1. Introduction

Mode-locked fiber lasers represent a dissipative nonlinear system [1]. Steady state solutions of stable pulses exist due to the compensation of dispersion and nonlinearity but also dissipation and saturable gain. For theoretical modeling of these passively mode-locked lasers the Ginzburg Landau equation (GLE) or several extensions like the Swift-Hohenberg equation are often used [1], where the actions within the cavity are averaged based in the assumption of small changes during one roundtrip [2]. These equations form stable solutions, which describe the outcome of the experimental laser for a large range of parameters quite well. These solutions are called stable attractors and are known to depend on the system parameters (e.g. net-cavity dispersion) rather than initial conditions [1]. However, in the framework of multiple pulse operation of a passively mode-locked lasers described by the GLE, dependencies on the initial condition have been found [3].

Beside the research based on the GLE, the numerical approach of following the pulse inside a cavity by transmitting though each element has been developed and applied to Ti:Sa lasers [4]. Such one-dimensional models with non-distributed parameters have been found to describe the experimental results of Ti:Sa lasers quite well with the drawback of lacking analytical access compared to the GLE approach. However, they could be used to study intra-cavity propagation dynamics [5], which are mainly determined by the net cavity dispersion.

For fiber lasers three regimes are known: Soliton mode-locking is characterized by negative net-cavity dispersion, where the pulse shape is that of a fundamental soliton (sech²). This regime is well-known for Er-doped fiber lasers, where the fiber itself provides anomalous dispersion. Due to the soliton area theorem, the output energies of such lasers are limited to ∼0.1 nJ. In analogy to long haul communication lines, dispersion-managed mode-locking has been discovered and described, where the fibers dispersion is compensated and the net cavity dispersion is close to zero [6]. However, the break-through in terms of output energy of more than 1 nJ was obtained with the access of wave-breaking free mode-locking (also termed self-similar mode-locking) operating in the normal dispersion regime, which has accurately been described by a non-distributed model [7]. The possibility of achieving this new regime of mode-locking in Ti:Sa lasers has also been discussed [8], where the limited group-velocity dispersion hinders it’s direct access. In fact, the larger amount of nonlinear dispersive material in a fiber lasers makes it to an interesting realization of a dissipative nonlinear system with a variety of possible pulse shaping mechanisms not yet fully understood.

In this paper, we study different aspects of the numerical modeling of fiber lasers based on a non-distributed model, which makes fewer assumptions on the pulse changes during one roundtrip compared to the GLE approach and maintains the possibility of studying the intra-cavity pulse evolution. Firstly, the influence of initial conditions to access different attractors of an ultra-short pulse fiber laser for fixed system parameters is investigated. In a second part, a variation of the system parameters is done within the scope of different initial conditions and regimes of mode-locking. As a result, the pulse shaping in the normal dispersion regime is revised and it is shown that self-similar evolution leading to linearly chirped parabolic pulse occurs for a parameter range that is embedded in a more general regime of wave-breaking-free operation preferred for high energy operation.

2. Model

Our model is based on simulating every part of the oscillator (Fig. 1) separately by solving the nonlinear Schrödinger equation (NLSE, Eq. 1) with the split-step algorithm [9]. The NLSE describes the temporal and longitudinal dependence of the slowly varying (complex) pulse envelope A(z,T) along each nonlinear dispersive element (e.g. the fiber segments). The dispersion is expressed by the propagation constant β(ω), γ is the nonlinearity parameter and g the gain or loss. For the gain fiber, the spectral dependence of g(ω) is evaluated by a multiplication in the frequency domain.

Similar to the cavity described in the paper of Ilday et.al. [7] the gain fiber is followed by the saturable absorber with an infinitely fast response time, which is appropriate for any absorber based on the Kerr effect (Kerr lens or nonlinear polarization rotation). The losses and output coupling are summarized by a reduction of pulse energy by a factor of 10 after the SAM. The remaining power propagates through a dispersion compensation stage before entering the single mode fiber, which closes the loop. This cavity design is again illustrated in Fig. 1. The parameters for each element can be found in Table 1, which are similar to the ones use in Ref. [7].

