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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Optimal design of higher energy dissipative-soliton ﬁber lasers Huaxing Zhang, Shumin Zhang n, Xingliang Li, Mengmeng Han College of Physics Science and Information Engineering, Hebei Advanced Thin Films Laboratory, Hebei Normal University, Shijiazhuang 050024, China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 July 2014 Received in revised form 9 September 2014 Accepted 10 September 2014 Available online 22 September 2014

We have numerically investigated dissipative soliton (DS) ﬁber lasers using the cubic complex Ginzburg– Landau equation (CGLE), and analyzed the effects of different parameters, including the cavity length (CL), the spectral ﬁlter bandwidth (SFBW), nonlinear phase shifts (ΦNL), and the initial ﬁelds. The results showed that for any initial input ﬁelds, once the stable output pulse was formed, the output pulse energy would increase with decreasing CL, or increasing ΦNL. When the pump power and CL parameters were ﬁxed, there existed an optimal SFBW for which the highest output pulse energy could be obtained. In order to obtain higher pulse energy in a DS ﬁber laser, one can ﬁrst ﬁx the CL according to the required repetition frequency, then optimize the SFBW, and ﬁnally, increase the pump power, which corresponds to increasing the ΦNL. In simulations, using optimal parameters, a single pulse with energy of 50 nJ, pulse duration of 10 ps, dechirped pulse duration of 37 fs, and pulse peak power of 1.4 MW was obtained. & 2014 Elsevier B.V. All rights reserved.

Keywords: Fiber laser Spectral ﬁlter Dissipative soliton Cavity length Nonlinear phase shifts

1. Introduction Fiber lasers have attracted much attention because of their simple design, low cost, high stability and low alignment sensitivity [1–8]. In order to achieve short pulses from a ﬁber laser, the mode locking technique is usually used. Generally, mode locked ﬁber lasers are based on the balance of nonlinearity and dispersion, but the pulse energy in this type of the mode locked laser is limited to 0.1 nJ by the soliton area theorem [9,10]. To achieve higher energy pulses, dispersion-managed soliton lasers or stretched-pulse ﬁber lasers with pulse energies and peak powers up to several nanojoules and kilowatts have been proposed [11,12]. Recently, researchers have shown theoretically [9] and experimentally [13–15] that a new kind of soliton – dissipative solitons (DSs), which operate at large normal dispersion can have much higher pulse energies [7,16]. In 2006, an all-normal-dispersion (ANDi) femtosecond ﬁber laser was demonstrated by the Wise group [17], in which pulse shaping was based on spectral ﬁltering of a highly chirped pulse in the cavity. These lasers depended strongly on the dissipative processes, that is to say, the lasers depended on the balance between linear gain, loss, nonlinearity and diffraction dispersion to shape the pulse. There was no anomalous group velocity dispersion (GVD) in the cavity. In 2007, Chong et al. demonstrated Yb-doped ﬁber lasers with pulse energies above 20 nJ, and dechirped pulse peak powers of nearly

n

Corresponding author. Fax: þ 86 311 80787305. E-mail address: [email protected] (S. Zhang).

http://dx.doi.org/10.1016/j.optcom.2014.09.031 0030-4018/& 2014 Elsevier B.V. All rights reserved.

100 kW [18]. In 2008, Ruehl et al. reported an erbium-doped ﬁber oscillator mode-locked by nonlinear polarization evolution operating in the large normal dispersion regime, and producing highly chirped, ultrashort 10 nJ pulses with dechirped peak powers of 140 kW [19]. Recently, DS ﬁber lasers have been actively investigated [20–23]. Pulses with energies of 420 nJ have been achieved in an ANDi ﬁber laser [18]. However, in contrast to the area theorems for conservative optical solitons, the DS energies were found not to scale inversely with the pulse duration. However, there was an upper limit to the pulse energy. When the pulse energy exceeded this upper limit, wave breaking occurred and multiple pulses formed [24]. Lou et al. experimentally measured the characteristics of solitary pulses in an ANDi Yb-doped ﬁber laser, and observed energy quantization [25]. Zhu et al. studied a tunable, high-order harmonic and passively mode-locked 75 m long Yb-doped ﬁber laser with ANDi, and pointed out that the order of the harmonic mode-locked pulses depended on the cavity length [26]. In 2014, our group produced pulse bursts with a controllable number of pulses per burst directly from a mode-locked Yb-doped ﬁber laser [27]. From the above discussion, it is apparent that in order to obtain a high energy pulse from an ANDi laser, cavity parameters should be optimized. In 2012, Chichkov et al. have experimentally investigated the scaling properties of an ANDi laser. Using ytterbium-doped double-clad ﬁber as the gain medium, and by stepwise variation of the resonator dispersion, total ﬁber length, and the spectral ﬁlter bandwidth, they have obtained pulse with the pulse energy up to 84 nJ [28]. Chong et al. have reported the

