## 3 Noise Measurement and Reduction in Mode-Locked Lasers
### Carrier-Envelope Phase Noise
#### 3.2.2 Carrier-Envelope Phase Noise Stabilization Methods  
_Pump power modulation:_ Similarly, after achieving \(f_{\text{ceo}}\) detection with high sensitivity, precise control and stabilization of the pulse trains' carrier-envelope phase pulse is an important prerequisite for obtaining a low-noise optical carrier of the femtosecond laser. Since the carrier-envelope offset frequency of the laser is mainly influenced by power fluctuation of the pump laser, in the early years, in Ti:Sapphire femtosecond laser OFCs, an acousto-optic modulator was applied to modulate the pump power. To this end, carrier-envelope offset frequency could be phase-locked to radiofrequency references [107, 108, 109, 10]. Later, in the fiber femtosecond laser OFCs, pump current modulation was also applied to stabilize the carrier-envelope phase of the emitted pulse train [86, 110]. However, the main bottleneck of pump power modulation in carrier-envelope phase stabilization is that the  
Figure 5: Setup of active _f-2f_ interferometer. Reprint with permission from [100]. Copyright 2019 Optical Society of America. upper-state lifetime of the gain medium in the laser will restrain the carrier-envelope phase modulation bandwidth to the order of kHz. In literature [111], through a special design of the pump current modulation circuit and optimization of the parameters of the phase-locked loop, the carrier-envelope phase lock with a bandwidth of 1 MHz was achieved. **Feedforward method:** In 2010, [PERSON] et al. proposed to use an acousto-optic frequency shifter outside the laser cavity to shift the overall frequency of the OFC to achieve the \(f_{\text{coo}}\) modulation [112]. In this technique, \(f_{\text{coo}}\) is modulated outside the laser cavity. Thus, the modulation bandwidth can easily reach the MHz level, determined by the acousto-optic modulator. Using the feed-forward method, they achieved a residual carrier-envelope phase noise of 45 mrad, which corresponds to a residual timing jitter of 12 as. Furthermore, by optimizing the length of the nonlinear crystal in the \(f\)-\(2f\) interferometer and combining with the pump power feedback modulation, they achieved a carrier-envelope phase with residual noise of 20 mrad (10 as residual timing jitter) [113]. Later in 2012, [PERSON] et al. demonstrated long-term carrier-envelope phase stabilization using the feed-forward method [98]. **Cavity loss modulation:** On the other hand, modulating the cavity loss of the laser cavity also enables modulation of \(f_{\text{ceo}}\). Moreover, cavity loss modulation is not retrained by the gain medium's lifetime. In 2015, [PERSON] et al. altered the voltage on a thin piece of graphene in the erbium-doped fiber femtosecond laser to modulate the cavity loss with a high bandwidth. In this way, they demonstrated the phase-locking with a bandwidth over 1 MHz [95]. In addition, graphene-modulation-based \(f_{\text{ceo}}\) stabilization is widely used in different fiber lasers, such as all polarization-maintaining nonlinear amplification loop mirror lasers [94] and thulium-doped fiber lasers [97]. Recently, inserting an AM-EO modulator in a laser cavity to phase lock \(f_{\text{ceo}}\) through loss modulation was demonstrated by [PERSON] et al. [114]. **Phase lock to \(f_{\text{rep}}\):** The above-mentioned techniques all require a radio frequency source as a reference. Therefore, the stabilized \(f_{\text{ceo}}\) is at a certain offset frequency, which refers to that the optical phase and pulse' envelope is still periodically mismatched. To solve this issue, in 2018, [PERSON] et al. successfully locked the carrier-envelope offset frequency to the repetition rate [115]. Through balanced detection and dual-beat phase locking, both the \(f_{\text{rep}}\)\(+f_{\text{ceo}}\) and \(f_{\text{rep}}\)\(-f_{\text{ceo}}\) signals are locked to the pulse train's repetition rate. No radio references are involved and therefore electric fields under each pulse envelope are shot-to-shot reproducible. **Passively CEP-stable combs:** In 2011, [PERSON] et al.