## 3 Noise Measurement and Reduction in Mode-Locked Lasers
### Tining Jitter
#### 3.1.1 Timing Jitter Measurement Methods
As shown in Figure 2a, after mixing these beats, the timing difference between two pulse trains can be converted into voltage fluctuations of the electronical signal from the RF mixer. Using this approach, the timing jitter PSD could be characterized with yoctosecond level resolution. The key point in this technique is to fully drive the mixer's LO port in order to minimize its conversion loss. Carefully using electrical amplifiers or transformers to boost up the input power is feasible. Compared with the BOC technique, the OH technique setup benefits from its ultra-high-resolution due to the two following two reasons:  
* In BOC setups, because of the involvement of the sum-frequency process, which is a second-order nonlinear process, the SNR of the error signal would increase as the square of the input power. In contrast, in the OH technique setup, the SNR would just simply increase as linear as the input power. * Due to the absence of the sum-frequency process, the OH technique doesn't suffer from nonlinear conversion efficiency. Besides timing jitter measurement of laser oscillators, the OH technique could be applied in the measurement of optical fibers' timing jitter as well [73]. Figure 3 shows a result of timing jitter PSD measurement of 5-m single-mode fiber using the OH technique. At \(>\)10 kHz Fourier frequency, the measurement is limited by electronic noise from amplifiers. At \(<\)10 kHz Fourier frequency, the PSD characterizes a slow drift with a slop of \(f^{-1}\). The  
Figure 1: Timing jitter PSD of twin soliton molecule in mode-locked Ti:sapphire laser measured by BOC technique. Figure 2: Experimental setup comparison of (**a**) OH technique and (**b**) DOH technique. measurement resolution is at \(10^{-10}\) fs\({}^{2}\)/Hz level. The integrated timing jitter is 64.7 from 10 MHz to 10 Hz. _Delayed optical heterodyne technique:_ Delayed optical heterodyne (DOH) technique can be regarded as a modified version of the optical heterodyne technique. In DOH setups, a fiber delay line, and an acoustic optical frequency shifter (AOFS) are implemented to artificially create a'reference laser' with a certain frequency offset from the laser under test (LUT), as shown in Figure 2b [74]. The fiber delay line used in this setup is typically \(\sim\)100-m long that amplifies the frequency noise of the laser by a factor of \(\tau\), where \(\tau\) is the delay time of the fiber. Due to the dispersion of the long fiber, the two distinct wavelengths with a narrow bandwidth (\(\sim\)1 nm) are required for heterodyne beats generation. In the same manner as the OH technique, after two heterodyne beats (at twice the center frequency of AOFS's driving frequency) are detected, the timing jitter PSD could be obtained through mixing these two heterodyne beats. Figure 4 shows the timing jitter PSD of a NALM Er-fiber mode-locked laser characterized by the DOH technique [45]. In this case, the measurement resolution is \(10^{-7}\) fs\({}^{2}\)/Hz level at 100 kHz offset frequency. For the reason that the DOH technique is capable of getting rid of reference lasers, this technique is comparatively low-cost and simple. Recently timing jitter PSD measurement of supercontinuum spectra [75], microcombs [76], monolithic fiber resonators [77] and a real-time jitter meter [78] were demonstrated by the DOH technique. Figure 4: Timing jitter spectrum of a NALM Er-fiber mode-locked laser characterized by DOH technique. Reprint with permission from [45]. Copyright 2020 Optical Society of America. Figure 3: Timing jitter PSD of 5-m long single mode fiber using OH technique.