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Abstract
We have numerically investigated broadband high-energy similariton fiber lasers, demonstrated that the self-similar evolution of pulses can locate in a segment of photonic crystal fiber without gain-bandwidth limitation. The effects of various parameters, including the cavity length, the spectral filter bandwidth, the pump power, the length of the photonic crystal fiber and the output coupling ratio have also been studied in detail. Using the optimal parameters, a single pulse with spectral width of 186.6 nm, pulse energy of 23.8 nJ, dechirped pulse duration of 22.5 fs and dechirped pulse peak power of 1.26 MW was obtained. We believe that this detailed analysis of the behaviour of pulses in the similariton regime may have major implications in the development of broadband high-energy fiber lasers.
© 2017 Optical Society of America
1. Introduction
Though passively mode-locked soliton fiber lasers are compact and robust compared with solid lasers, it is difficult to achieve high-power optical pulses since small mode field area of the fiber can produce large nonlinear phase shifts. Excessive nonlinearity can then result in optical wave breaking [1,2]. In contrast to solitons, a new class of solutions to the nonlinear Schrödinger equation with gain – the self-similar pulse, which has a parabolic temporal profile, can tolerate strong nonlinearity without wave breaking. Besides being free of wave breaking, self-similar pulses also have linear chirp which can lead to highly efficient pulse compression. As a result, self-similar pulses have attracted much attention since Fermann et al. applied self-similarity methods to study pulse propagation in normal-dispersion fiber amplifiers [3]. Subsequently, Kruglov et al. also used self-similarity methods [4], and confirmed that similariton pulses can be compressed by 2 orders of magnitude in a dispersion-decreasing Raman-pumped fiber amplifier [5]. The self-similar propagation of ultrashort parabolic pulses in a laser resonator was theoretically predicted, and experimentally observed by Ilday et al. [6]. Oktem et al. reported a dispersion-managed fiber laser in which the mode-locked pulse evolved as a similariton pulse in the gain segment, and this pulse was subsequently referred to as the amplifier similariton [7]. Renninger et al. observed parabolic amplifier similaritons inside a normal-dispersion fiber laser, and had demonstrated that amplifier similariton evolution can also yield practical features such as parabolic output pulses with high energies [8]. Though parabolic amplifier-similariton lasers can produce high energy pulses with high pump power, the spectral width is restricted by the bandwidth of the gain medium [9,10], which presents a clear challenge to the generation of ultrafast (<50 fs) pulses. In order to overcome this drawback, a broadband amplifier-similariton fiber laser with nonlinear polarization rotation (NPR) as mode locker was proposed. The property that lasers with self-similar evolution of the pulse in the gain medium can tolerate strong spectral breathing can be used to eliminate the gain-bandwidth limitation to obtain ultrafast pulses, and ~200-nm broad spectra with a dechirped pulse duration of ~20 fs was generated through passive nonlinear propagation in a normal-dispersion and highly nonlinear fiber [9]. In addition to this, Tang et al. have demonstrated that the self-similar evolution of pulses can locate in a segment of dispersion-compensating fiber in a passively mode-locked erbium-doped fiber laser featuring a large normal dispersion [11], and pointed out that self-similar pulse energy is associated with the location of the output coupler.
Moreover, due to both the high efficiency of ytterbium and the normal material dispersion at 1-μm wavelength, 1-μm fiber laser sources are more favorable for energy scaling through nonlinearity management than 1.5-μm fiber lasers [12–14]. It is apparent that in order to obtain higher energy pulses from a broadband similariton 1-μm fiber laser, cavity parameters should be optimized. Furthermore, since the NPR mode-locked laser tends to be environmentally unstable [15], a real and passive mode locker, graphene, is wavelength independent and usually used in the cavity [16]. Hendry et al. have demonstrated that graphene exhibits a very strong nonlinear optical response [17]. These characters suggest that graphene may be of benefit for the formation of broadband pulses in the cavity at relative lower pump power. This was the initial motivation of our work.
In this paper, we report the optimal design of a broadband similariton fiber laser with graphene as a saturable absorber and a segment of photonic crystal fiber as a nonlinear medium. Using numerical simulations based on the cubic complex Ginzburg-Landau equation (CGLE), we have investigated the influence of different parameters including the cavity length, the spectral filter bandwidth, the pump power, the length of the photonic crystal fiber and the output coupling ratio on the output pulse of the laser. The simulation results show that each parameter value has a certain range which allows the pulse at the output retained a fully parabolic profile and that the chirp was sufficiently linear.
