=\frac{\Delta\phi_{\text{CEO}}}{2\pi}f_{\text{CEO}} \tag{2}\]  
The carrier-envelope offset frequency _f_CEO constitutes a global frequency shift of the comb spectrum, such that the comb lines are not exact harmonics of the repetition rate _f_rep. This is described by the previous Equation 1, where _f_CEO is located by definition in the frequency interval \(0\leq f_{\rm CEO}\leq f_{\rm rep}/2\). The CEO frequency is not part of the comb optical spectrum and thus cannot be directly observed. The standard method to detect it was initially proposed by [PERSON] _et al._[11], and applied later for the first time by [PERSON] _et al._[13] and [PERSON] _et al._[38]. It makes use of nonlinear interferometry to perform a heterodyne beat between two harmonic spectral ranges of a comb spectrum. This requires a broad enough comb spectrum such that the two harmonic components end up in the same wavelength range for the heterodyne beat to be detected. The most common method uses _f_-to-_2\(f\) interferometry [11], in which a red portion of the comb spectrum is frequency-doubled and heterodyned with the existing blue comb components as displayed in Figure 2, which requires the comb spectrum to cover at least one frequency octave (_i.e._, ranging from frequency_f_ to frequency _2_f_). Alternatively, nonlinear interferometry involving higher-order nonlinearities can be used with a narrower spectrum, such as _2_f_-to-_3\(f\) that requires a spectrum covering only 2/3 of an octave, but two nonlinear processes are needed to produce second and third harmonic components [11,39].  
Figure 1: (**a**) Time domain pulse train emitted by a modelocked laser with the slippage \(\Delta\phi_{\rm CEO}\) occurring from one pulse to the next between the carrier phase (shown by the thick red line) and the pulse envelope (blue line). (**b**) Spectral representation of a frequency comb in the frequency domain. The carrier-envelope offset frequency \(f_{\rm CEO}\) directly results from the carrier-envelope phase slippage \(\Delta\phi_{\rm CEO}\) of the time domain from the properties of the Fourier transform that links the time and frequency representations.  
Figure 2: Principle of the _f_-to-_2\(f\) interferometry method used to generate the CEO beat from an octave-spanning comb spectrum. A red component of the comb spectrum at \(\
u_{1}=f_{\rm CEO}\ +\ N_{1}\cdot f_{\rm rep}\) is frequency-doubled and heterodyned with a blue component at \(\
u_{2}=f_{\rm CEO}\ +\ 2N_{1}\cdot f_{\rm rep}\) to produce a beat at frequency \(f_{\rm CEO}\). In practice, the CEO beat does not result from the beat between a single pair of lines, but two finite spectral ranges (within the bandwidth of the frequency-doubling process) contribute and the CEO beat thus arises from the superimposition of beat signals between a multitude of lines pairs.