## 2 Theoretical Models
### Comb-Line Noise and Carrier-Envelope Phase Noise  
After the pulse train from a mode-locked laser is Fourier transformed, a discrete, comb-shaped spectrum can be obtained in the optical frequency domain. The frequency/phase noise of each comb modes is an important parameter to evaluate the performance of an OFC. The linewidth of comb modes is mainly affected by two noise terms: Schawlow-Townes linewidth limit and technical noise terms. In this sub-section, only the conclusive formalisms of these noise terms will be listed. A detailed derivation of the formalisms could be found in Refs. [46; 47]. In 1958, [PERSON] and [PERSON] proposed the concept of the linewidth limit of electromagnetic waves oscillating in an optical cavity. They named this limit the Schawlow-Townes linewidth limit [46]. The physical origin of the Schawlow-Townes linewidth limit is the spontaneous emission in the laser cavity. However, due to the existence of other technical noise sources, in most lasers, the linewidth of the electromagnetic wave is much larger than the Schawlow-Townes linewidth limit. The expression of the linewidth limit is:  
\[S_{\upsilon n}^{ASE,ST}(f)=2\frac{f_{rep}^{2}}{(2\pi)^{2}}\Bigg{(}\frac{(1+n_{ sp})h\upsilon_{0}G}{P_{circ}}\Bigg{)}\]  
where, \(n_{sp}\) is the amplified spontaneous emission factor; \(G\) is the intracavity gain; \(P_{circ}\) is the intracavity pulse power. In OFCs, the frequency noise PSD of comb modes introduced by a certain technical noise source \(X\), can be described by the following formula [47]:  
\[S_{\upsilon n}^{X}=\Big{(}\upsilon_{n}-\
u_{fix}^{X}\Big{)}^{2}S_{r}^{X}(f)\]  
where \(\upsilon_{n}\) is the optical frequency of the comb modes; \(\
u_{fix}^{X}\) is the \"fixed point\" for a given parameter \(X\); \(S_{r}^{X}(f)\) is the PSD of the fractional repetition-rate fluctuations driven by the fluctuations in the parameter \(X\), in the unit of \(1/\mathrm{Hz}\). For comb modes' frequency noise generated by pump fluctuation, cavity length fluctuation (caused by environment), cavity loss fluctuation (caused by environment), and timing jitter (caused by amplified spontaneous radiation), \(S_{r}^{X}(f)\) can be expressed by the following formalisms, respectively:  
\[S_{r}^{pump}(f)=B\frac{1}{1+\left(f/f_{3B}\right)^{2}}S_{RINpump}\]  
\[S_{r}^{length}(f)=\Bigg{(}\frac{f_{rep}}{\
u_{group}^{L}}\Bigg{)}^{2}S_{ length}(f)\]  
\[S_{r}^{loss}(f)=B\frac{1}{1+\left(f/f_{3B}\right)^{2}}S_{loss}(f)\]  
\[S_{r}^{ASE,\text{ timing }}(f)=2f_{rep}^{2}\left(\frac{(1+n_{sp})h\upsilon_{0}G}{ P_{circ}}\right)\times\left[\hat{\mu}_{rms}^{2}+\left(\frac{\beta_{2}}{4D_{g} \omega_{rms}}\right)\right]\]  
where, \(S_{RINpump}\) is the PSD of pump laser's intensity noise; \(B\) is the rate of repetition frequency change introduced by the pump power; \(f_{3 dB}\) is the 3-dB roll-off frequency, which is typically 6 kHz for Er-doped fiber lasers; \(S_{length}\) (\(f\)) and \(S_{loss}\) (\(f\)) are the PSDs of cavity length fluctuation and cavity loss, respectively; \(\
u_{group}^{L}\) is the group velocity of pulse propagation in the laser cavity; \(\beta_{2}\) is the net dispersion in the cavity; \(D_{g}\) is the gain dispersion, which is typically 0.065 ps for Er-fiber lasers; \(t_{rms}\) and \(\omega_{rms}\) are the rms values of the pulse width and spectral width in the cavity, respectively. Using the above formalisms, the contribution of various technical noise sources to every comb mode could be evaluated. It should be noted that the 0 th frequency comb tooth is the carrier-envelope offset frequency. Thus, the carrier-envelope offset frequency noise originating from various technical noise sources can be obtained when \(\upsilon_{n}\) =\(f_{\text{ceo}}\). For comb modes in the optical range, the main origin of frequency noise is the fluctuation of the laser cavity length. However, for carrier-envelope phase noise, the case is rather complicated. Noise origin is distinct in different Fourier frequency ranges.