 

Fig. 1. Illustration of the fiber laser cavity elements used for the simulations.

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Tab. 1. Parameters of the simulation

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The net cavity dispersion is calculated by β₂ ^net=β₂ ^Gain∙L^Gain+β₂ ^SMF∙L^SMF+β₂ ^DC∙L^DC and is only changed by the dispersion compensation β₂ ^DC∙L^DC throughout the paper. The pulse energy changes dramatically during the roundtrip and is given at a certain position in the cavity by the balance of saturable gain for the given bandwidth and energy, and the losses and saturable absorber. Each element, which is not given by a linear expression, is modeled by Eq. 1. The saturable absorber is modeled by a reflectivity given in Eq. 2, where R_unsat is the unsaturable reflectance, R_sat is the saturable reflectance, P_sat is the saturation power and P the instantaneous pulse peak power.

The gain is saturated according to Eq. 3, where g₀ is the small signal gain (assuming 30 dB), E_sat the saturation energy due to limited pumping and E the pulse energy.

3. Basin of attractors

To show the variety of attractors for a given set of laser parameters that can also be found for the non-distributed model, several simulations have been done. The parameters used are partly shown in Tab. 1 and fixed to β₂ ^net=+0.004 ps² and E_sat=100 pJ. The simulations use a temporal resolution of 19.5 fs, 1024 data points and a central wavelength 1030 nm.

3.1 Initial conditions

To access the different solutions, several initial conditions have been used and the transition to their attractor is shown a Poincare map in Fig. 2. There, the RMS width of the spectral domain with respect to the temporal domain after each roundtrip (after the SMF) is plotted and gives access to the time-spectral relation of the pulse even if the complete phase cannot be shown by this graph. The blue and black graphs show the evolution starting from quantum noise, where each line starts from different noise. For the red lines, a Gaussian pulse with different initial chirp settings and spectral width is the initial condition. The purple line starts from a transform-limited Gaussian pulse with a spectral width of 27.13 nm. All simulations have been checked to converge to the same attractor when repeated with higher temporal or spectral resolution (for quantum noise, only the temporal resolution is increased).

As a result of these simulations one can see that there are at least three different attractors accessible, which form the basin of attractors for the fixed set of laser parameters. The evolution of the initial conditions to these attractors is shown in Fig. 2 and Fig. 3. The first attractor (attractor A1) is obtained by the quantum noise initial conditions following the blue lines in Fig. 2 (see also Fig. 3(a)). Its parameters in front of the gain fiber are Δλ_RMS=5.45 nm and ΔT_RMS=322 fs. The following section will reveal that with this solution, the pulse evolves in stretched-pulse mode intra cavity. Interestingly, the black line also starts from quantum noise, however, the final solution is a linear chirped pulse with the parameters of Δλ_RMS=10.35 nm and ΔT_RMS=1.57 ps in front of the gain fiber (Fig. 3(d)). This second attractor (attractor A2) is also obtained for the initial conditions of a stretched Gaussian pulse (Fig. 3(c)) shown as the red lines in Fig. 2 and will be identified as a region of wave-breaking free intra-cavity pulse evolution. Finally, the third attractor (attractor A3) forms an oscillating solution (purple line in Fig. 2) shown in Fig. 3(b) in detail. Such oscillating solutions [10] are not scope of this paper and will not be discussed further.

 

Fig. 2. Basin of attractors presented in the space of temporal and spectral width (RMS values) for the parameters of E_sat=100 pJ and β₂=0.004 ps². The lines connect the measured values after each roundtrip (after the SMF). The arrows indicate the evolution from the initial condition to the attractor marked with yellow circles. (gray: below transform-limited condition). A description of the attractors A1 to A3 is given in the text.

