Induced dark solitary pulse in an anomalous dispersion cavity fiber laser

Guodong Shao; Yufeng Song; Jun Guo; Luming Zhao; Deyuan Shen; Dingyuan Tang

doi:10.1364/OE.23.028430

1. Introduction

Soliton formation is an interesting nonlinear wave phenomenon that has been observed in diverse physical systems and been extensively studied [1,2]. In nonlinear optics, the optical soliton formation in single mode fibers (SMF) has attracted great attention both due to its fundamental importance and potential applications in optical signal processing and optical communications. It is well known that light propagation in SMFs is governed by the nonlinear Schrodinger equation (NLSE), which supports either the bright or dark optical soliton formation depending on the sign of the fiber dispersion [3]. In practice, a SMF actually supports two orthogonally polarized modes owing to unavoidable fiber birefringence, generated by the fiber bending and/or technical imperfection of the fiber drawing. Therefore, the light propagation in a practical SMF involves the coupling between the two polarization modes. It has been theoretically shown that depending on the strength of fiber birefringence, there is either coherent coupling, where four-wave-mixing (FWM) between the polarization modes also takes place, or incoherent coupling, where FWM is negligible [3]. The polarization coupling between the fiber modes could lead to formation of various nonlinear optical effects, e.g. under the incoherent polarization coupling, polarization domains and walls could be formed [4]; Under the coherent polarization coupling the dark-bright vector solitons had also been theoretically predicted [5,6].

Fiber laser cavity is a fascinating nonlinear system that possesses both the features of light propagation in SMFs and in a nonlinear cavity. The operation of a fiber laser is governed by the complex Ginzburg-Laudau Equation (GLE) [7]. However, when a fiber laser is under the steady state operation, and if the effective gain bandwidth is sufficiently broad so that its bandwidth limiting effect could be ignored, the light circulation in a fiber laser cavity would be equivalent to that in a SMF. Hence, a fiber laser provides an ideal testbed for the experimental study of the various optical soliton phenomena. Apart from the bright solitons, SMFs also support dark solitons. The dark soliton formation in normal dispersion SMFs was experimentally confirmed previously [8]. Recently, the dark soliton formation in fiber lasers has also been demonstrated [9]. Moreover, like the bright solitons formed in fiber lasers, ultrahigh repetition rate dark soliton pulse trains have also been generated [10]. Recently, the observation of bright-dark pulse pair was also reported in a fiber laser [11]. However, we note that so far all the dark solitons formed in SMFs or fiber lasers are in the normal dispersion regime. In this letter we report on a novel effect of induced dark solitary pulse that is formed in a net anomalous dispersion cavity fiber laser. We show experimentally that under coherent polarization coupling, even in the anomalous cavity dispersion regime a dark solitary pulse could be induced in a fiber laser. We also provide results of numerical simulations that support the experimental observations.

2. Experimental setup

Our fiber ring laser setup is schematically shown in Fig. 1. The fiber ring has a total length of 22.6 m, consisting of a piece of 3 m Erbium doped fiber (EDF) with a group velocity dispersion (GVD) parameter of −48 ps/nm/km, 13.5 m single mode fiber (SMF-28) with a GVD parameter of 18 ps/nm/km and 6.1 m dispersion compensation fiber (DCF) with a GVD parameter of −4 ps/nm/km. The fiber ring cavity is estimated to have an average net anomalous GVD parameter of 3.3 ps/nm/km. The fiber laser is pumped by a 1480 nm single mode Raman fiber laser whose maximum output power is 5 W. A polarization independent isolator is inserted in the cavity to force the unidirectional circulation of light in the cavity. Besides, an intra-cavity polarization controller (PC) is used to fine-tune the linear cavity birefringence. A wavelength division multiplexer (WDM) is used to couple the pumping light into the cavity, and a 10% fiber output coupler is used to output the light. The pumping coupling efficiency from the pump source to the cavity is about 60%. All the components used in our experiment have very low polarization dependent loss (PDL) (WDM: 0.01 dB, Isolator: 0.04 dB, Coupler: 0.01 dB). Therefore, the PDL induced mode locking is unlikely to occur in our laser [12]. An external cavity polarization beam splitter (PBS) is used to experimentally resolve the two orthogonal polarization components of the laser emission. To this end a polarization controller is inserted before the external polarization beam splitter to balance the linear polarization change caused by the lead-fibers. The two orthogonal polarization components are simultaneously monitored with a high-speed electronic detection system made of two 40 GHz photo-detectors and a 33 GHz bandwidth real-time oscilloscope.

Fig. 1 A schematic of the Erbium-doped fiber ring laser. EDF: Erbium-doped fiber. SMF: Single mode fiber. DCF: Dispersion compensation fiber. WDM: Wavelength division multiplexer. PC: Polarization controller. ISO: Isolator. OC: Output coupler. OSA: Optical spectrum analyzer.

