## Abstract

Novel features in stretched-pulse and similariton mode-locked regimes of Yb-doped fiber laser with photonic bandgap fiber used for dispersion compensation are found by means of numerical simulations. We show that the mode-locked pulse may become shorter with increasing third-order dispersion. Analytical estimations explain observed behavior through resonant interaction of the main pulse with dispersive waves involving both resonant sidebands and zero-group-velocity dispersion waves. Switching between the stretched-pulse and the similariton regimes is also studied.

©2007 Optical Society of America

## 1. Introduction

Ytterbium- (Yb-) doped fiber lasers and amplifiers are considered attractive because of the broadband gain and high doping concentrations enabling high single pass gain, as well as due to many biological and medical applications in the spectral domain near λ=1μm [1]. Recently, significant progress has been achieved in creating integrated all-fiber cavity configuration in which a photonic bandgap fiber (PBF) has been used for compensation of the group velocity dispersion (GVD) [2–5]. Although impressive results have been demonstrated, it has been noticed that performance has been severely affected by the third-order dispersion (TOD) of the PBF. Significant analysis of the TOD in the passive mode locking has been done for the soliton regime of generation [5–8] however the role of the TOD for the cases of the stretched-pulse (SP) [9] and similariton [10] mode locking regimes remains unexplored.

In the present paper we study SP and similariton mode-locking regimes by means of numerical simulations for the parameters corresponding to the hollow-core and solid-core PBF. Previous theoretical and experimental works have elucidated a harmful role of the TOD for the case of soliton regime of mode locking with the expected result that the TOD increases the mode-locking pulse duration [6]. However, recent studies of the short pulse fiber amplifiers have witnessed nontrivial interplay between the fiber nonlinearity and the TOD of the stretcher, amplifier and compressor. In particular, Shah et al. have demonstrated pulse compression which relies on the compensation of the TOD via self-phase modulation of the cubicon pulses in a fiber amplifier [11]. Furthermore, it has been found that an optimal nonlinear phase shift reduces the pulse duration in the presence of the TOD [12].

Here we demonstrate that, contrary to common belief, increasing TOD may lead to shortening of the pulse generated in the SP regime. More specifically, the pulse with the smallest duration is achieved at certain TOD, which is determined by the net cavity GVD. The pulse shortening can be explained through spectrum enhancement on the blue side due to resonant interaction of the main peak with the phase-velocity and group-velocity matched frequencies. We observe different spectrum behavior in the SP regime for the negative and positive cavity GVD. We investigate how the TOD affects the transition from the SP to the similariton regime and argue that under action of the TOD, similariton acquires features of cubicon [11] due to contribution of a cubic component into the phase.

## 2. Laser model

Ring cavity configuration which is presented in Fig. 1 is quite popular design solution in development of the all-fiber lasers [2, 9, 10, 13]. In our case, mode-locking mechanism relies on the nonlinear polarization evolution [2, 9]. The cavity consists of relatively short section of the Yb-doped fiber (30 cm) followed by a meter of the dispersion compensating PBF. The pump is injected through the single mode fiber (SMF). The cavity length *L* and net cavity GVD are controlled by varying length of the SMF segment between 1.5 m and 2.5 m. Light propagation in the SMF and PBF is modeled by nonlinear Schrödinger equation, which for the case of Yb-doped fiber transforms to

where *z* is the propagation coordinate, *t* is the time in the moving reference frame, *A*(*z*,*t*) is the complex field amplitude, *β _{2}* is the GVD parameter,

*β*is the TOD,

_{3}*γ*is the nonlinear parameter taken as 0.005 W

^{-1}/m for SMF, whereas for the hollow-core PBF this parameter is taken 1000 times smaller [14]. The last term in Eq. (1) corresponds to the gain in the Yb-doped fiber, where

*g*

_{0}is the small signal gain parameter (5.5 m

^{-1}) with the Lorentzian frequency dependence and a bandwidth of 40 nm,

*E*is the pulse energy and

_{pulse}*E*is the gain saturation energy (taken as 3nJ). Mode-locking mechanism associated with the nonlinear polarization evolution has been modeled as an intensity dependent transmission with its particular form not essential for the simulation results. A standard split-step Fourier algorithm has been used in simulations with a white noise as initial field. The total loss of the cavity has been chosen as 10 dB corresponding to the losses at different interfaces as well as to the light output from the cavity. The pump parameter has been chosen to generate nanojoule pulses but not high enough to cause overdriving or multiple pulsing. While the dispersion of microstructured fiber can be tailored to the desired characteristics, we take the GVD and TOD parameters for the solid-core and hollow-core PBFs close to those reported in [2, 3, 4].

