# Influence of Spectral Filtration on Pulse Dynamics in Ring-Cavity Mamyshev Oscillator
## 4 Transition from Harmonic Mode-Locking to a Single Pulse Generation  
An interesting and practically important question remains, namely the dynamics of transition from harmonic mode-locking to a single-pulse generation, as well as the ability to control it. In this section, we investigate these dynamics resulting from the gradual increase in the spectral separation \(\Delta\lambda\) between the filters. The filter width was fixed and equal to 0.6 nm, while their central wavelengths symmetrically changed as a step-wise function. Note that filter parameters in the first step should satisfy conditions of Faraday instability discussed above. Figure 4 demonstrates how initial \"white\" Gaussian noise evolves to a single pulse when spectral separation between the filters varies from 0.7 to 2.8 nm. After each step of \(\Delta\lambda\), the intracavity field evolved during 50 roundtrips till the next step. The resulting temporal distribution of the output laser field measured at each cavity roundtrip is shown in the figure. We also observed that transient evolution of the optical field from harmonic mode-locking to single-pulse generation is accompanied by formation of soliton molecules with complex self-organization-soliton pairs, triplets, and 2 + 1 molecules. Figure 4: Evolution of the output field corresponding to varying spectral separation between the filters from 0.7 nm to 2.8 nm. Filters width = 0.6 nm. Figure 3: (**a**) The repetition rate of the output pulse train predicted in Floquet stability analysis and calculated using the full model. (**b**) Evolution of the pulse numbers in simulations during the first 50 round trips. Filter width = 0.6 nm, \(\Delta\lambda\) = 0.7 nm. (**c**,**d**) Characteristic temporal shape of the output field in the harmonic mode-locking and single-pulse regimes. Figure 5 illustrates how a form of the initial field and the way of tuning the filters' spectral separation can affect the resulting output field. If a seed pulse is used at the first round trip, filter separation \(\Delta\lambda=1.43\) nm leads to stable generation of a single pulse (Figure 5a). Starting from the noise with \(\Delta\lambda=1.43\) does not not lead to any mode-locked regime (Figure 5b). Self-starting of mode-locking can be initiated if the initial spectral separation satisfies the condition of Faraday instability (\(\Delta\lambda=0.7\) nm, Figure 5c-e). However, single pulse generation is not reached at \(\Delta\lambda=1.43\) nm anymore. Variation of the frequency step with which the wavelengths of the gratings move away from each other changes the pulse dynamics and leads to hysteresis. The same \(\Delta\lambda=1.43\) nm corresponds to a single multi-pulse generation or even the absence of mode-locked state. For a more complete picture of the output field dynamics, we performed numerical simulations of the Mamyshev oscillator, starting from 100 different realizations of initial \"white\" noise and \(\Delta\lambda=0.7\) nm. After 50 roundtrips of the field propagation, the spectral separation \(\Delta\lambda\) was chosen randomly in a given interval from 1.3 to 4 nm, outside the Faraday instability region, and simulations continued from the current field during the next 50 roundtrips. If the field energy converged to zero, the process started again from the noise and \(\Delta\lambda=0.7\) nm. The total number of steps along \(\Delta\lambda\) did not exceed 5. By changing the filter separation as described above, more than 27,000 lasing regimes were collected, including unstable noisy regimes. The variety of the obtained single- and multipulse regimes in the plane of \(\Delta\lambda\) and output field energy is shown in Figure 6. Single pulse generation can be achieved in one step of \(\Delta\lambda\), for spectral separation close to 2 nm, or in several steps for \(\Delta\lambda\geq 3\) nm (Figure 6a).