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3. Experimental results

Under low pump power the laser always emits continuous wave (CW). However, depending on the net cavity birefringence, which can be altered experimentally by changing the orientation of the intracavity polarization controller, the laser emission could be in different polarization states. In the case of very weak net cavity birefringence, the laser could emit either an elliptically polarized beam where the two polarization modes have the same oscillation wavelength, or a linearly polarized beam which can be attributed to the effect of polarization instability [13]. However, the most common state is that the laser emits CW along the two orthogonal polarization modes simultaneously, and each CW has different oscillation wavelengths. Experimentally it is identified that the wavelength difference of them varies with the net cavity birefringence. Specifically, strong birefringence leads to big wavelength difference. Hence experimentally we used the wavelength difference as an indicator on the strength of the net cavity birefringence and operated the laser under different pump strength and net cavity birefringence.

Increasing the pumping power to about 1.0 W (30 dBm), by tuning the intra-cavity PC, a kind of soliton-dark pulse pair emission similar to that shown in Fig. 2(a) is obtained. In a previous paper, we have reported the phenomenon and explained its formation mechanism [14]. Briefly, because the bright solitons and the CW in the opposite polarization direction have different central wavelengths, there is incoherent polarization coupling between them. Consequently, an effect similar to the polarization domain formation takes place, leading to that corresponding to each bright soliton a broad dark pulse is generated on the CW background. We note that the formed dark pulse is much broader than the bright soliton, and the stronger the pump power the narrower the dark pulses. When we keep increasing the pumping power to about 3.2 W (35 dBm), in the meantime also slightly tuning the intra-cavity PC to make the net cavity birefringence very weak, a special phenomenon as shown in Fig. 2 is further observed. Figure 2(a) shows the oscilloscope traces of the polarization resolved laser emission. It shows that the laser emits stable bright soliton-dark pulse pairs, as the phenomenon reported in [14]. However, different from that, associated with each of the bright solitons there is a narrow dark pulse formed on the bottom of each of the broad dark pulses. Figure 2(b) shows the zoom-in of one bright-dark pulse pair. The broad dark pulse is about 250 ps wide. On the bottom of the dark pulse there is an additional narrower dark pulse of about 30 ps wide, which is the bandwidth limitation of our oscilloscope. The narrow dark pulse synchronizes with the bright soliton in the orthogonal polarization direction. Figure 2(c) is the autocorrelation trace of the bright soliton, whose FWHM width is 1.46 ps, suggesting that the width of the bright soliton is about 944 fs if a Sech-shape pulse profile is assumed.

Fig. 2 Experimental results of induced dark solitary pulse formation in a fiber laser. (a) The polarization resolved oscilloscope traces of the laser emission. (b) The zoom-in oscilloscope traces of a soliton-dark pulse pair. (c) Autocorrelation trace of the bright solitons. (d) The Polarization-resolved spectra.

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Figure 2(d) shows the polarization-resolved optical spectra of the laser emission. The central wavelength of CW light (in vertical axis) is 1575.2 nm. The central wavelength of the bright soliton (the broadened spectrum in horizontal axis) is about 1575.5 nm, which can be estimated by fitting the spectrum profile. A Kelly sideband locates at 1588.95 nm, which indicates that the bright pulses in the horizontal axis have fully evolved into solitons [15]. Besides, we can clearly see two peak-dip spectral structures locating at 1570.45 nm and 1580.6 nm, which stands for the energy exchange between the two polarization components and is the symbol of coherent coupling between the bright solitons and the dark pulses [16]. To confirm that the narrow dark pulse is induced by the coherent polarization coupling of the bright soliton, we have experimentally deliberately changed the cavity birefringence so that the two polarization components are in the incoherent coupling. In this case the broad dark pulse still exists, while the narrow dark pulse become unmeasurable. In the meantime, no coherent energy exchange spectral sidebands appear on the optical spectra. The experimental result clearly suggests that the weak dark pulse could be an induced dark soliton by the bright soliton through the coherent polarization coupling. We note that there is also some spectral broadening of the CW beam. It is due to the modulation instability effect. This is also reflected by the noise background of the oscilloscope trace of the CW beam.

