1. Introduction
A mode-locked fiber laser exhibits a wide variety of com- plex dynamics including Q-switch, single pulse and multiple pulse operation [1–4]. The pulse’s stability is an important issue and must be considered for practical applications.
However, several mechanisms limit the pulse energy and prevent single-pulse oscillation [5–8]. As a consequence, multiple pulse operation often occurs above some pumping threshold. Several mechanisms contribute to this behavior such as the excess of chromatic dispersion in the fiber, the nonlinear losses and the high power in the cavity. It arises in passive mode-locking when a dissipative single soliton becomes unstable. A double balance, between nonlinear- ity and dispersion, and losses and gain must be satisfied for any pulsed laser emission [9]. So, managing these effects becomes crucial for stabilizing the pulsed fiber laser.
Since the theoretical prediction of the existence of bound solitons [10], there have been considerable works in multi- ple-pulse fiber lasers. The latter can manifest through differ- ent patterns. Tang et al report for the first time experimental
observation of the bound state in a passively mode-locked fiber laser [11]. The observed bound solitons are due to a direct interaction between the solitons in the laser. This result was confirmed by the works of Grelu [12]. In both experi- ments the mode-locking is achieved by the nonlinear polariza- tion rotation (NPR).Using the same technique and by scaling up the pump power, patterns similar to the states of the mat- ter have been observed. Indeed, soliton gas, soliton liquid, soliton polycrystal and soliton crystal were identified [2, 13]. Multi-order bunched soliton pulses have been reported in erbium-doped fiber laser mode locked by the nonlinear polarization rotation [14]. Harmonic mode-locking (HML) of solitons or packets of solitons is another dynamic involv- ing multiple pulses. Indeed, HML has been reported with a single pulse [15], or with packets of two, three or more pulses [16, 17]. Multiple pulses have been experimentally reported in figure-of-eight lasers [18, 19]. These experimental works point out the universal nature of the multi-soliton regime [19].
Multiple noise-like pulses have been demonstrated as a recur- rent feature in fiber lasers. They have been reported both in a figure-of-eight laser [20] and in a ring configuration, mode Khmaies Guesmi1,2, Faouzi Bahloul1, Mohamed Salhi2, François Sanchez2
and Rabah Attia1
1 Laboratoire Systèmes Electroniques et Réseaux de Communications (SERCOM), Ecole Polytechnique de Tunisie, EPT, B.P. 743, 2078, Université de Carthage, Tunisia
2 Laboratoire de Photonique d’Angers, E.A. 4464, Université d’Angers, 2 Bd Lavoisier, 49045 Angers Cedex 01, France
E-mail: [email protected] Received 30 January 2015, revised 19 February 2015 Accepted for publication 20 February 2015
Published 18 March 2015 Abstract
We report a theoretical investigation of multi-pulse emission of a microstructured figure-of- eight fiber laser operating in passive mode-locking. The proposed laser is mode locked by the nonlinear amplifying loop mirror (NALM). We study, in this paper, the hysteresis dependence and the number of pulses in steady state as a function of both the small signal gain and the nonlinear coefficient of microstructured fiber. The numerical simulation confirms that the pulse splitting is a consequence of the energy quantization in anomalous dispersion. Moreover, our results suggest that the hysteresis phenomenon is an intrinsic feature of the mode-locked fiber lasers independently of the exact mode-locking mechanism. Finally, we identify that the nonlinear coefficient of microstructured fiber plays a key role in the formation of multi-soliton.
Keywords: figure-of-eight, microstructured optical fiber, mode locked (Some figures may appear in colour only in the online journal)
K Guesmi et al
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locked by the nonlinear polarization rotation and operating at high-order harmonic noise-like pulsing [21]. Topological insulators have been used as an effective saturable absorber to achieve mode-locking and to obtain multi-soliton patterns and noise-like pulsing [22].