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Fig. 3. Examples of the convergence to the different attractors in the temporal domain: (a) attractor A1, (b) attractor A3, (c)+(d) attractor A2. (linear scale: 0≥max)

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3.2 Gain-ramping

The two attractors accessed by quantum noise as initial condition should be briefly discussed in terms of gain-ramping. This is done by linearly increasing the E_sat from 10 pJ to the value of 100 pJ (as used in the previous section) by 10 pJ steps every 20 roundtrips. The result in Fig. 4(a) shows that the same initial condition previously converging to attractor A2 is now converging to attractor A1 by a complete different trajectory compared to the quantum noise shown in Fig. 2 (blue lines). The transient temporal evolution is shown in Fig. 4(b). From an experimental point of view, such a gain ramping is a better description of the experimental conditions (transient behavior of the pump diode). Nevertheless, it seems that there exists no defined way of simulating self-starting mode-locking with quantum noise as initial condition with a single run of simulation in this model at least for specific regions of system parameters. Other methods of simulating the gain and gain fiber more appropriate are the use of a combined rate-equation model including ASE or adding noise each time the pulse passes through the amplifier [11].

 

Fig. 4. (a) Poincare map of the transient evolution from quantum noise with (black line) and without (red line) gain ramping. (b) Transient evolution in the temporal domain from quantum noise to the steady state solution using gain ramping, where attractor A1 is obtained instead of attractor A2 (compare to Fig. 3(d)). (linear scale: 0 ≥max)

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3.3. Unstable attractors

Another aspect of these numerical simulations is the stability of the attractors. All solutions discussed so far are stable, meaning that the stationary pulse does not change after several thousand roundtrips and even stabilizes if a small amount of noise is added to the solution. However, there are also situations where this is not the case. For the following example the cavity parameters are set to β₂ ^net=+0.007 ps², E_sat=4 nJ, P_sat=15 kW and thereby resemble the parameters used in Ref. [7]. The initial condition in front of the gain fiber is a linearly stretched Gaussian pulse with a bandwidth of Δλ_FWHM=32.2 nm, an initial chirp of 0.1 ps² and an energy of 1 nJ. Figure 5 shows the result. The pulse evolves into a parabolic temporal and spectral profile after a small amount of roundtrips. However, increasing the number of roundtrips leads to a breakup of the solution into noise. The simulation has been repeated by further increasing the temporal and spectral resolutions and bandwidth. Furthermore, the adaptive condition for the nonlinear propagation within the cavity elements has been increased. As a result the number of roundtrips before breakup could be changed, however, finally the solution turns into noise even without adding noise initially to check for stability. It has to be mentioned that the attractor was not accessible using quantum noise as an input as described in Ref. [7], which indicates that for this set of parameters the attractor cannot be termed as stable.

 

Fig. 5. (a) Spectrogram of the transient evolution of a Gaussian initial pulse to the unstable attractor of a parabolic pulse turning into noise [movie: 1.5MB] and (b) the corresponding transient temporal evolution. (linear scale: 0 -15-13-8252-i003" xmlns:xlink="http://www.w3.org/1999/xlink"/> ≥max) [Media 1]

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4. Variation of system parameters

In the previous section we showed that for a given set of parameters of the mode-locked laser, the numerical solution depends on the initial condition. In this section the different pulse regimes of the laser are analyzed by changing the parameters of E_sat from 100 pJ to 400 pJ (100 pJ setps) and β₂ ^net from -0.002 ps² to +0.01 ps² (0.005 ps² steps) for two different initial conditions in front of the SMF: a transform-limited Gaussian pulse with E₀=66 pJ and T₀=100 fs, and quantum noise (all other parameters are given in Tab. 1). For each set of laser parameters, exactly the same initial condition of quantum noise has been used to prevent uncertainties as already shown in section 2 for different noise.