H. Zhang et al. / Optics Communications 335 (2015) 212–217

properties of an ANDi femtosecond ﬁber lasers [16,21]. In this paper, we report the optimal design of DS ﬁber lasers by using numerical simulations based on the cubic complex Ginzburg– Landau equation (CGLE). We demonstrate the effects of different parameters, such as cavity length (CL), spectral ﬁlter bandwidth (SFBW), nonlinear phase shifts (ΦNL) and the initial ﬁelds. The results show that for a laser of given cavity length, with any initial ﬁelds, there always exists an optimal SFBW, for which the highest pulse energies can be obtained. Using the optimal parameters, we have obtained, in simulations, pulses with pulse energy of 50 nJ and a pulse duration of 10 ps. This pulse was dechirped to 37 fs, and then had a 1.4 MW peak power. We have also theoretically studied DS ﬁber lasers with different CLs of 5, 6, 8, 12 and 16 m, and have found optimal SFBWs of 12, 10, 8, 6 and 4 nm, respectively.

213

2. Numerical model Optical pulse evolution in DS mode-locked ﬁber laser cavities can be theoretically described using the CGLE with a cubic saturable absorber term as follows: 2 ∂Aðz; τÞ 1 β2 ∂2 Aðz; τÞ ¼ gðEpulse ÞAðz; τÞ þ i þ iγ Aðz; τÞ Aðz; τÞ; ∂z Ω 2 ∂τ2 ð1Þ where A is the electric ﬁeld envelope, τ is the time from the center of the pulse, z is the propagation distance, Ω is related to the SFBW, β2 is the second order group velocity dispersion, γ is the Kerr nonlinear coefﬁcient, and Epulse is the pulse energy. Epulse can be expressed as Z Epulse ¼

T R =2 T R =2

Aðz; τÞ2 dτ;

ð2Þ

TR is the cavity round-trip time. In Eq. (1), g(Epulse) is the net gain, which is nonzero only for the gain ﬁber. Gain saturation is modeled according to gðEpulse Þ ¼

Fig. 1. Schematic of the model. SMF: single-mode ﬁber; OC: output coupler; WDM: wavelength division multiplexer.

g0 : 1 þEpulse =Esat

ð3Þ

where g0 is the small-signal gain taken to be 30 dB, and Esat is the gain saturation energy (taken to be 1.44 nJ). The CGLE includes the effects of GVD, Kerr nonlinearity, and saturated gain with a Lorentzian gain shape. The bandwidth is taken to be 80 nm. In the simulations, higher order dispersion and higher order nonlinear effects were ignored.

Fig. 2. DS spectra (a) and pulse shapes (b) for ﬁve different values of the CL. Also shown are the pulse duration (red) and spectral width (black) as functions of the CL (c), and the pulse energy (red) and pulse peak power (black) as functions of the CL (d). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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Fig. 1 shows a schematic of the model. It included a 4-m-long single-mode ﬁber (SMF1), a 2-m-long Yb-doped gain ﬁber, and one more piece of single mode ﬁber, 2 m long (SMF2). The dispersion and nonlinear coefﬁcients of the ﬁber used in the simulation were β2 ¼21 ps2/km and γ ¼4.5 W 1/km. A saturable absorber in the cavity was modeled to generate the pulse with a monotonically increasing transfer function, T ¼1 l0/[1þ P(τ)/Psat], where l0 ¼80% is the unsaturated loss, P(τ) is the instantaneous pulse power, and Psat ¼1000 W is the saturation power. The output coupler placed after the saturable absorber had a 70:30 coupling ratio. The output coupler is followed by a Gaussian spectral ﬁlter with 8 nm bandwidth. A Gaussian shaped pulse with pulse duration of 100 fs and pulse peak power of 200 W was used as the initial pulse. The split-step Fourier method was used to solve Eq. (1).

