We numerically studied the polarization dynamics in dissipative soliton lasers mode-locked by nonlinear polarization rotation (NPR). It was found that the polarization states of the intracavity dissipative soliton vary with time across the pulse. Depending on output coupling ratios, the polarization states of the pulse peak before the polarizer can be either nearly circular or nearly linear polarizations. The polarization dependent component in NPR is found to play a role of spectral filter under high and medium output coupling. However, NPR may work as a weak optical limiter under low output coupling, when additional spectral filtering is necessary to maintain steady mode-locking state.
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Passive mode-locking fiber laser has seen great progresses in the last two decades, for its advantages in compactness, high stability, and so on. Various regimes have been proposed to increase the output pulse energy , including soliton lasers, stretched-pulse lasers, and self-similar lasers. Recently, the novel dissipative soliton fiber lasers, also named as all normal dispersion fiber lasers [2–5], have been proposed and extensively investigated, where the spectral filtering is found to play a crucial role in maintaining short pulse duration with high energy. Different from other regimes, pulse shaping in these lasers depends strongly on the dissipative processes as well as phase modulation .
Dissipative soliton lasers have been demonstrated with different saturable absorbers (SAs). Most reports are based on nonlinear polarization rotation (NPR) technique, for its fast responding and immunity to thermal damage as an artificial SA. With this scheme sub-100 fs pulses at watt-level powers from single mode fibers have been reported, whose performance are comparable to those of solid-state femtosecond lasers .
In the NPR mode-locked fiber lasers, phase locking between the two polarization modes can never be locked, so it is interesting to investigate the evolution dynamics of the polarization state of the vector solitons formed in these lasers . The laser regimes have been found to affect the pulse polarization evolution in the cavity. In soliton fiber lasers, the polarization state is found to be invariant with time at a fixed intracavity location in stable soliton operation state . However, in self-similar fiber lasers, it is found that the polarization states are different across the pulse from peak to wings . To the best of our knowledge, even though many theoretical and simulation works have been done to study the properties of the dissipative soliton lasers [4, 5], no study has been carried out on the polarization dynamics in dissipative soliton lasers. It would be worth investigating the mechanism of NPR mode locking in dissipative soliton lasers, and determining the distinct features of the polarization dynamics in such lasers.
Here we numerically studied the polarization dynamics under different output coupling ratios in dissipative soliton lasers mode-locked by NPR. Different from those in soliton fiber lasers and self-similar fiber lasers, the polarization states vary for different parts across the pulse, and the polarization states of the pulse peak before the polarizer can be either nearly circular or nearly linear polarizations for different output couplings. Also, the pulse shaping of NPR in these lasers is revealed. It is found that the polarization dependent components in NPR play a role of spectral filter under high and medium output coupling. However, additional spectral filtering is necessary to maintain steady mode-locking state under low output coupling, as NPR may work as a weak optical limiter in this case.
2. Model of NPR mode-locked dissipative soliton fiber lasers
The scheme diagram of a NPR mode-locked dissipative soliton fiber laser is shown in Fig. 1 . It is made up of three pieces of fiber (SMF1, Yb-doped fiber, SMF2), waveplates, polarization beam splitter (PBS), and a Gaussian filter. The unidirectional operation of the ring cavity is enforced by the inline isolator. The output is extracted from the PBS ejection port.
Numerical studies on pulse propagation in fiber sections are based on coupled Ginzburg-Landau equation model for the orthogonal electric field polarization states, Ax and Ay:7].
The functions of waveplates are described with Jones matrixes (angles are defined with respect to the x axis, which is assumed as one of fiber eigen polarization axes). The PBS is oriented to make the x-component of the injecting pulses be ejected.
Parameters in the simulations are: SMF1, length L= 200 cm, β 2= 23 fs2/mm, β 3= 25.4 fs3/mm, γ= 0.0044 W−1m−1; Yb-doped fiber, L= 50 cm, β 2= 23.7 fs2/mm, β 3= 25.4 fs3/mm, γ= 0.0048 W−1m−1, g 0= 3.45, Esat = 720 pJ; SMF2, L= 100 cm, β 2= 23 fs2/mm, β 3= 25.4 fs3/mm, γ = 0.0044 W−1m−1. The −3 dB bandwidth of the Gaussian filter is 12 nm. In the following simulation, these parameters will be kept unless otherwise stated.
3. Simulation results and discussions
3.1 polarization dynamics under high output coupling
The dissipative soliton fiber laser gets mode-locking with orientations of waveplates at θ QWP1= 1.530, θ HWP= 2.871, θ QWP2= 2.487. The steady state at PBS is shown in Fig. 2 , in which x-component is ejected by the PBS as output. The output pulse energy is 3.51 nJ, taking ~80.9% of the total energy at the PBS. Its central wavelength is 1031.0 nm, with −3 dB spectral bandwidth of 12.0 nm. It is shown that the intracavity pulses with lower pulse energy have smoother spectrum, as predicted in Ref . The tiny asymmetry of the spectrum, as observed in experiments , results from the third order dispersion in the fibers. The temporal full width at half maximum (FWHM) of the output pulse is 4.46 ps, indicating that the output pulses are highly chirped.