The first paragraph starts with an analysis of the convergence, which identifies if a stable solution exists for the given set of parameters. The temporal and spectral characterization of these stable solutions is focus of the second paragraph. The last paragraph of this section deals with the regime of mode-locking and therefore the pulse shape especially for normal (positive) net-cavity dispersion.

4.1 Convergence

Figure 6 shows the number of roundtrips (N) before the solution has been converged. The condition for convergence was that the relative change ε in the field N defined by Eq. (4) should be below 10^-6 for at least 300 roundtrips. However, solutions like bound states (double pulses) also converged and have been removed from the analysis manually as done also for oscillating solutions.

It again can be seen from the comparison of Fig. 6. (a) and (b) that different attractors exist for different system parameters. Moreover, for some parameters, the initial condition of quantum noise does not reach a stable solution whereas it does for the Gaussian pulse. If a solution is obtained in both cases, the outcome is still (slightly) different even for large values of net-cavity dispersion and saturation energy. This is especially interesting as the attractors show the same intra-cavity pulse regime as will be discussed in section 3.2.

 

Fig. 6. Number of roundtrips until convergence starting from noise (a) and from a Gaussian pulse (b). (linear scale: 0 3000 [white: unstable])

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4.2 Characterization of the solutions

Figure 7 shows the temporal width (FWHM) after the SMF for the given set of parameters. The corresponding spectral width (FWHM) is shown in Fig. 8.

 

Fig. 7. Pulse duration (FWHM) after the SMF of the converged solution that started from noise (a) and from a Gaussian pulse (b). (linear scale: 0 10 ps [white: unstable])

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Fig. 8. Spectral bandwidth (FWHM) after the SMF of the converged solution that started from noise (a) and from a Gaussian pulse (b). (linear scale: 5 nm 35 nm [white: unstable])

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For this range of parameters two fundamental different intra-cavity regimes are found and are directly visible in Fig. 7, where the regimes are distinguable by the temporal width. For negative and small values of positive dispersion, the width after the SMF is quite small compared to the values obtained for larger positive net-cavity dispersion. Figure 9(a) and (b) shows the temporal intra-cavity pulse evolution for two representative points in the parameters space of the regimes. In Fig. 9(a), where E_Sat=100 pJ and β₂=-0.002 ps², the pulse width has two minimum points, which is well-known for stretched-pulse mode-locking [6]. In contrast, for E_Sat=400 pJ, β₂=+0.0045 ps² Fig. 9(b) shows that only one minimum exists for the pulse width inside the cavity. The pulse spectrum and pulse shape is almost parabolic, thus, this regime is also known as self-similar mode-locking [7].

 

Fig. 9. Intra-cavity pulse evolution for (a) E_sat=100 pJ, β₂=-0.002ps², (b) E_sat=200 pJ, β₂=+0.003 ps², and (c) E_sat=2000 pJ, β₂=+0.03 ps². (logarithmic scale: -30 dB 0 dB)

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4.3. Pulse shape for normal dispersion

To investigate the pulse shaping in more detail, another parameter of the pulses temporal and spectral shape has been analysed. The kurtosis k has already been used to characterize parabolic pulses [12]. It is defined by Eq. 5, where μ₄ is the fourth centered momentum and σ⁴ the square of the variance. For parabolic shapes it has a value of k=-0.86 and k=0 for a Gaussian pulse shape.

The temporal and spectral kurtosis is shown in Fig. 10(a) and 10(b), respectively, for positive values of β₂ ^net=0.0035 ⋯ 0.01 and E_sat=100 ⋯ 400 pJ, where Gaussian and quantum noise initial condition result converge to the same attractor. However, as can be seen in this graph, only for a small region around β₂ ^net∼0.004 and E_sat=300⋯400 pJ the kurtosis is close to -0.86 in the temporal as well as the spectral domain. It indicates that only in this region the chirp is linear and images the parabolic shaped spectrum in the temporal domain. This is what is known from self-similar propagation [13]. It is also well-known that for a fast convergence to the self-similar evolution in the SMF inside the cavity, the pulse parameters of the pulse entering the SMF have to be optimum. For passive fibers such a condition was discussed by [14]. It is therefore clear that only a limited range of intra-cavity pulse conditions lead to pulses evolving truly self-similar in the SMF and explains the small region of equal kurtosis in the temporal and spectral domain. Furthermore, it has to mention that even if the spectral bandwidth in this small region is comparable to the gain bandwidth, a transition of parabolic pulses to a solitary-wave pulse shape as described in [15] could not be observed within the cavity.

