Dark soliton fiber lasers Dingyuan Tang,1,2,* Jun Guo,3 Yufeng Song,3,4 Han Zhang,4 Luming Zhao,1,2 and Deyuan Shen1,2 1
2
School of Physics and Electronic Engineering, Jiangsu Normal University, China Jiangsu Key Laboratory of Advanced Laser Materials and Devices, Jiangsu Normal University, China 3 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 4 College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China * [email protected]
Abstract: Dark soliton formation and soliton dynamics in all-normal dispersion cavity fiber ring lasers without an anti-saturable absorber in cavity is studied both theoretically and numerically. It is shown that under suitable conditions the dark solitons formed could be described by the nonlinear Schrödinger equation. The dark soliton formation in an allnormal-dispersion cavity erbium-doped fiber ring laser without an antisaturable absorber in cavity is first experimentally demonstrated. Individual dark solitons are experimentally identified. Excellent agreement between theory and experiment is observed. ©2014 Optical Society of America OCIS codes: (060.4370) Nonlinear optics, fibers; (060.5530) Pulse propagation and temporal solitons.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Y. S. Kivishar and G. Agrawal, Optical Solitons: From fiber to photonic crystals, Academic Press, 2003. Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012). L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). P. Emplit, J. P. Hamaide, F. Reynaud, C. Groehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62(6), 374–379 (1987). S. M. J. Kelly, K. Smith, K. J. Blow, and N. J. Doran, “Average soliton dynamics of a high-gain erbium fiber laser,” Opt. Lett. 16(17), 1337–1339 (1991). C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17(6), 417–419 (1992). I. N. Iii, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16(8), 539–541 (1991). D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005). H. Zhang, D. Y. Tang, L. M. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A 80(4), 045803 (2009). D. Y. Tang, L. Li, Y. F. Song, H. Zhang, and D. Y. Shen, “Evidence of dark solitons in all-normal dispersion fiber lasers,” Phys. Rev. A 88(1), 013849 (2013). M. J. Ablowitz, T. P. Horikis, S. D. Nixon, and D. J. Frantzeskakis, “Dark solitons in mode-locked lasers,” Opt. Lett. 36(6), 793–795 (2011). V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823– 828 (1973). S. A. Gredeskul and Y. S. Kivshar, “Generation of dark solitons in optical fibers,” Phys. Rev. Lett. 62(8), 977 (1989). S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49(19), 1224–1226 (1986).
1. Introduction Soliton formation is an attractive topic that has been extensively investigated in diverse branches of physics such as nonlinear optics, condensed matter physics, fluid mechanics, and particle physics. In nonlinear optics the study on optical soliton formation and soliton dynamics have attracted great attention over the past two decades [1, 2]. Optical solitons can be formed either in the time or spatial domains. While the optical spatial solitons have been
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19831
realized under a diversity of nonlinear effects, the optical temporal solitons are mostly observed in the light propagation in the single mode fibers (SMFs). It is now well known that the optical temporal soliton formation in SMFs is governed by the nonlinear Schrödinger equation (NLSE). In particular, while the bright solitons are formed in the anomalous dispersion SMFs, the dark solitons, these are the intensity dips embedded in a CW background, are formed in the normal dispersion SMFs. Experimentally, both the bright and dark soliton formation in SMFs has been demonstrated [3,4]. Moreover, bright soliton formation in mode locked fiber lasers has also been theoretically studied and experimentally demonstrated [5–8]. Although the existence of laser gain and cavity output affect the detailed parameters of the formed solitons in a fiber laser, it was found that the essential dynamics of the solitons are still governed by the NLSE. In view of the dark soliton formation in normal dispersion SMFs, one would also expect that the dark solitons could be formed in the normal dispersion cavity fiber lasers. However, so far the research on the dark soliton fiber lasers was less addressed. In a recent paper we reported the experimental observation of dark pulse emission of a normal dispersion cavity fiber laser [9]. Based on the features of the observed dark pulses, it was postulated that the observed dark pulses could be related to the dark solitons in the laser. This postulation was confirmed by further experimental study on the dark pulses [10]. Using