## 3 Noise Measurement and Reduction in Mode-Locked Lasers  
### Tining Jitter  
#### 3.1.1 Timing Jitter Measurement Methods  
Compared with traditional microwave oscillators, the timing jitter of the optical pulse train emitted from mode-locked lasers shows superior low timing jitter. Thus, a high-precision, high-resolution timing jitter measurement technique is an essential prerequisite for their further applications. _Electronic methods:_ Date back to 1986, [PERSON] firstly measured RF spectra of the pulse train's repetition rate and its high harmonics from a mode-locked laser to estimate the mode-locked lasers' timing jitter [49]. Years after that, the phase detector method was proposed. With the participation of a reference mode-locked laser, the timing jitter PSD can be retrieved from the phase error between the two repetition rates' harmonics of the laser under test (LUT) and the reference laser (REF). Timing jitter of Ti:sapphire mode-locked lasers [50] and various types of fiber mode-locked lasers [51; 52; 53; 54; 55] have been characterized by this technique. The aforementioned timing jitter detection method with direct optical-electric conversion progress has been well-introduced in Ref. [56]. _Balanced optical cross-correlation:_ The bottleneck of dealing with the harmonics generated from the photodiode is that the signal-to-noise ratio (SNR) of the detected harmonic signal is restrained by the optical-electric conversion process. With the increase of input power, shot noise and amplitude modulation to phase modulation (AM-PM) conversion noise induced by nonlinearities from photodiodes during direct optical-electric conversion gradually becomes an issue, which generally limits the retrieved PSD at 10 kHz offset frequency [57]. To overcome this, a balanced optical cross-correlator (BOC) was demonstrated in 2003 in order to measure the relative timing jitter between a Ti:sapphire laser and a Cr:forsterite laser [58]. Timing jitter measurement based on the BOC technique requires two identical mode-locked lasers as well. A nonlinear crystal in the correlator converts the timing information of the two pulses into the intensity fluctuation of the sum frequency signal. In this way, timing jitter could be characterized with attosecond-level resolution. Since then, the BOC technique has been widely applied in timing jitter measurement of various types of mode-locked lasers, e.g., solid-state mode-locked lasers [59; 60; 61], fiber mode-locked lasers [62; 63; 64; 65; 66], and fiber amplifiers [67; 68] due to its practicality. Later in 2016, [PERSON] et al. improved the BOC technique and proposed the polarization-noise-suppressed balanced optical correlation (PNS-BOC) technique [69]. Through the insertion of a birefringence crystal before the optical correlator, the crosstalk between the pulses with orthogonal polarization states can be suppressed, leading to an improved SNR of error signals. The BOC technique can also be applied in relative timing instability analysis of twin-pulse soliton molecules and counter-propagating solitons in microcavity [70]. Recently, [PERSON] et al. probed the relative timing jitter in a closely separated stationary doublet soliton molecule produced by a Kerr-lens mode-locked Ti:sapphire laser by adopting the BOC technique [71]. An upper estimate of 60 as intramolecular timing jitter with \(5\,\mathrm{z}\mathrm{s}/\sqrt{\mathrm{Hz}}\) temporal resolution is achieved. The corresponding results are shown in Figure 1.  
_Optical heterodyne technique:_ Timing jitter measurement using optical heterodyne (OH) technique was firstly demonstrated by [PERSON] et al. [72]. Two heterodyne beats (at the center frequency of \(\Delta f_{\text{coo}}\)) at two distinct wavelengths between two independent femtosecond lasers, acting as laser under test (LUT) and reference laser (REF), are detected by two photodetectors, respectively.