In addition to the induced dark solitons, under the coherent polarization coupling, experimentally we have also observed the induced bright soliton formation inside the wide dark pulse, as shown in Fig. 3. Figure 3(a) shows again the polarization resolved oscilloscope traces of the laser emission. Note that in the cavity multiple bright solitons are formed. Associated with each of the bright solitons a weak bright soliton is induced in the orthogonal polarization. Figure 3(b) is the polarization resolved optical spectra. Note that the soliton wavelength and the CW wavelength are not the same. Hence, there is incoherent polarization coupling between them. This explains why each bright soliton also induces a broad dark pulse on the CW beam. The polarization-resolved spectra further show that the induced bright solitons have the same central wavelength as that of the inducing bright solitons. This could be identified by drawing the spectral profiles of the pulses and comparing their Kelly sideband positions. The Kelly sidebands on both polarization directions are clearly visible and they have the same locations. Moreover, the energy exchange spectral sidebands are also visible on the optical spectra, suggesting that the weak bright solitons are coherently induced. We point out that formation of induced bright solitons in mode locked fiber lasers were reported before [17]. We found that the induced bright solitons have very similar features to those reported previously.

Fig. 3 The induced bright soliton formation inside the wide dark pulse of bright-dark pulse emission state. (a) Laser emissions along the two orthogonal polarization directions. (b) The corresponding polarization resolved spectra.

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For the formation mechanism of the narrow dark solitary pulse inside the dark pulse, we notice that D. N. Christodoulides theoretically proved the existence of bright-dark soliton pair by the coherent coupling between polarization modes in [5]. Therefore, we conjecture that in our experiment, the formation of the narrow dark solitary pulse in the wide dark pulse is induced by the bright soliton in its orthogonal polarization mode through coherent coupling. Besides, this is a special phenomenon because in SMF environment, only bright soliton can propagate steady in the anomalous dispersion regime and dark soliton in normal dispersion regime. In our experiment, we find the dark solitary wave also can exist in anomalous dispersion regime. We suggest it’s due to the coherent coupling between two polarization modes and dissipative properties of fiber laser.

4. Numerical simulations and discussion

We have investigated both experimentally and theoretically the polarization domain formation in quasi-vectorial cavity fiber laser and shown that it is an intrinsic feature of the incoherent polarization coupling in the fiber lasers [18,19]. The bright soliton-dark pulse pair formation of the laser under relatively low pump power or even high pump power but incoherent polarization coupling, could be considered as a special case of the polarization domain formation [14]. Here due to the influence of the strong nonlinear self-phase modulation the domains are represented in a different form. What is interesting here is that once the polarization coupling becomes coherent, in addition to the broad dark pulse formation, a weak dark or bright soliton is further induced. The formation of induced dark soliton under the polarization coupling was theoretically predicted before [20,21], and formation of induced bright solitons in a mode locked fiber laser was also experimentally reported [17]. However, to the best of our knowledge, no induced dark solitons in the anomalous dispersion regime are experimentally observed. In order to understand how the dark soliton is formed, we numerical simulated the operation of the fiber ring laser. Our simulation is based on the coupled complex Ginzburg-Landau equations (CGLE), which describe the light propagation in a weakly birefringent fiber cavity,

\begin{array}{l} \frac{\partial u}{\partial z} = i β u - δ \frac{\partial u}{\partial t} - \frac{i k''}{2} \frac{\partial^{2} u}{\partial t^{2}} + i γ ({| u |}^{2} + \frac{2}{3} {| v |}^{2}) u + \frac{i γ}{3} v^{2} u^{*} + \frac{g}{2} u + \frac{g}{2 Ω_{g}^{2}} \frac{\partial^{2} u}{\partial t^{2}} \\ \frac{\partial v}{\partial z} = - i β v + δ \frac{\partial v}{\partial t} - \frac{i k''}{2} \frac{\partial^{2} v}{\partial t^{2}} + i γ ({| v |}^{2} + \frac{2}{3} {| u |}^{2}) v + \frac{i γ}{3} u^{2} v^{*} + \frac{g}{2} v + \frac{g}{2 Ω_{g}^{2}} \frac{\partial^{2} v}{\partial t^{2}} \end{array}

where u and v describe the optical fields of the two orthogonal polarization components in the fiber cavity.

2 β = 2 π Δ n / λ = 2 π / L_{b}

is the wave number difference between these two polarization modes.

2 δ = 2 β λ / 2 π c

is the group velocity difference, which are related with the value of beat length L_b and stand for the cavity birefringence. k” is the second-order dispersion coefficient, γ represents the nonlinearity of the fiber cavity, g is the gain coefficient of the gain fiber and Ω_g is the effective gain bandwidth. The gain saturation is defined as

g = G \exp [- \frac{\int ({| u |}^{2} + {| v |}^{2}) d t}{E_{0}}]

where G is the small signal gain coefficient and E₀ is the gain saturation energy, which is related to the pumping power in the experiment.

