# Influence of Spectral Filtration on Pulse Dynamics in Ring-Cavity Mamyshev Oscillator
## 1 Introduction  
Fiber Mamyshev oscillators (MO) attract a significant amount of attention from the laser community [1, 2, 3, 4, 5, 6]. MOs combine advantages of fiber technologies, including compactness, efficiency,and excellent beam profile quality, and outstanding performance in terms of energy and recording spectral width of generated pulses. External compression of the pulses generated inside the MO cavity allows megawatt peak powers to be achieved [7].  
One of the major issues of MO design is the self-starting of a pulsed regime. Since an MO cavity is spectrally closed to continuous waves, to start the pulsed regime, one has to excite sufficient optical power fluctuation. The first experimental demonstrations of the MO setup included a seed pulse source [8, 9] as a mode-locking initiator. However, the need to use additional sources of ultrashort pulses significantly reduces the commercial appeal for such devices in real applications. Attempts to get rid of an external lasing source have resulted in the use of a starting arm inside the MO cavity. The starting arm commonly contains part of the mode-locking laser, for instance, a saturable absorber [10] or polarization-dependent elements [11]. The requirements for initial power fluctuations are not high; therefore, noisy Q-switched regime generated via the starting arm is sufficient to achieve mode-locking of MO cavity. Transition from a Q-switched to a mode-locked regime is provided by modification of the laser cavity, for example, via mechanical flipping of a mirror.  
A recent approach to MO self-starting is based on spectral filters with overlapping bandwidths [12, 13]. MO cavities with such filters have revealed reach dynamics of pulsing regimes, including soliton rains and harmonic mode locking [14]. At first sight, MOs containing only spectral filters have superior advantages over alternatives. However, in the work [10], the authors confirm that they did not achieve stable mode locking despite the fact that they had access to tuning the parameters of the spectral filters. In the work [15], the authors emphasize the need to occasionally mechanically shake the fiber cavity toachieve a stable mode-locking regime. Modulation of pumping power and application of a proper strategy to adjust the pumping power level and filter passband in linear MO cavity could solve the self-starting issue [16].  
Here, we perform extensive research on the influence of spectral filtering on MO performance. It is known that dissipative Faraday instability may initiate multi-pulse generation inside fiber lasers in the case of zig-zag spectrally dependent losses [17; 18; 19]. Faraday instability may provide multi-pulse generation even for relatively low-power initial noise. We demonstrate that Floquet spectrum analysis is a powerful tool to examine self-starting possibility of the MO in this case. By analyzing the position and intensity of the maximum of the Floquet spectrum, it is possible to predict whether self-starting condition is satisfied and what the temporal separation will be between the mode-locked pulses at the laser output. We also studied transient dynamics from multi- to single-pulse generation, gradually increasing the spectral separation between the filters at the fixed pump power. The observed rich optical field dynamics, including hysteresis phenomena and formation of soliton molecules, makes the system attractive for further developing of new intelligent methods for active control of MO parameters.