2. Numerical model
Pulse propagation in the laser cavity described above was modeled using the CGLE as:
where U(z,t) is the slowly varying electric field envelope, z is the propagation distance of the pulse, t is the pulse local time, Ω is related to the filter bandwidth, β2 is the second order group velocity dispersion and γ is the Kerr nonlinear coefficient. Higher-order dispersion and higher-order nonlinearity were ignored. The pulse energy Epulse can be calculated as:
where TR is the cavity round trip time. The saturated gain coefficient of the Yb-doped fiber was modeled as:
where g0 is the small signal gain, which is non-zero only for the gain fiber, and Esat is the gain saturation energy. A formula of intensity-dependent nonlinear absorption was used to model the role of the saturable absorber [18] as:
where Ipulse and Isat are the instantaneous intensity of the pulse and the saturation intensity, respectively. αS is the nonlinear saturable absorption loss and αNS is the nonlinear nonsaturable absorption loss (see Table 1). The nonlinear saturable absorption loss is also called modulation depth.
Table 1. Main parameters of the monolayer graphene saturable absorber in our simulationsa
In order to quantify the difference between a similariton and the output pulse actually obtained, a misfit parameter M between the pulse intensity profile |U(t)|2 and a parabolic fit |Up(t)|2 was calculated [19]:
Generally, M = 0.14 for a Gaussian pulse, while M ≤0.06 corresponds to a parabolic pulse [20]. In this paper, we arbitrarily assume a misfit parameter of M ≤0.04 to define a pulse that is sufficiently parabolic to ensure the formation of self-similar pulses [21].
The cavity structure for the laser simulated is illustrated in Fig. 1.
Fig. 1 Schematic of the similariton laser. SMF: Single mode fiber; YDF: Yb-doped fiber; PCF: Photonic crystal fiber; SA: Monolayer graphene saturable absorber; SF: Spectral filter.
The laser cavity used as a starting point contained 200 cm of single mode fiber (SMF-1), 80 cm of Yb-doped fiber (YDF), another 20 cm of single mode fiber (SMF-2) and 100 cm of photonic crystal fiber (PCF). To make our simulation as realistic as possible, we used the same parameters for the saturable absorber (SA) as were investigated in [18] (see Table 1). The output port placed before the SA had a 70% output. The SA was followed by a Gaussian spectral filter with a bandwidth of 10 nm. The gain bandwidth, dispersion and nonlinear coefficients of the fiber were taken to be 80 nm, β2SMF = β2YDF = 23 ps2/km, β2PCF = 7 ps2/km, γSMF = γYDF = 4.5 (W·km)−1, and γPCF = 40.5 (W·km)−1. The small signal gain, the gain saturation energy and the splicing loss used in the simulation were g0 = 30 dB, Esat = 1.44 nJ and 35%, respectively. To accelerate the convergence of the calculation, a Gaussian pulse with a FWHM of T0 = 100 fs, peak power of P0 = 100 W and central wavelength of λ = 1030 nm was used as the initial field [22]. In [22], we have demonstrated that both the white noise and the Gaussian shaped pulse as initial field would not affect conclusions of system design. The governing CGLE was solved with a standard symmetric split-step propagation algorithm, in which the linear terms were solved in the Fourier domain and the nonlinear terms were solved with a fourth-order Runge-Kutta algorithm.
3. Numerical simulations
Using the above model and parameters, we simulated the output pulses as shown in Fig. 2. The misfit parameter M of the output pulse just before the SA was M = 0.0203, which identified the pulse as having a nearly parabolic shape [see Figs. 2(a)–2(f)]. The most notable feature of Fig. 2(b) is that the spectrum of the pulse has a spectral modulation with many peaks. We can understand it as follows, in general, the same chirp occurs at two values of T as shown in Fig. 2(c), it indicates that, at the same instantaneous frequency, there exists two waves, which with different phases, since these two waves can interfere constructively or destructively depending on their relative phase difference, the multipeak spectral modulation formed. Furthermore, since the number of peaks depends on maximum phase shift, the highly nonlinear coefficient of the PCF will increase the number of peaks [23–25]. From Figs. 2(e) and 2(f) we can see that, unlike in conventional amplifier-similariton fiber lasers in which the similariton is a local nonlinear attractor in the gain segment [26,27], in our simulation, since the PCF has a highly nonlinear coefficient, the evolution to a similariton is located in the PCF. In order to further investigate the dynamic features of our similariton fiber laser, we numerically analyzed the effects of the cavity length, the spectral filter bandwidth (SFBW), the length of the PCF, the small signal gain, and the output coupling ratio (OCR).