3. Simulation results

power both decreased monotonically with the increase in CL (Fig. 2(d)). We can qualitatively understand these observations in the following way. During pulse propagation in the ﬁber laser, with small pump power, the dispersion and the self-phase modulation (SPM) play a dominant role in the pulse dynamics. The SPM generates new frequency components that are red-shifted near the leading edge and blue-shifted near the trailing edge of the pulse. Since the laser wavelength is in the normal GVD regime, the red-shifted components travel faster than the blue-shifted components, which results in pulse broadening. Since an increase in the CL corresponds to an increase in the GVD effect and a decrease in the SPM effect, an increase in the CL results in an increase in the output pulse duration and a decrease in the spectral width, pulse energy and the pulse peak power. These simulation results are consistent with theory [29]. From Fig. 2(a) we can see that the spectrum has steep edges, which is a typical spectral characteristic of DSs.

3.1. The effect of the cavity length 3.2. The effect of the spectral ﬁlter bandwidth In order to investigate dynamic features of the DS ﬁber laser, we numerically analyzed the effect of the cavity length, CL, the spectral ﬁlter bandwidth, SFBW, the nonlinear phase shifts, ΦNL, and the initial ﬁelds. To investigate the effect of changing the cavity length, we varied CL from 8 to 12 m by changing the length of SMF1 while holding constant the SFBW (8 nm) and the pump power (g0 ¼ 30 dB). The change in SMF1 resulted in a corresponding change in the net GVD from 0.168 to 0.252 ps2. The output spectral width became narrower (Fig. 2(a) and (c) (black curve)) and the pulse duration (Fig. 2(b) and (c) (red curve)) became broader with the increase in CL. In addition, the pulse energy and the pulse peak

To gain broader understanding of the DS mode-locking laser dynamics, we further investigated the effect of the spectral ﬁlter bandwidth (SFBW). Simulations were performed by varying the SFBW from 4 nm to 16 nm while the other parameters, CL and g0, were held ﬁxed at 8 m and 30 dB, respectively. When the SFBW was increased, the central intensity of the spectrum gradually increased (Fig. 3(a)), whereas the spectral width decreased (see the black curve of Fig. 3(c)), and the pulse duration increased (see Fig. 3(b) and (c) (red curve)). The variation of the pulse energy and the pulse peak power versus the SFBW are shown in Fig. 3(d). It may be seen that increasing the SFBW decreased the pulse peak

Fig. 3. DS spectra (a) and pulse shapes (b) for four different values of SFBW. Also shown are the pulse duration (red) and spectral width (black) as functions of the SFBW (c), and the pulse energy (red) and pulse peak power (black) as functions of the SFBW (d). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

H. Zhang et al. / Optics Communications 335 (2015) 212–217

power. Interestingly, however, the variation of the pulse energy was not monotonic in that there existed an optimum value of the SFBW, at which the pulse energy reached its maximum value. Further study showed that this optimum SFBW was related to the CL when the pump power was kept constant, and the longer of the cavity, the narrower the optimum SFBW. Details are shown in Table 1. 3.3. The effect of the nonlinear phase shifts In this section, we investigated the effect of ΦNL on the output pulse. Fig. 4 shows the trends as ΦNL varies while the SFBW and CL are held constant at 8 nm and 8 m, respectively. Since ΦNL can be calculated using the equation ФNL ¼γPpeakLeff, where ФNL is the nonlinear phase shift, Ppeak is the intracavity pulse peak power, and Leff is the effective interaction length of the ﬁber, there exist several methods to change ΦNL, such as altering the pump power which is equivalent to adjusting g0 of the gain ﬁber, adjusting the output coupling ratio or changing the length of the gain ﬁber. In our work, the values of ΦNL were varied from 1π to 58π by changing the pump power (g0), while the CL was ﬁxed at 8 m with its corresponding optimum SFBW value of 8 nm. Comparing Figs. 2 and 4 we can see that, for the spectral width, pulse energy, and the pulse peak power, reduction of CL produces the same qualitative trend as increasing ΦNL. However, the pulse duration initially Table 1 Laser CLs corresponding to their optimal SFBWs. CL (m) Optimal SFBW (nm)

