## 2 Optical Frequency Comb Self-Referencing
### Coherent Supercontinuum Spectrum Generation  
External spectral broadening at the output of the modelocked laser is needed to achieve the broad spectrum that is required for CEO detection by nonlinear interferometry. The only laser system that is able to directly generate an octave-spanning spectrum in some circumstances is Ti:Sapphire as a result of its extremely broad gain bandwidth and high intra-cavity nonlinearities [40]. In addition, the coherence between the two spectral parts involved in the heterodyne beat must be preserved to enable a CEO beat with a sufficient signal-to-noise ratio (SNR) to be generated. The spectral broadening is obtained by supercontinuum (SC) generation in a nonlinear medium, such as highly nonlinear fiber (HNLF) or a photonic crystal fiber (PCF). SC generation relies on different physical mechanisms such as dispersion and nonlinear effects like self- or cross-phase modulation, Raman scattering, four wave mixing, and self-steepening to produce new frequency components in the comb spectrum while ideally preserving the coherence of the spectrum. Spectral broadening in nonlinear fibers used in the anomalous dispersion regime (_i.e._, with a negative group delay dispersion coefficient \(\beta_{2}\)) or close to the zero-dispersion wavelength is particularly efficient as short and intense pulses can be maintained over a long propagation distance. In the presence of negative group velocity dispersion (GVD) and self-phase modulation (SPM) only, soliton propagation can be achieved in an optical fiber, in which GVD and SPM perfectly balance each other leading to a stationary pulse propagating along the fiber as a fundamental soliton. Higher-order solitons can also propagate in a quasi-stationary regime, where the pulse undergoes a periodic change with strong spectral and temporal effects, but retrieves its initial shape after a soliton period \(z_{sol}=(\pi/2)L_{D}\). This length depends on the magnitude of GVD through the dispersive length \(L_{D}=T_{0}^{2}/|\beta_{2}|\) and on the pulse duration \(T_{0}\)[41]. However, in presence of other nonlinearities and higher order dispersion terms that are always present in optical fibers, a \(N\)th order soliton splits into \(N\) fundamental solitons after a fiber length of about \(L_{\mathrm{fiss}}=L_{D}/N\)[41]. This phenomenon is referred to as soliton fission. The resulting fundamental solitons then propagate as distinct and independent pulses with their own group velocity and are red-shifted by Raman scattering. In the normal dispersion regime at wavelengths below the zero-dispersion, dispersive waves are produced from a resonant energy transfer from the solitons, which extends the spectrum towards short wavelengths. The length of the nonlinear fiber is generally optimized such that the intense Raman soliton at long wavelengths and the dispersive wave at short wavelengths are separated by exactly one frequency octave. Therefore these spectral components efficiently contribute to the CEO beat generated in an _f_-to-_2\(f\) interferometer. An example of SC spectrum generated in a PCF from the output of a GHz Yb:CALGO (CaGdAlO\({}_{4}\)) DPSSL is shown in Figure 3. Numerical simulations of the SC spectrum can be performed using a split-step Fourier-transform algorithm [41] that takes into account parameters such as the dispersion curve and nonlinear coefficients of the fiber and the properties of the pulses launched into the fiber. The results are generally in a good agreement with the experimentally observed SC spectrum when the fiber parameters are known with a sufficient precision (see Figure 3). The soliton order \(N_{\mathrm{sol}}\) of the initial pulse depends on the ratio between the dispersive length \(L_{D}\) and the nonlinear length \(L_{NL}=1/(\gamma P_{0})\), where \(P_{0}\) is the pulse peak power and \(\gamma\) the nonlinear coefficient:  
\[N_{\mathrm{sol}}=\sqrt{L_{D}/L_{NL}} \tag{3}\]The soliton order \(N_{\rm sol}\) constitutes a useful metric to estimate the degree of coherence of the generated SC spectrum.