Fig. 2 The similariton pulse (a), spectrum (b), chirp (c) and dechirped pulse (d) at the output port, the evolution of the pulse duration and spectral bandwidth (e), and the evolution of the misfit parameter M (f) along the cavity.
3.1 The effect of the cavity length
To investigate the effects of the cavity length, we varied the length of SMF-1 from 120 cm to 360 cm (corresponding to changing the net cavity dispersion from 0.058 ps2 to 0.113 ps2), while holding constant the other variables (PCF = 100 cm, g0 = 30 dB, OCR = 70%, SFBW = 10 nm).
With increasing cavity length, the pulse duration became broader and the spectral width became narrower [Fig. 3(a)]. In addition, the pulse energy increased while the pulse peak power slightly decreased [Fig. 3(b)]. We can qualitatively understand this in the following way. A pulse propagating in an optical fiber is dominated, at low pump power, by the effect of group velocity dispersion (GVD) and self-phase modulation (SPM). The SPM introduces new frequency components that are red-shifted near the pulse leading edge and blue-shifted near the pulse trailing edge. Different frequency components in fiber have different propagation velocities. In the normal dispersion regime, red-shifted components travel faster than blue-shifted components. Since an increase in the cavity length corresponds to an increase in the effect of the GVD and a decrease in the SPM, this results in an increase in the pulse duration and decrease in the spectral width as the length of SMF-1 increases. For the length of SMF-1 lying in the range from 161 cm to 283 cm, the misfit parameter M was always smaller than 0.04 [Fig. 3(c)], corresponding to a high quality parabolic pulse.
Fig. 3 The pulse duration and spectral width (a), the pulse energy and pulse peak power (b), and the misfit parameter M (c) as a function of the length of SMF-1.
3.2 The effect of the spectral filter bandwidth
Secondly, we varied the SFBW from 8.5 nm to 14.5 nm, while keeping other variables constant (SMF-1 = 200 cm, PCF = 100 cm, OCR = 70%, g0 = 30 dB). The influence of changes in the SFBW is shown in Fig. 4. With increasing SFBW, the pulse duration and the pulse energy both increased, while the spectral width decreased and the pulse peak power remained roughly unchanged [Figs. 4(a) and 4(b)]. This is qualitatively similar to increasing the cavity length. The misfit parameter M of the pulse was smaller than 0.04 in the SFBW range from 8.7 nm to 12.5 nm [Fig. 4(c)].
Fig. 4 The pulse duration and spectral width (a), the pulse energy and pulse peak power (b), and the misfit parameter M (c) as a function of the SFBW.
3.3 The effect of the pump power
Thirdly we changed the pump power by varying g0 from 21 dB to 39 dB, and maintaining other parameters constant (SMF-1 = 200 cm, PCF = 100 cm, SFBW = 10 nm, OCR = 70%). The results are shown in Fig. 5. With increasing pump power, both the pulse duration and the spectral width increased [Fig. 5(a)], as did the pulse energy and the pulse peak power [Fig. 5(b)]. This is consistent with the characteristics of a self-similar pulse [7]. From Fig. 5(c) we can also see that the output pulse maintained a parabolic shape (M ≤ 0.04) when the pump power was changed from 24.9 dB to 36.4 dB.
Fig. 5 The pulse duration and spectral width (a), the pulse energy and pulse peak power (b), and the misfit parameter M (c) as a function of the pump power.
3.4 The effect of the photonic crystal fiber
In the simulation, we used the PCF (characteristics of NL-1050-NEG-1 fiber from NKT Photonics) as a nonlinear medium in order to increase the spectral width. To gain broader understanding of the similariton fiber laser dynamics, we varied the length of the PCF from 50 cm to 140 cm, while fixing the other variables (SMF-1 = 200 cm, SFBW = 10 nm, g0 = 30 dB, OCR = 70%).