Multiple pulsing in fiber lasers is generally accompanied by multistability and hysteresis phenomena resulting from the sensitivity to initial conditions. In the NPR technique, Komarov et al used the model consisting of an iterative equation taking into account the Kerr nonlinearity, the phase plates, the polarizer, the partial differential equation related to the gain, the group velocity dispersion (GVD) and additional frequency selective losses [23]. The numerical simulation demonstrates the bistability between the mode lock and the continuous regime (CW), the pump power hysteresis and the multistability. The experimental results given by Haboucha et al showed that the theoretical results were favorably com- pared with experimental data in the normal dispersion regime while systematic differences exist in the anomalous case [24]. Tang et al have experimentally found the pump power hysteresis during the formation and annihilation of solitons in a unidirectional fiber ring laser [25]. Zavyalov et al reported numerically a hysteresis phenomenon of soliton molecules and bistability in a mode-locked fiber laser. They identified the crucial impact of saturation gain in the existence of these phenomena [26]. In [27] the authors suggested that the accu- mulation of excessive pulse chirp associated with the non- linear polarization effect in large normal dispersion plays a key role in the existence of multistability. These results were demonstrated numerically in [28].
In this paper, we propose, for the first time to the best of our knowledge, the study of the hysteresis and multistability in a figure-of-eight microstructured fiber laser. We generate numerically up to thirteen pulses for small signal gain. We investigate these phenomena as a function of both the small signal gain and the nonlinear coefficient of the microstruc- tured fiber. In contrast with results obtained in the NPR case [23–25], we point out the existence of narrow instabil- ity regions between the n and n + 1 stable pulse emission domains. Energy quantization is identified as responsible for pulse splitting in the anomalous dispersion regime. Finally, we show that the nonlinear coefficient of microstructured fiber has a strong impact on the pulse evolution.
2. Numerical model with gain saturation
The proposed figure-of-eight microstructured fiber laser is illustrated in figure 1. The key element in this configuration is the microstructured optical fiber (MOF) inserted in the non- linear amplifying loop mirror (NALM). It is used to control the dispersion and nonlinearity of the cavity. In addition to the MOF, the NALM contains an erbium-doped fiber amplifier EDFA and pieces of a standard single-mode fiber (SMF). The asymmetrical 90/10 coupler is used to connect the NALM to the unidirectional ring (UR).The UR is composed of an opti- cal isolator in order to ensure unidirectional propagation, a dispersion shifted fiber (DSF) and an optical coupler with 50% output coupling. The total cavity length is 21 m which corresponds to a fundamental repetition rate of fR = 9.5 MHz.
The fiber laser operates in the anomalous dispersion regime with β2 totalL = −0.06 ps2.
In such a configuration the NALM is responsible for the mode-locking. Pulses from the UR are split into two signals, propagating through the NALM in opposite directions. These pulses undergo different nonlinear phase shifts because of the use of an asymmetric coupler and because one pulse is first amplified in the erbium-doped fiber and then propagates through the MOF, while the other one propagates through the MOF before being amplified by the EDFA. Therefore, the interference condition for recombining the two pulses in the central coupler depends on the signal intensity. Mode-locking is realized when high signal intensity constructively interferes in the clockwise direction of the UR while low signal inten- sity efficiently interferes in the anti-clockwise direction and is therefore blocked by the optical isolator. The steady state is obtained after hundreds of round trips in the cavity.
For our analysis, we consider the case of isotropic fibers.
In this situation the fibers do not affect the polarization state of the electric fields. The fibers are assumed to exhibit optical Kerr nonlinearity, group velocity dispersion, losses and also saturable gain for the EDFA. The use of the following scalar modified nonlinear Schrödinger equation to describe the field evolution in the fiber laser is therefore justified:
⎛
⎝⎜⎜ ⎞
⎠⎟⎟
ω β
τ
α γ
∂
∂ + + ∂
∂ − − =
U z
j j g U g
U j U U 2
( )
2 ,
g2 2 2
2
(1)2
Figure 1. Figure-of-eight microstructured fiber laser.
cient of the fiber, ωg is the bandwidth of the laser gain and g describes the gain function of EDFA. It is given by:
= +
g g
1 EE
0
p sat
(2) where g0 is the small signal gain which characterizes the pumping level, Esat is the saturation energy and EP is the pulse energy given by:
∫
=
−
E U d ,t
T T
P
2 2
2
R R
(3)
where TR is the cavity round-trip time. g is equal to zero in passive fibers.