an ultrahigh speed real-time electronic detection system the individual dark solitons formed in the fiber lasers could be clearly identified. The high speed real-time measurement also clarified why associated with a broad optical spectral profile a broad dark pulse was formed in the fiber lasers. It was actually a bunch of the dark solitons. The previous dark soliton fiber lasers have a cavity design that an artificial anti-saturable absorber is automatically incorporated. Numerical simulations have shown that with an antisaturable absorber in cavity, stable black solitons will always be formed in a fiber laser [9– 11]. In addition, an anti-saturable absorber in cavity also introduces detrimental effects. Not only the black solitons have the tendency of bunching, it also causes strong CW background intensity fluctuation. In this letter we show theoretically that dark solitons could also be formed in normal dispersion cavity fiber lasers without an anti-saturable absorber in cavity. Especially, under suitable conditions dark solitons formed in such a fiber laser could be described by the nonlinear Schrödinger equation. The dark soliton formation in such a fiber laser was numerically simulated. It was found that an arbitrary intensity dip on the CW laser emission could simultaneously generate multiple dark soliton pairs with different darkness and pulse widths. An all-normal-dispersion cavity erbium-doped fiber laser without an antisaturable absorber in cavity is also constructed. Formation of individual dark solitons in the laser is experimentally identified. The observed dark soliton formation and soliton dynamics are well in agreement of the theoretical and numerical predictions. 2. Simulations Under the condition that the cavity length is far shorter than the dispersion and nonlinear length, and the optical field is resonant with the cavity, the operation of a fiber ring laser is described by the extended NLS equation. i
∂u β 2 ∂ 2 u g −α g ∂ 2u 2 − +γ u u −i u −i =0 2 ∂z 2 ∂t 2 2Ω 2g ∂t 2
(1)
where u is the slowly varying amplitude of the light field, β2 is the average cavity dispersion parameter, γ is the nonlinearity of the fiber, α is the cavity loss coefficient, g is the effective laser gain coefficient, and Ωg is the effective bandwidth of the laser gain. The gain saturation of the laser is described by g=
g0 1 + u dt Es 2
(2)
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19832
where g0 is a small signal gain and Es is the saturation energy. Eq. (1) can be further normalized with LNL = 1 γ P0 , Ld = T02 β 2 , U P0 = u , η = sgn ( β 2 ) , ξ = z / Ld , τ = t / T0 , N 2 = Ld LNL = 1 to a dimensionless form:
i where G = gLd = g 0 Ld
∂U η ∂ 2U (G − L) G ∂ 2U 2 − + U U − i U − i =0 ∂ξ 2 ∂τ 2 2 2Ω 2 ∂τ 2
(1 + U
2
)
(3)
dt I s , L = α L , Ω = Ω g T0 I s = Es P0 T0 are the normalized gain d
coefficient, loss coefficient, bandwidth and saturation energy, respectively. We note that for a laser under steady state operation the cavity losses are always balanced by the saturated gain. Hence, the forth term of Eq. (3) is practically zero. Moreover, if the spectral bandwidth of the laser emission is far narrower than the effective laser gain bandwidth, one could also ignore the last term of the equation. Consequently, Eq. (3) reduces to the normalized NLSE. The result suggests that under appropriate conditions a fiber laser could exhibit the NLSE dynamics. However, it is to bear in mind that a laser is in nature a dissipative system. The physics is that the laser would behave effectively like a NLSE system when the cavity losses are balanced by the saturated laser gain. The soliton solutions admitted by the NLSE were theoretically extensively studied. In particular, using the inverse scattering technique Zakharov and Shabat have found the following analytical solution [12] U (ξ ,τ ) = u0 [cos φ tan h ( u0 cos φ (τ − u0ξ sin φ ) ) + i sin φ ]exp ( iu02ξ )
(4)
where u0 is a CW background field strength, and φ (0 ≤ φ ≤ π/2) is an arbitrary phase factor. According to Eq. (4) it is obvious that under a fixed CW level, depending on the φ value, dark solitons with different darkness and pulse width could be formed in a NLSE system. To illustrate the feature, we have shown in Fig. 1 the dark solitons calculated from Eq. (4) with different φ values. The one with φ = 0 has the narrowest pulse width, and its peak also reaches the zero intensity. The dark soliton is known as the black soliton. All the other solitons with φ ≠ 0 have a shallower depth. They are known as the grey solitons. The larger the φ, the shallower the depth and the broader the dark pulse width.
Fig. 1. Normalized intensity of the dark solitons calculated from Eq. (4) with different values ofφ.