We have numerically solved the Eqs. (1) and (2) using the split-step method and details on the numerical techniques used are reported previously [22]. To model the induced dark or bright soliton formation in the laser, we have set the initial state as a soliton-dark pulse pair, which is similar to the experimental situation. The initial bright soliton has the form of $u = A \cdot sech (B \cdot t)$ , while for the initial dark pulse we have either the form of $v = C \cdot \tan h (D \cdot t)$ , where there is a phase jump in the center of the dark pulse; or $v = C \cdot \sqrt{(1 - {sech}^{2} (D \cdot t))}$ , where there is no phase jump. Here A and C stand for the bright soliton intensity and dark pulse depth, usually A is set much bigger than C because the bright soliton peak power is much higher than the CW power in our experiment. B and D stand for the width of bright soliton and the dark pulse. B is set much smaller than D, in accordance with the experimental results that the dark pulse is much wider than the bright solitons. In all our simulations we have possibly used the actual laser cavity parameters if possible, e.g. the cavity length L = 22.6 m and the average cavity dispersion is 3.3 ps/nm/km. In our experiment results, the coherent coupling between two orthogonal polarization components can only happen under very weak cavity birefringence. So we set L_b = 50 km, which means the cavity birefringence is very weak.

First, we numerically simulated the case of dark pulse with a phase jumping $v = C \cdot \tan h (D \cdot t)$ . Under appropriate parameters selection, we can obtain a stable state laser emission as shown in Fig. 4. Like what observed in the experiment, a narrow pulse width bright soliton polarized along one polarization eigenmode of the fiber laser could induce a weak dark soliton on the bottom of a broad dark, and consequently form a coupled bright-dark soliton pair. Both the broad and narrow dark pulses are stable and they are trapped with the bright solitons and propagate together in the cavity. To confirm that the narrow dark soliton is induced by the coherent coupling, we also deliberately removed the four-wave-mixing (FWM) terms from the CGLEs and did the simulations with the same parameters. No induced dark soliton could be formed inside the broad dark pulse. Therefore, we conclude that the formation of the induced dark solitary pulse is a result of the coherent polarization coupling between the two polarization modes.

Fig. 4 (a) Stable soliton-dark pulse emission of the laser when the input dark pulse has a phase jump. The saturation energy E₀ = 0.1 pJ and the small signal gain G = 640 m⁻¹. (b) Zoom-in of the dark pulse. The induced dark soliton is on the bottom of the broad dark pulse.

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We then repeated the numerical simulations for the case of no phase jump dark pulse, $v = C \cdot \sqrt{(1 - {sech}^{2} (D \cdot t))}$ . We used exactly the same simulation parameters as the previous one. The results are shown in Fig. 5. In this case instead of that an induced dark solitary pulse is formed, a bright solitary pulse is formed inside the broad dark pulse. The numerically simulated results are well in qualitative agreement with the experimental observations.

Fig. 5 (a) Stable soliton-dark pulse emission of the fiber laser when the input dark pulse has no phase jump. The saturation energy E₀ = 0.1 pJ and small signal gain G = 640 m⁻¹. (b) Zoom-in of the dark pulse. The induced bright soliton is on the bottom of the broad dark pulse.

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Based on the numerical simulations it is to see that as a result of the coherent polarization coupling, a bright soliton in an anomalous dispersion cavity fiber laser could induce either an dark soliton or a bright soliton on the bottom of a broad dark pulse polarized along the orthogonal polarization, depending on if there is a phase jump in the broad dark pulse or not. Finally, we point out that strictly speaking all solitons formed in a fiber laser should be dissipative solitons as a laser is intrinsically a dissipative system due to the gain and losses balance [23]. In view of that a bright dissipative soliton could either be formed in the normal or anomalous cavity dispersion regimes, it should also be possible that a dissipative dark soliton could be formed either in the normal or anomalous dispersion regime.

5. Conclusion

In conclusion, we have shown experimentally that in a quasi-vectorial cavity fiber laser, under the coherent polarization coupling, a bright soliton polarized along one polarization eigenmode of the cavity could either induce the formation of a weak dark soliton or a bright soliton on the bottom of a broad dark pulse in the orthogonal polarization direction in a net anomalous dispersion cavity fiber laser. In particular, the induced solitons are stable in the cavity. Numerical simulations based on the coupled complex GLEs have well confirmed the experimental observations. The numerical simulations have further revealed that whether a dark or bright soliton is induced depends on if there is a phase jump in the broad dark pulse.

Acknowledgment

The research is partially supported by the funds of Priority Academic Program Development of Jiangsu Higher Education Institutions (PADP), China, by Minister of Education (MOE) Singapore, under Grant No. 35/12, and AOARD under Agreement No. FA2386-13-1-4096.

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Induced dark solitary pulse in an anomalous dispersion cavity fiber laser

Abstract

1. Introduction

2. Experimental setup

3. Experimental results

4. Numerical simulations and discussion

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (5)

Equations (2)

Optics Express