# Influence of Spectral Filtration on Pulse Dynamics in Ring-Cavity Mamyshev Oscillator
## 3 Analysis of Faraday Instability  
To find conditions for self-starting pulsed generation, we performed the linear Floquet stability analysis of stationary homogeneous solution and calculated spectrum of Faraday instability [17; 18; 21]. Two independent small perturbations were added to each spectral mode of a stationary solution and the optical field evolved during one cavity round trip. At the end of the round trip, a 2 \(\times\) 2 matrix, indicating relative change of the real and imaginary parts of perturbation, was obtained. The matrix was numerically diagonalized, and the Floquet multipliers \(F(\omega)\) were calculated as the eigenvalues of the matrix. If  
Figure 1: Simplified scheme of the Mamyshev oscillator used in numerical simulations.  
the modulus of the largest eigenvalue is less than one \(F_{max}(\omega)<1\), the amplitude of modulation does not grow. The average growth exponent for each mode can be calculated as \(ln(F_{max}(\omega))<0\), which corresponds to attenuation of the perturbation. In the opposite case, instability arises (\(ln(F_{max}(\omega))>0\)). Figure 2 shows the Floquet multipliers spectrum \(F_{max}(\omega)\), where a white line indicates a frequency, corresponding to the maximum growing rate of initial small signal modulation. This frequency defines the repetition rate of the pulses arising after stationary wave breaking.  
We then performed full NLSE-based numerical modeling of the Mamyshev oscillator, starting from \"white\" Gaussian noise at the first round trip, and compared the simulation results with theory predictions (Figure 3a). The results of the full modeling confirm that proper choice of filters bandwidth and spectral offset \(\Delta\lambda\) between them (filters overlapping) leads to harmonic mode-locking of the fiber laser. Note that the repetition rate of the pulses is in good qualitative agreement with the theoretical predictions (Figure 3a). Figure 3b illustrates how Faraday instability occurs during the first 50 cavity round trips. The green line depicts the number of pulses at the laser output, averaged over 100 realizations of the initial noise. At the first stage (from 1 to 12 round trip), the number of pulses in the temporal numerical window decreases and the optical field tends toward a continuous wave. Then, at the beginning of the second stage, the continuous wave becomes unstable and breaks into a strain of the pulses. The typical shape of the generated pulse train is shown in Figure 3c. If we continue to increase the spectral offset between the filters so that they no longer overlap, formation of a single pulse may occur (Figure 3d). The shape of a single pulse is close to parabolic, indicating similariton formation and evolution inside the laser cavity. Filter parameters, corresponding to a single pulse generation, are shown by black arrows in Figure 2. The instability gain equals zero (\(F_{max}=1\)), because the filters does not overlap and continuous wave generation is suppressed.  
Figure 2: Faraday instability spectra \(F_{max}(\omega)\) for varying filters bandwidth (**a**–**c**) and spectral separation (**d**–**f**).  
Therefore, if parameters of the tunable filters (bandwidth and spectral separation) are chosen in such a way that Faraday instability occurs (Figure 2), self-start of pulse train generation is initiated. Further transition from pulse train to a single pulse and conditions of successful transition are studied in the next section.