 

Fig. 10. Temporal and spectral kurtosis of the converged solution after the SMF. (linear scale: -1.11 -0.2, -0.86)

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Figure 11 again shows the deviation from a perfect parabolic shape in the temporal domain indicated by the kurtosis values in Fig. 10 for of three representative parameters: E_sat=400 pJ, β₂ ^net=0.0045 ps² being in the region of a kurtosis close to -0.86 in the spectral and temporal domain; and E_sat=400 pJ, β₂ ^net=0.0045 ps² and E_sat=100 pJ and β₂ ^net=0.01ps² far outside this region. To prove if a linear chirp is obtained within and outside the region of self-similar evolution Fig. 11 shows the spectrograms of the pulse obtained after the SMF for the same three sets of parameters. For Fig. 11(b) and (c) variations from the linear chirp in the pulse and spectral wings are obvious. It proves that only for the aforementioned self-similar regime, truly linear chirped pulses are obtained. Such pulses are especially of practical relevance due to the advantages for high peak power amplification [16]. However, in the whole wave-breaking free regime, the linear chirp is dominant in the solution and recompression of the pulses close to transform-limit should be possible.

 

Fig. 11. Temporal pulse profiles normalized in time to their FWHM for β₂ ^net=0.0045 ps², E_sat=400 pJ (red), β₂ ^net=0.01 ps², E_sat=400 pJ (blue) and β₂ ^net=0.01 ps², E_sat=100 pJ (green) shown in linear (a) and logarithmic scale (b) in comparison with a perfect parabolic shape (black). (Spectral profiles are not shown but exhibit similar features.)

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In summary it is important to revise and keep in mind that the “wave-breaking free mode-locking” is the more general term and embeds “self-similar mode-locking”, where a linear chirped parabolic pulse is indeed obtained due to self-similar evolution in the SMF. Furthermore, if other cavity geometries (e.g. linear cavities [17]) are considered one has to check again if self-similar evolution is possible at all within a wave-breaking free regime.

 

Fig. 12. Spectrograms of the steady state solution for (a) β₂ ^net=0.0045 ps², E_sat=400 pJ, (b) β₂ ^net=0.01 ps², E_sat=400 pJ and (c) β₂ ^net=0.01 ps², E_sat=100 pJ. (Spectrogram resolution: 600 fs). (linear scale: 0 max)

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5. Conclusion

In conclusion, we have shown that for the non-distributed model to describe a mode-locked fiber laser, multiple attractors are accessible by different initial conditions even for fixed system parameters. Due to faster convergence Gaussian initial conditions are often favored. However, especially in the transient region between mode-locking regimes, different solutions can be obtained by e.g. quantum noise as the starting condition. We have demonstrated that the solution can converge to multiple attractors even when using different initial quantum noise fields, but also have shown one example where gain ramping removed this uncertainty. Therefore, the question remains if properties like self-starting can be analyzed by this method. Also pointed out is the stability of attractors if accessed by a special initial condition. In a second part, we have carefully analyzed the different regimes of mode-locking obtained by varying the intra-cavity dispersion and saturation energy of the gain fiber. We have revised the characteristics of stretched pulse mode-locking and showed additionally that a regime producing linearly chirped parabolic pulses indicating self-similar mode-locking is embedded in a regime termed wave-breaking free mode-locking.