_{sat}## 3. Results and discussion

In the case of negative net total cavity GVD the SP regime with its distinctive pulse breathing dynamics [9] is observed in simulations. When the TOD parameter increases gradually, rather nontrivial behavior is found that is presented in Fig. 2. Here, the pulse profiles after amplification in the Yb-doped fiber [Fig. 2(a)] and their transform-limited counterparts [Fig. 2(b)] are shown in the absence of the TOD in the PBF (red dashed line) and for TOD=500 fs^{3}/mm (blue solid line). To prove that we observe SP regime in simulations, in Fig. 2(c) we present the pulse phase at two locations inside the cavity for the case of PBF with TOD=500 fs^{3}/mm. The dotted curve is the phase for the pulse before Yb-doped fiber, and solid curve is the phase of the pulse after amplification shown by the solid curve in Fig. 2(a). It is seen that the phase profile flips, i.e., the chirp changes its sign during pulse propagation inside the cavity, which is an indication of the SP regime.

Comparing pulse characteristics for the cases with and without TOD in the PBF we come to the following observation. Contrary to the common belief that increasing TOD leads to an increase in pulse duration [6], the pulse in the case with significant TOD is shorter than the pulse for the case without TOD. More specifically, from the transform limited pulses in Fig. 2(b) we estimate FWHM as 118 fs (without TOD) and 85 fs (with TOD).

The spectra corresponding to the pulses in Fig. 2(a) are presented in Fig. 2(d). It is seen that Kelly spectral sidebands [15] become more pronounced and asymmetric in the presence of TOD and the spectral maximum is shifted in the anti-Stokes direction. To estimate positions of the Kelly sidebands in the frequency space [shown in Fig. 2(e)], we adopted theory developed for the case of soliton regime of generation [7], where the order of the sideband *N* versus frequency offset *Δω _{N}* is given by

The difference from the soliton regime is that in our case of stretched-pulse regime we have used the spectral width *ΔΩ* instead of the inverse pulse width τ^{-1} because the spectrum remains nearly unchanged during pulse circulation in the cavity whereas the pulse width in the SP regime varies significantly. It is seen from Fig. 2(e) that numerical data (solid and dashed lines) and the analytical results (open circles) agree very well.

Figure 2(f) presents spectral dependence of the total cavity GVD calculated as *∑*(*β _{2,k}*+

*β*

_{3,k}*Δω*)

*L*, where the subscript

_{k}*k*marks particular segments of the cavity. It is clear that the contribution of the TOD is reflected as the slope of the line in Fig. 2(f) while the intersection with the vertical axis (points P and R) is determined by the GVD at the carrier frequency. For the SP regime under consideration, the intersection point R is in the negative domain. With increasing TOD the line is rotating around point R, the behavior is illustrated by the arrow in Fig. 2(f). The key to understanding the spectrum behavior is to notice how the zero GVD point, marked by triangle in Fig. 2(f), shifts with increasing TOD and approaches zero frequency offset from the blue side. For small TOD, zero GVD point lies far from the center of the spectrum: in Fig. 2(f) zero point is at 15THz where the spectrum is below -60dB and interaction with the dispersive waves is inefficient. The situation changes when TOD exceeds 300 fs

^{3}/mm, where the zero GVD point lies close to the first sideband on the blue side of the spectrum and with farther increase approaches spectrum maximum. In the range close to TOD=500 fs

^{3}/mm corresponding to the solid line in Fig. 2(f), the zero GVD point lies between spectrum maximum and first blue sideband and this is a range where we observe the shortest pulses in simulations. This can be attributed to the fact that the main spectrum intensity should be high enough for effective interaction with the dispersive waves.

It should be noted that pulse shortening can be attributed to the spectral shift (0.3 THz) seen in Fig. 2(d) towards the anti-Stokes direction, where GVD value is smaller. However, only GVD change cannot explain the magnitude of the pulse shortening indicating principal role of the TOD in the observed behavior.