The polarization dynamics at different intracavity locations are shown in Fig. 3 . The polarization states at every time slices are represented as points on the surface of the Poincare sphere. Here the polarization states at the time slices between the peak and 1/20 intensity of the peak are shown. The QWP2 transfers the linear polarization after PBS into elliptical polarization before injecting into the fiber section for NPR. The polarization states are invariant with time after QWP2 as shown in Fig. 3a. In the fiber section the NPR happens, and different parts of the pulse get various polarization rotations for the differences in intensity (Fig. 3b). The following QWP1 and HWP will further transfer the polarization states to get steady mode-locking (Figs. 3c and 3d). Figure 3d shows that the pulse peak is of nearly circular polarization, as reported in self-similar lasers .
Note that spectral filtering effect would be introduced by the polarization dependent component PBS here. Figure 4 shows the transmission curves of the PBS and the combination of PBS and Gaussian filter, in temporal and spectral domains respectively. It can be seen that the PBS works as a spectral filter to some degree. Different wavelength components, corresponding to different time slices as the pulses are highly chirped, are of different polarization states before the polarizer- PBS (shown in Fig. 3d). So the transmission at PBS will be different for different wavelengths, which makes a spectral filter.
With the same set of waveplate orientations but no Gaussian filter inserted, steady mode-locking can also be achieved. However, the spectral bandwidth of the output pulses is 5.8 nm, narrower than former result. And the temporal FWHM is broadened to be 13.83 ps. Experimentally, dissipative soliton fiber lasers mode-locked by NPR without intentional filters in the cavity have also been reported [9, 10], where mode-locking is assisted with the artificial spectral filter [11, 12]. It should be noted that the spectral filtering introduced by NPR is not always good for steady mode-locking in dissipative soliton lasers, which will be discussed in the following subsection 3.3.
3.2 polarization dynamics under medium output coupling
Steady mode-locking can also be achieved with the same parameters but different orientations of the waveplates. With θ QWP1= 0.854, θ HWP= 2.620, θ QWP2= 2.458, pulse energy of the output ejected by PBS is 3.46 nJ, taking 52.7% of the total energy at the PBS. The pulses and corresponding spectra are shown in Fig. 5a and 5b. The central wavelength is 1031.0 nm, and the −3 dB bandwidth is 15.1 nm. The corresponding temporal FWHM is 5.97 ps. Compared with the output spectrum under high output coupling, the spectral bandwidth gets broader, which may result from higher energy of the intracavity pulse.
In Fig. 5c, the polarization state traces of the time slices at the peak and 1/20 intensity of the peak are shown. After HWP, the points on the Poincare sphere are close to the +S2 axis (within a black dash circle in Fig. 5c), which indicates that the polarization states are transferred to be nearly linear polarization with ~45° azimuth to the passing axis of the polarizer. Thus the transmission of the PBS is expected to be around 50%, agreeing with the above claim.
Similarly, the PBS here can also act as a spectral filter, and mode-locking without inserting the Gaussian filter at the same waveplate orientations can also be achieved.
3.3 polarization dynamics under low output coupling
With waveplate orientations of θ QWP1= 2.741, θ HWP= 0.840, θ QWP2= 2.291, the dissipative soliton laser gets mode-locking with output pulse energy of 1.03 nJ, taking 10.3% of the total energy at the PBS. Figure 6a and 6b show the pulses and corresponding spectra at the PBS. The central wavelength of the output is 1031.1 nm, and −3 dB bandwidth is 18.4 nm. The corresponding temporal FWHM is 4.90 ps. The output spectrum gets smoother in this case, and the spectral bandwidth is much broader. Compared with former results, it can be seen that broader spectrum is achieved at the cost of lower output pulse energy.
Under low output coupling, the polarization dynamics of the time slices at the peak and 1/20 intensity of the peak are shown in Fig. 6c. The polarization states of the pulse peak after HWP are transferred to be nearly linear polarization with ~70° azimuth to the passing axis of the polarizer, which suggests that the output coupling ratio would be low in this case.
From the NPR mode-locking theory, the polarization states before the polarizer should be transferred to make sure that the pulse peak gets higher transmission than the wings. However, it is found that in dissipative soliton lasers under low output coupling, the transmission of the polarizer may be lower at the pulse peak. Figure 7 shows the transmission curves of the PBS and the combination of PBS and Gaussian filter. It can be seen that the NPR alone acts as a weak optical limiter here, which is harmful for mode-locking. By neglecting the Gaussian filter in the simulation model, but with the same waveplate orientations as above, no mode-locking can be achieved. So in this case, additional spectral filter is necessary for starting and maintaining steady mode-locking.
In conclusion, the polarization dynamics under different output coupling ratios in dissipative soliton lasers mode-locked by NPR are numerically studied. It is found that the polarization dynamics are distinct, inherently different from those in soliton fiber lasers and self-similar fiber lasers. In dissipative soliton lasers, the polarization states vary with time across the pulse, and the polarization states of the pulse peak before the polarizer depend on the output coupling ratio: under high output coupling, they are nearly circular polarizations; but in the medium and low output coupling case, they are nearly linear polarizations. Moreover, we study the pulse shaping of NPR in such lasers. It is found that under high and medium output coupling, the polarization dependent components in NPR play a role of spectral filter to some degree. However, under low output coupling, the NPR may work as a weak optical limiter, thus additional spectral filtering is necessary to maintain steady mode-locking state.
Lingjie Kong thanks W. H. Renninger, S. Lefrancois, and L. M. Zhao for helpful discussions. Portions of this work were supported by the National Science Foundation of China (NSFC 61077032).
References and Links
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