## 3 Noise Measurement and Reduction in Mode-Locked Lasers
### Carrier-Envelope Phase Noise  
#### 3.2.1 Carrier-Envelope Phase Noise Measurement Methods  
Pulse train's carrier-envelope phase performs the periodically sweeping between the pulse electric field's peak and the envelope's peak in time domain during the pulse's propagation. The carrier-envelope phase mainly accounts for the non-zero net dispersion in the femtosecond laser, which causes the difference between the pulses' group velocity and the phase velocity. The carrier-envelope phase of the femtosecond pulse is introduced by various noise sources, such as the intracavity amplified spontaneous emission, cavity loss, pump noise, and the fluctuation of the laser cavity length. In this sub-section, carrier-envelope phase noise measurement techniques are presented. _f-2 if interferometry:_ The most used method in carrier-envelope offset frequency detection is the self-referencing based on the _f-2f_ interferometer [10]. Around the 2000s, the \(f_{\text{ceo}}\) signal of Ti:Sapphire femtosecond lasers using _f-2f_ interferometry was firstly reported by [PERSON] et al., where the non-common path (NCP) _f-2f_ interferometer configuration was applied. Generally speaking, the heterodyne between \(\
u_{\text{2n}}\) and \(2\
u_{\text{n}}\) components from octave-spanning supercontinuum optical spectrum results in the detection of \(f_{\text{ceo}}\). The major issue of non-common-path configuration is that environmental fluctuations can easily introduce non-common-mode noises on both two arms of the interferometer, which will be coupled to \(f_{\text{ceo}}\). Therefore, the establishing of a common path (CP) interferometer can eliminate non-common mode noise. In this setup, after highly nonlinear fibers (HNLF) or photonic crystal fiber (PCF), the high frequency and low-frequency parts are directed into a dispersion compensation fiber to compensate the relative group delay, ensuring that these two spectral parts overlap in the time domain. In some cases, the PPLN waveguide in the interferometer compensates for the relative group delay. CP design could eliminate the influence of non-common mode noise with all-fiber configuration, which is typically used in all-polarization-maintaining OFC systems [86]. However, the signal-to-noise ratio of the \(f_{\text{ceo}}\) signal is very sensitive to the overlap of the two frequency parts. The accuracy of cutting and splicing the dispersion compensation fiber is very restricted. Then, a quasi-common path (QCP) configuration was designed [87; 88]. Based on the common path design, a dichroic mirror is used to separate the high frequency and low-frequency parts. The position of the mirror behind the dichroic mirror guarantees the two pulses' accurate time overlap. This design combines the characteristics of adjustable temporal delay and rejection of non-common noise. The high-SNR \(f_{\text{ceo}}\) signal is significant for its stabilization. Besides the temporal overlap between \(\
u_{\text{2n}}\) and \(2\
u_{\text{n}}\) components, the coherence degradation during the generation process [89] and shot noise generated by the photodetector [90] all need to be carefully considered and optimized for high-SNR \(f_{\text{ceo}}\) detection. In addition, the use of balanced photoelectric detection can eliminate fluctuations in the amplitude of the supercontinuum spectrum to improve the signal-to-noise ratio of the \(f_{\text{ceo}}\) signal [91; 92]. For the measurement of \(f_{\text{ceo}}\) PSD, the sensitivity is usually limited by the shot noise or thermal noise of the detector at 1 MHz offset frequency. The signal-to-noise ratio of the \(f_{\text{ceo}}\) signal of fiber lasers can reach 40 dB at 100 kHz resolution bandwidth [86; 93; 94; 95; 96; 97]. For solid-state femtosecond lasers, due to their intrinsic low-noise characteristics, the signal-to-noise ratio of the \(f_{\text{ceo}}\) signal can reach 60 dB [98; 99]. _Active f-2f interferometry:_ An improved version of _f-2f_ interferometer, the active _f-2f_ interferometer was proposed in 2018 by [PERSON] et al. [100].