Effective strategy for accessing the multi-pulse regime of mode-locked fiber lasers

. We unveil a strategy for configuring mode-locked fiber lasers by means of a tunable bandpass filter, which allows multi-pulse structures to be generated without distortions in their intensity profiles, with significantly reduced pumping power levels.


Introduction
Mode-locked fiber lasers (MLFLs) are highly valued for their ability to generate robust multi-pulse structures, such as soliton molecules, molecular complexes, or soliton crystals [1][2][3].However, to generate such structures, it is necessary to access the multi-pulse regime of the laser, under operational conditions that favor the coexistence and interaction of several solitons in the laser cavity.The usual procedure for accessing the multi-pulse regime of a MLFL is to gradually increase the laser pumping power, starting from the mode-locking threshold where only one pulse is present in the cavity, up to the pumping power where that pulse fragments into several pulses.But this procedure increases propagation instabilities and makes the cavity more prone to generate multi-pulse structures with distortions in their phase and intensity profiles [4].
In this paper, we unveil the configuration of a MLFL, which uses the effects of a tunable bandpass filter to reduce the laser pumping power by half, thus enabling the generation of multi-pulse structures without temporal profile distortions.

Numerical modeling
Our laser model is a unidirectional ring cavity that combines the following four major components placed in the order indicated in Fig. 1: a bandpass filter (BPF) with tunable bandwidth BPF, a section of Dispersion-Compensating Fiber (DCF) to adjust the average cavity dispersion, a section of Erbium Doped fiber (EDF, pumped by a laser diode through a WDM coupler), and a SESAM (SEmi-conductor Saturable Absorber Mirror) used to insert nonlinear losses and favor laser mode locking.
where β2, β3, γ, g, and α designate the second-order dispersion (SOD), third-order dispersion (TOD), Kerr nonlinearity, gain, and linear attenuation parameters, respectively.The term R[ψ], which describes the Raman effect, is given by  represents the Raman susceptibility, and ρ is the fractional contribution of the Raman scattering to the total nonlinearity, with ρ = 0.18.In Eq. ( 1), g = 0 in the case of the passive fiber sections (SMF), whereas for the active fiber (EDF) 0 g  [5].The NLSE (1) is numerically solved by means of the split-step Fourier method [5].The action of the SESAM on an incident light field is modelled by the following monotonous transfer function for the optical power:

P T P T T T P P P     
where T describes the instantaneous transmission of the SESAM, T0 is its transmitivity at low signal, and ΔT the absorption contrast, while Pi (PO) designates the instantaneous input (output) optical power.The BPF is modeled by the following Gaussian function: , where ΔΩ=2BPFis the filter's bandwidth.Our numerical simulations are performed using the following parameters: The parameters of the SESAM are : T0 = 0.7, ΔT = 0.

Conventional approach
The conventional approach to access the multi-pulse regime of a MLFL is to first adjust the laser PP (pumping power) slightly above the mode-locking threshold to generate a single pulse in the cavity.Then, the PP is increased gradually until the single pulse present in the cavity becomes unstable and fragments into two (or more) pulses.Fig. 2 illustrates the evolution of the temporal intensity of the intra-cavity electric field recorded at the input of the BPF, when the PP is increased from 21mW to 33mW.The panel (a) of Fig. 2 shows that in a cavity equipped with a BPF of 4nm bandwidth, the switch to the multi-pulse regime occurs at a PP of 25.6mW, which leads to the fragmentation of the initial pulse into two pulses.By continuing to increase the PP, a new fragmentation process occurs at 29.3mW, followed by a restructuring of the intracavity field into three pulses.Quite in contrast, panel (b) of Fig. 2 shows that in the PP range considered, i.e., between 21mW and 33mW, the multi-pulse regime is inaccessible with BPF=12nm.In fact, we found that the PP required to access the multipulse regime does not vary linearly as a function of BPF, but rather varies as a staircase displaying critical value C (~ 11nm for our cavity), as illustrated in panel (a) of Fig. 3.We then observe that for BPF > C (i.e. for a weak or non-existent filtering effect), the PP of access to the multi-pulse regime is relatively high, which exacerbates nonlinear effects and causes distortions in the generated pulse profiles, as shown in panel (b) of Fig. 3.

2D-parameter approach
Knowing that the conventional approach only relies on the adjustment of the PP, here we present an approach based on the adjustment of two cavity parameters, namely, the PP and the bandwidth of the BPF.For the sake of clarity, let us illustrate this 2D approach on the cavity configuration considered in panel (b) of Fig. 3 where BPF=12nm and where the conventional approach requires a PP of 49mW to access the multi-pulse regime.Our 2D approach takes place in two steps: (i) First, we set the filter bandwidth to a value opt <C, for which the multi-pulse regime is accessible by the conventional approach with the lowest possible PP.The curve of panel (a) in Fig. 3 shows that opt=8nm is a good choice.Then, from the mode-locking threshold, we gradually increase the PP up to the fragmentation point, say PPF (which is here equal to 25mW).In the second step of the procedure, the PP is maintained at the value PPF.
(ii) The second step begins with a laser already in multi-pulse regime but with a bandwidth opt <C Here, we gradually increase the filter's bandwidth from opt to BPF.We emphasize that such bandwidth adjustment must be done gradually in order to prevent the laser from exiting the multi-pulse regime.
The 2D approach described above is illustrated in panel (a) of Fig. 4, while panel (b) of Fig. 4 shows the temporal profile of the resulting structure.

Conclusion
Comparing the pulse profiles in Fig. 4(b) with those in Fig. 3(b) clearly highlights the main advantage of the 2D approach.Indeed, for the same cavity configuration, the 2D approach requires a pumping power that is half that required for the conventional approach to access the multi-pulse regime and generates better pulse profiles.

Fig. 2 :
Fig. 2: Evolution in color scale of the temporal intracavity intensity as a function of the PP for BPF =4nm and BPF = 12nm.