5 12

6 10

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decreases as ΦNL increases. With a further increase in ΦNL, the pulse duration increases with ΦNL. That is to say, there exists a value of ΦNL, at which the pulse duration reaches its minimum value. We can understand this as follows. Unlike the conventional soliton, in which the balance is between nonlinearity and dispersion, in the DS ﬁber lasers, this single balance will be replaced by a composite balance between nonlinearity, dispersion, gain and loss. This balance should be exact in order to produce stationary localized solutions [20]. In fact, small values of ΦNL corresponds to small pump power, in this case, the behavior of the temporal and spectral proﬁle mainly depends on whether GVD or nonlinearity dominates in the cavity [29]. The dominance of nonlinearity extensively increases the pulse chirp and hence widens the spectrum. In contrast, dominance of the GVD widens the temporal pulse. At the beginning of increasing the pump power, the nonlinearity increases with the pump power and therefore the temporal pulse width becomes narrower with increasing ΦNL [30]. Then with a further increase in the pump power, the effect of the gain replaces the nonlinearity effect and plays a dominant role in the dynamics. As a result, there is a reversal in the evolution of the pulse duration with a further increase in the ΦNL. However, the dechirped pulse duration decreases and the peak power increases gradually with the pump power as shown in Fig. 5, which is consistent with Ref. [16]. In the simulations, the pulses generated with a maximum energy of 50 nJ and 10 ps pulse duration, can be dechirped to 37 fs duration and 1.4 MW pulse peak power. 3.4. The effect of the initial ﬁelds

8 8

12 6

16 4

Finally, we investigated the effect of the initial ﬁelds on the output pulse. Fig. 6 shows the temporal (a) and spectral (b)

Fig. 4. DS spectra (a) and pulse shapes (b) for four different values of ΦNL. Also shown are the pulse duration (red) and spectral width (black) as functions of ΦNL (c), and the pulse energy (red) and pulse peak power (black) as functions of ΦNL (d). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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Fig. 5. The dechirped pulses with ΦNL values of 1π, 8π, 17π, 24π.

Fig. 6. DS pulse (a) and spectral (b) evolution for white noise (red curve) and Guassian shaped pulse (black dotted curve). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

evolution along the round-trip number (NR) in laser resonator for the initial ﬁeld of white noise (red curve) and a Gaussian shaped pulse with pulse duration of 100 fs (black dotted curve), respectively. The cavity parameters are the same as that in Section 3.1, except holding the cavity length at 8 m. From Fig. 6 we can see that after repeating different cycles (for Gaussian shaped pulse NR ¼15, and for white noise NR ¼100), both the white noise and the Gaussian shaped pulse will evolve into the same pulse shape. Further study showed that the initial ﬁelds did not affect our conclusions of system design.

4. Conclusion In conclusion, we have numerically demonstrated the effects of CL, SFBW, ΦNL, and the initial ﬁelds on DS passively mode-locked ﬁber lasers using the CGLE and the split-step Fourier method. The results showed that for a ﬁxed cavity-length laser, with any initial ﬁelds, once the stable output pulse is formed, there always exists an optimal SFBW, for which the highest pulse energy can be obtained. In the simulation, by using optimal parameters, a pulse with 50 nJ energy, 10 ps pulse duration have been obtained.

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This simulation can be used as a guide for the establishment of high pulse energy DS ﬁber lasers. Acknowledgments This research was supported by grants from the National Natural Science Foundation of China (Grant nos. 11074065, 11374089 and 61308016), the Natural Science Foundation of Hebei Province (Grant nos. F2012205076 and A2012205023), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20101303110003) and the Technology Key Project of Colleges and Universities of Hebei Province (Grant nos. ZH2011107 and ZD20131014). References [1] N. Akhmediev, J.M. Soto-Crespo, Ph. Grelu, Phys. Lett. A 372 (2008) 3124. [2] Z.C. Luo, M. Liu, H. Liu, X.W. Zheng, A.P. Luo, C.J. Zhao, H. Zhang, S.C. Wen, W.C. Xu, Opt. Lett. 38 (2013) 5212. [3] X. Liu, Phys. Rev. A 81 (2010) 023811. [4] X. Wu, D.Y. Tang, H. Zhang, L.M. Zhao, Opt. Express 17 (2009) 5580. [5] Z.C. Luo, Q.Y. Ning, H.L. Mo, H. Cui, J. Liu, L.J. Wu, A.P. Luo, W.C. Xu, Opt. Express 21 (2013) 10199. [6] A. Cabasse, G. Martel, J.L. Oudar, Opt. Express 17 (2009) 9537.

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