## 3 Noise Measurement and Reduction in Mode-Locked Lasers
### Comb-Line Noise  
#### 3.3.1 Comb-Line Noise Measurement Methods  
In Section 2, we pointed out that in mode-locked lasers, the comb-line noise, which refers to the frequency noise of optical modes, is mainly determined by the cavity length fluctuation. In this subsection, several comb-line noise measurements methods are introduced. _Beat with a CW laser:_ The most typical method to characterize the frequency noise of a certain optical mode in an OFC is to beat it with a narrow-linewidth CW laser. The use of a photodetector enables detection of the beat signal in the radio frequency range. The noise of the beat signal is related to the frequency noise of the optical comb mode when the noise of the reference CW laser is much lower than the optical modes under test. The comb-line noise power spectrum at MHz offset frequency can be obtained by this technique [86, 94, 95, 114, 118, 119]. However, narrow linewidth CW lasers only exist at certain wavelengths and are not available in many wavelengths. _Asynchronous optical sampling:_ Early in 2007, [PERSON] et al. analyzed the relative phase noise of optical modes in mode-locked lasers [120]. Their setup was based on two lasers with a slight repetition rate difference. Through numerical analysis of the asynchronous optical sampling signal, a linewidth of 50 kHz of the optical modes was measured. This approach was also capable of unraveling the correlations of the phases of different modes. _Multiple fringe-side locking method:_ [PERSON] et al. proposed a multiple fringe-side locking method to characterize the frequency noise spectrum of optical modes, requiring no additional reference lasers [121]. They used the transmission curve of a low finesse Fabry-Perot cavity as a high-precision phase discriminator. The comb-line noise spectrum can be characterized up to 1 MHz offset frequency. _Delayed optical heterodyne technique:_ Besides the optical cavity, the fiber delay line could also serve as a sensitive phase discriminator for frequency noise measurement in OFCs. Recently, [PERSON] et al. applied the DOH technique in an asymmetric fiber delay line interferometer to characterize the frequency noise of comb modes [45]. The delayed self-heterodyne process can magnify the frequency noise of the laser under test by a factor of \(\tau\), which is the round-trip delay time between the two arms in the interferometer. Through demodulation of the heterodyne beat note, the frequency noise spectrum of a single comb line could be retrieved, as shown in Figure 7.  
Figure 6: Representative performance of carrier-envelope phase noise reduction methods presenting as locking bandwidth vs. residual phase noise. The different color-shaded regions indicate noise of free-running comb originating from environmental noise, pump noise, quantum noise and SNR limit. _Correlation between \(\mathbf{n}\times f_{\textit{rep}}\) noise and \(f_{\textit{ceo}}\) noise:_ It has been reported by serval groups that, comb-line noise spectrum in OFCs is not the simple sum of their \(\mathbf{n}\times f_{\textit{rep}}\) and \(f_{\textit{ceo}}\) noise spectra [45; 103; 122; 123]. There is anti-correlation between \(\mathbf{n}\times f_{\textit{rep}}\) noise and \(f_{\textit{ceo}}\) noise, which can be described by the following equation:  
\[S_{\Delta v_{\mu}\Delta v_{\mu}}=S_{\Delta f_{\textit{ceo}}\Delta f_{\textit{ ceo}}}+S_{N\cdot\Delta f_{\textit{rep}}N\cdot\Delta f_{\textit{rep}}}+ \Gamma_{\Delta}(\omega)\times\sqrt{S_{\Delta f_{\textit{ceo}}\Delta f_{\textit{ ceo}}}\cdot S_{N\cdot\Delta f_{\textit{rep}}N\cdot\Delta f_{\textit{rep}}}}\]  
where \(S_{\Delta v_{\mu}\Delta v_{\mu}}\), \(S_{N\cdot\Delta f_{\textit{rep}}N\cdot\Delta f_{\textit{rep}}}\) and \(S_{\Delta f_{\textit{ceo}}\Delta f_{\textit{ceo}}}\) are frequency noise spectra of \(\mathbf{v}_{\mathrm{n}}\) noise, \(\mathbf{n}\times f_{\textit{rep}}\) noise, and \(f_{\textit{ceo}}\) noise, respectively, \(\Gamma_{\Delta}\) (\(\omega\)) is the sum of complex coherence.