It may be seen from Fig. 6 that as the length of the PCF increased, the spectral width and pulse duration also increased [Fig. 6(a)], while the pulse energy and the pulse peak power decreased. Since increasing the length of the PCF is equivalent to increasing the effect of the SPM, the changes in the pulse duration and the spectral width are similar to the effects of increasing the pump power [Compare with Figs. 5(a) and 6(a)]. On the other hand, the pump power did not change when the PCF was increased. Since the nonlinear coefficient of the PCF is much larger (about 9 times) than that of the SMF, the effect of the SPM becomes much stronger as the PCF length increases, and it requires much more energy to introduce new frequency components, which results in the observed decrease in the pulse energy and peak power [Fig. 6(b)]. The values of the misfit parameter M were less than 0.04 when the length of the PCF lay in the range from 65 cm to 120 cm.
Fig. 6 The pulse duration and spectral width (a), the pulse energy and pulse peak power (b), and the misfit parameter M (c) as a function of the length of the PCF.
3.5 The effect of the output coupling ratio
Finally, we changed the output coupling ratio from 50% to 90%, while holding other parameters unchanged (SMF-1 = 200 cm, PCF = 100 cm, g0 = 30 dB, SFBW = 10 nm). It may be seen from Fig. 7 that as the OCR increased, both the pulse duration and the spectral width gradually decreased [Fig. 7(a)]. The variation of the pulse energy and the pulse peak power versus the OCR are shown in Fig. 7(b). It may be seen that increasing the OCR initially increased the pulse peak power and the pulse energy. Interestingly, however, the variations were not monotonic in that there existed optimum values of the output coupling ratio, at which the pulse energy and the pulse peak power respectively reached their maximum values. Further study showed that the optimum value of the OCR was related to the PCF length, the SFBW and the pump power. When the pump power or the SFBW was increased while keeping the length of the PCF constant, the optimum value of the OCR became larger. Details are shown in Table 2. In the chosen range for the output coupling ratio, the misfit parameter M was less than 0.04 when the output coupling ratio was less than about 75% [Fig. 7(c)].
Fig. 7 The pulse duration and spectral width (a), the pulse energy and pulse peak power (b), and the misfit parameter M (c) as a function of the output coupling ratio.
Table 2. Optimal output coupling ratio with different pump powers and spectral filter bandwidths. The length of the PCF was fixed at 100 cm. (The symbol “—” means our simulation failed to converge in this situation)
4. Generation of broadband self-similar pulses with high pulse energy
From the above investigation we can see that a self-similar parabolic pulse can be obtained by maintaining a composite balance between nonlinearity, dispersion, gain and loss. In order to obtain a self-similar pulse with a broadband spectrum, a short SMF, a narrower SFBW, a longer PCF, and a higher pump power are required. On the other hand, if the goal is to obtain a self-similar pulse with high pulse energy, a long SMF, a wider SFBW, a shorter PCF, a higher pump power and an optimum OCR should be chosen. Consequently we can obtain an optimal design for a broadband high energy similariton fiber laser as follows. For a laser with a fixed SMF length and a fixed SFBW, one can increase the pump power to enhance the pulse energy and the spectral width, and choose the optimal OCR to obtain the pulse with the maximum pulse energy. However, when the above parameters are chosen, it is necessary to ensure that the pulse retains its parabolic intensity profile, since every parameter has only a certain range within which the output pulse retains a parabolic shape.
In the simulations, using the optimal parameters (LSMF-1 = 174 cm, LPCF = 94 cm, g0 = 110 dB, SFBW = 20 nm, OCR = 90%), a single pulse with a spectral width of 186.6 nm which exceeds the 80-nm gain bandwidth, pulse energy of 23.8 nJ, dechirped pulse duration of 22.5 fs, and dechirped pulse peak power of 1.26 MW was obtained as shown in Figs. 8(a)–8(d). In this part, the optimal design conclusions of similariton fiber laser are clear, although the small signal gain g0 had a great value.
Fig. 8 The similariton pulse (a), spectrum (b), chirp (c), dechirped pulse (d) with the optimal cavity parameters.
5. Conclusion
In summary, we have investigated the influence of various parameters on the generation of a broadband similariton. Our simulations show that there are significant ranges of parameter values which allow the misfit parameter M to be less than 0.04, thus ensuring the formation of self-similar pulses. Further experimental investigation of the broadband similariton fiber laser is underway.
Funding
National Natural Science Foundation of China (NSFC) (11374089, 61605040); Natural Science Foundation of Hebei Province (NSFHP) (F2017205162, F2017205060, F2016205124); Program for High-Level Talents of Colleges and Universities in Hebei Province (PHLTCUHP) (BJ2017020); Science Foundation of Hebei Normal University (SFHNU) (L2016B07).
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