The implementation of the numerical simulation in the fig- ure-of-eight microstructured fiber laser is taken thanks to the symmetrized split-step Fourier method detailed in our previ- ous paper [29].
3. Results and analysis
Based on our previous numerical algorithm [29], we optimize the operation of the cavity in terms of energy and the number of generated solitons. The main objective is the investigation of hysteresis phenomena and multistability in the figure-of- eight fiber laser and the role of the microstructured fiber.
Numerical simulations have shown that a highly asymmetrical coupler must be used between the UR and the NALM in order to have a large number of pulses in the cavity. For the same reason the fiber laser must operate in anomalous dispersion regime (β2totalL = −0.06 ps2) which favors multiple pulsing.
Thus, by properly adjusting the laser parameters, self-starting mode-locking can be achieved. The parameters used in our simulations are listed in table 1. The spectral gain bandwidth of the amplifier is ωg = 15.7 ps−1.
Let us note that splicing losses together with other unde- sired localized losses are neglected in our approach. Thus, the cavity produces multiple ultrashort pulses with pulse energy of about 22 pJ.
3.1. Hysteresis phenomena and multistability in the multiple pulses regime
In this section, we have fixed Esat to 20 pJ and we are inter- ested in the dynamics of the fiber laser as a function of the
in figure 2. The hysteresis phenomena and multistability are clearly pointed out.
As can be seen in figure 2, by increasing the small signal gain, the fiber laser emits a single pulse just above threshold.
Then, if g0 is increased, additional pulses appear one by one up to 13 pulses per cavity round trip. The resulting soliton pattern can be organized or randomly distributed [2, 19]. For example, from g0 = 0.5 m−1 to g0 = 0.7 m−1, the fiber laser emits bound states of soliton as illustrated in figure 3. Indeed, the optical spectrum is strongly modulated, thus proving the mutual coherence between the two pulses [11, 12]. Above g0
= 0.7 m−1, the generated pulses are randomly distributed along the cavity. During the numerical simulations we have observed narrow instability regions localized in the transition zone from states of n-pulse to n + 1-pulse (see figure 4). Such behavior has been obtained experimentally in [27] and numerically in [28] in large normal dispersion for unidirectional fiber ring laser. Bale et al also observed this phenomenon [30]. For clarity, these zones are not represented in figure 2. They are reported in figure 4. It is expected that this instability acts as a precursor to the transition from n- to n + 1-soliton mode-lock- ing regime. Now, at the level of 13 pulses, which is the high- est number of pulses that has ever been achieved numerically, to our best knowledge, in figure-of-eight fiber laser, if the small signal gain decreases, the dynamics of multiple soliton are different. Indeed, the switching values of the small signal gain depend on whether the gain is increased or decreased.
Moreover, the number of pulses disappears one by one. Note that at the end of the decreasing process, the laser is not able to operate in the single-pulse regime. Thus, these results con- firm the existence of hysteresis phenomena and multistability versus the small signal gain in the figure-of-eight fiber laser.
We have tested several configurations in both the normal and anomalous dispersion regime and for different parameters and we have noticed that the hysteresis cycle and multistability occur in a very wide range of parameters of the fiber laser.
This is analogous to what is obtained in the fiber ring laser either experimentally or theoretically [23–28]. In fact, our result is very important since it suggests that hysteresis phe- nomena and multistability are general features of fiber lasers independent of the exact mode-locking mechanism (nonlinear amplifying loop mirror or nonlinear polarization evolution).