More general pictures illustrating pulse shortening with increasing TOD are shown in Fig. 3. Here we present transform-limited pulse FWHM for two types of dispersion compensating PBF: for the hollow-core case in Fig. 3(a) and for the solid-core case in Fig. 3(b). It is assumed that the pulses are perfectly phase compensated by the extra-cavity means, for example by using a spatial light modulator [16]. It is worth noting that the curves for two kinds of PBF demonstrate similar behavior: the shortest pulse is obtained at some certain TOD after which the pulse broadens with farther increase of TOD. Note that for a cavity with compensated GVD, i.e. GVD=0 at the carrier frequency, the pulse shortening effect vanishes as shown in Fig. 3 (lowest curves marked with diamonds). With increasing negative cavity GVD the minimal pulse is observed at larger TOD. Such monotonous dependence of the position of local minimum on the magnitude of the cavity net GVD can be understood from Fig. 2(f): at larger GVD the spectrum is narrower and larger TOD is needed to make the GVD line steeper to move zero-GVD frequency in the region of the pronounced sidebands.

It should be noted that record short pulses in Yb-doped fiber laser have been obtained in a cavity with both compensated GVD and TOD [17], what is in line with Fig. 3. Nevertheless, when the GVD is non-zero, the third-order dispersion can balance GVD action and assist in obtaining shorter light pulses. That might have been the case in the experiment [18].

Spectrum enhancement through resonant interaction of dispersive waves has been reported earlier in fiber lasers for the soliton regime of mode-locking [8]. Very recently Herda et al. observed both asymmetric main peak and sidebands in the same soliton regime of generation with a PBF in the cavity [5]. Thus, relevance of our results on pulse shortening with increased TOD may be expected for the soliton-like mode-locked regime.

Let us consider now the situation of positive nominal cavity GVD, the case corresponding to the point P in Fig. 2(f). The positive GVD is achieved by the extension of the SMF section, i.e., SMF length is larger than 2.2m in the case of our model. For the case of zero or negligible TOD we observe transformation of the SP regime into similariton regime, which is a rather known scenario. However, with increasing TOD the mode-locked pulse establishing from the noise appears to be asymmetric with a quite pronounced cubic component in the pulse phase. Recently, the name “cubicon” has been introduced in the context of optical fiber amplifiers [9] to reflect cubic contribution into the phase of the similariton pulses under influence of the TOD. We find it appropriate here to extend the use of the term “cubicon” to the case of the mode-locked pulses. In Fig. 4, the temporal profiles (a) and the corresponding spectra (b) are shown for the case of positive GVD and for different TOD values. For small TOD (i.e. when TOD is taken into account only in the SMF and Yb-doped fibers), the pulse looks quite symmetric, and the top part of the spectrum demonstrates some tilt (solid curves marked with dots in Fig. 4). With increasing TOD in the PBF fiber, the pulse acquires cubicon-like features. The pulse becomes more asymmetric, tilt becomes highly pronounced so that the spectrum takes triangular-like shape shown by dashed curves in Fig. 4.

We have found that at fixed positive GVD of the cavity, increase of TOD leads to switching from the above described similariton-cubicon regime to the SP regime with much narrower pulse and broader spectrum with pronounced asymmetric sidebands (bold solid lines in Fig. 4). Again we find that shorter pulses are obtained for the larger TOD when all other parameters fixed. The spectrum looks asymmetric and the main peak is shifted in the red direction. As in the case of negative GVD, the spectrum features can be understood from analysis of Fig. 2(f): at positive GVD of the cavity, the zero-GVD dispersion waves are in resonance with the red wing of the spectrum. This causes the spectral shift towards the anomalous dispersion domain what favors generation of shorter pulses. However we have not observed spectrum broadening with increasing TOD in the case of positive GVD. At larger TOD due to shift of the intersection point to the right in Fig. 2(f), the spectrum becomes narrower and more centered relative to zero axis.

It should be noted that in the domain intermediate between the stretched-pulse and similariton regimes we observed secondary instabilities of the mode-locked pulses (both SP pulses and similaritons) established from the noise. The TOD influences the instabilities in the similar way as it has been described before for the soliton generation regime [19, 20].

## 4. Conclusion

In conclusion, our extensive numerical simulations show that resonant interaction of both phase-velocity matched and GVD-matched dispersive waves with the mode-locked pulse may lead to spectrum broadening which means obtaining shorter mode-locked pulses with increasing TOD. We have presented results for the case of positive TOD while analogous behavior should be observed for the negative TOD as well. It is found also that similariton-cubicon regime loses stability under action of TOD, and strong TOD stabilizes stretched-pulse regime well in the domain of positive GVD.

## Acknowledgment

One of the authors (YL) thanks Dr. F. Ö. Ilday for help in developing numerical procedure.

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