3.2. Pulse energy quantization
It is well known that in the anomalous dispersion regime, the pulse energy of a dissipative soliton is limited by the area
theorem [5]. Therefore, soliton energy quantization limits the generation of pulses with large energy. In fact, the pulse energy can be determined by the peak power and the pulse width. Our numerical simulations show that the energy per pulse varies between about 17 and about 22 pJ. The energy given by the pump provides the amplification of the pedestal of pulses. The domains of maximum peak power coincide with the domains of minimum pulse duration. The highest peak power is about 60 W and the smallest pulse duration is about 250 fs.
Let us now consider the evolution of the pulse energy as a function of the small signal gain. At the beginning of the formation of the n-soliton state the energy per pulse is low, corresponding to pulses with small pedestals. At the end of the n-soliton regime, the energy becomes higher and the pedestals of solitons larger. Because of an energy limiting effect, the n solitons spontaneously split into n + 1 solitons with lower
energies. Just before the n + 1 soliton state, a narrow unstable region occurs. The new train of pulses undergoes the same evolution when the small signal gain increases, and hence it splits into a new pulse train of n + 2 pulses. This feature is illustrated in figure 4.
3.3. Impact of the nonlinear coefficient of the microstructured fiber
As can be seen in figure 5, a high nonlinear coefficient of the microstructured fiber can improve the pulse production in the cavity. Indeed, this result is in good agreement with our previous study [29] concerning the effect of the MOF non- linear coefficient on the generation of a bound soliton pair.
The microstructured fiber provides a precise control of the dispersion and the nonlinearity in the cavity, and hence, this propriety can improve the dissipative pulse production which
Figure 2. Numerical observation of the pulse generation as a function of the small signal gain.
0,0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 2,4 2,7 3,0 3,3 0
2 4 6 8 10 12 14
Pulsenumber(n)
Small signal gain(m-1)
Descending process Ascending process
Figure 3. Bound soliton pair obtained at g0 = 0.6 m−1 (a) temporal response (b) the corresponding optical spectrum.
1530 1540 1550 1560 1570
-70 -60 -50 -40 -30 -20
Spectrumintensity(dBm)
Wavelength (nm) (b)
-20 -15 -10 -5 0 5 10 15 20
0 5 10 15 20 25 30
Power(W)
Time (ps) (a)
is a direct result of a balance between dispersion and the Kerr effect on one hand and loss and gain on the other hand.
Therefore, we have varied the effective area of the micro- structured fiber keeping constant the other parameters of the laser. Figure 5 shows the evolution of the steady state as a function of the effective area. We can observe that the num- ber of pulses increases with increasing the nonlinear coef- ficient of microstructured fiber, which scales inversely with the effective area (γ=2π λn2/ Aeff). This formation is charac- terized by a decrease in the energy per pulse. The increase of the number of pulses is due to the unbalanced average disper- sion by the nonlinear effects in the cavity. Indeed, for a given n-pulse state, if the nonlinear parameter changes then the regime becomes unstable. In our simulation, eleven pulses are obtained with an effective area of MOF equal to 10 µm2
while only two pulses are observed in 85 µm2 for g0 = 3 m−1 and Esat = 10 pJ (see figure 5).
4. Conclusion
In conclusion, we have numerically generated thirteen pulses in a figure-of-eight microstructured fiber laser. We have dem- onstrated the existence of small signal gain hysteresis and multistability. Furthermore, we have pointed out the existence of narrow instability regions between two adjacent multiple soliton states and we have studied the mechanism of multiple soliton generation. The energy quantization is clearly respon- sible for pulse splitting in anomalous dispersion regime. The multiple soliton generation in the laser can be optimized by the use of a microstructured fiber in the cavity.
Figure 4. Evolution of the pulse energy as a function of the small signal gain. The gray region corresponds to unstable regions.
0,2 0,4 0,6 0,8 1,0 1,2 1,4
17 18
Puls 19
Small signal gain (m-1)
Figure 5. Pulse evolution as a function of the effective area of the microstructured fiber for Esat = 10 pJ and g0 = 3 m−1.
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