A striking property of the dark solitons which is different from the bright solitons is that their formation has no threshold [13]. Any a small intensity dip on the CW background could evolve into one or multiple dark solitons. To highlight the property we have used Eq. (1) to
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19833
simulate the dark soliton formation in a fiber laser initialed by an arbitrarily weak intensity dip, as shown in Fig. 2. The dip is defined as u = A(1-B sech2(Ct)), in which A is the amplitude of an arbitrary small CW background, B is the depth of the initial dip, and C is the width of the dip. We considered a fiber laser with the following cavity parameters: averaged cavity dispersion β2 = 22 ps2/km, γ = 3 W−1km−1, cavity length l = 9 m, cavity output loss α = 10%, and small signal gain coefficient g0 = 0.8 m−1. Through appropriately setting the saturation energy Es which is controlled experimentally by the pump power, a stable operation state of the fiber laser can always be obtained.
Fig. 2. Numerically simulated dark soliton formation initialed by a dip of A2 = 0.02W, B = 1, C = 0.25. (a) Contour plot under Es = 0.9pJ. (b) Intensity and phase profiles of the dark solitons at the round trip of 1200.
From Fig. 2(a) it is to see that after a short distance of propagation, the intensity dip split into two pairs of dark pulses with different depths. The two dark pulses within each pair moved to opposite directions, and dark pulses of different pairs moved with different velocities. The pulses can collide with each other. However, after collisions they still remain their pulse profiles, suggesting that they are dark solitons. In Fig. 2(b) we have also shown the intensity and phase profiles of the formed dark pulses at the round trip of 1200. It is obvious that the dark solitons with different depth have different width and phase jumps. We emphasize the simultaneous formation of multiple dark solitons with different pulse widths and darkness from one initial dip. Numerically it is found that the number of soliton pairs formed and the darkness of the formed solitons varied with the initial dip properties and the saturation energy Es. However, once they are formed, all the dark pulses are stable in the cavity. The feature of the dark solitons is distinguished from those of the dark solitons formed in fiber lasers with an anti-saturable absorber in cavity [9–11], where only the black solitons are stable. 3. Experimental results
To experimentally study the NLSE type dark soliton formation and soliton dynamics, we constructed an all-normal dispersion erbium-doped fiber ring laser with a configuration as shown in Fig. 3. The fiber ring was made of 3 m erbium-doped fiber (EDF) with a group velocity dispersion (GVD) parameter of −48 ps/nm/km and 6 m dispersion shifted fiber with a GVD parameter of −4 ps/nm/km. In order to remove the nonlinear polarization rotation induced artificial anti-saturable absorption effect from the cavity, a polarization independent isolator was used, in addition, all the intracavity components were specially made with the dispersion shifted fiber and carefully selected so that they have ignorable polarization dependent losses.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19834
Fig. 3. Schematic configuration of the fiber laser. WDM: wavelength division multiplexer. DSF: dispersion shifted fiber. EDF: erbium doped fiber. PC: polarization controller.
Fig. 4. Dark pulse emission of the fiber laser experimentally observed. (a) A measured oscilloscope trace; (b) the corresponding optical spectrum.
Although there is no polarization selective component in the cavity, it was experimentally found that under strong pumping, the laser could still operate in a linear polarization state. We attribute the feature of the fiber laser operation to the effect of polarization instability of nonlinear light propagation in weakly birefringent fibers [14]. Under the linear polarization operation dark solitons were always observed in the fiber laser. Fig. 4 shows for example a typical case of the dark soliton emission of the laser. Fig. 4(a) is the measured oscilloscope trace of the laser emission, and Fig. 4(b) is the corresponding optical spectrum. We note that with and without the dark pulse formation the spectral bandwidth of the laser emission is obviously different. Different from the laser emission with an anti-saturable absorber in cavity, as reported in [9] where the CW background intensity exhibited strong fluctuations, the CW background of the current laser emission is stable. In addition, the dark solitons formed are well scattered in the cavity. There is no tendency of soliton bunching.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19835
In obtaining Fig. 4 we had carefully controlled the laser operation power so that the individual dark pulses could be well resolved by our detection system. Obviously, the dark pulses formed in the cavity had different darkness and pulse width, which is exactly as predicted by the NLSE and the numerical simulations. Based on the measured 3 dB spectral bandwidth of ~0.16 nm, if a transform-limited sech2 pulse profile is assumed, we estimate that the narrowest dark pulse should have a pulse width of ~17 ps. We note that taking into account the influence by the limited bandwidth of the detection system, the result shown in Fig. 4(a) is well in agreement of the estimation. Here it is worth mentioning that for the erbium-doped fiber lasers the effective gain bandwidth is normally larger than 10nm. From the experimental result shown in Fig. 4(b) the 3dB spectral bandwidth of the dark solitons formed is only about 0.16nm, which is far smaller than the laser gain bandwidth. Hence our above treatment on omitting the last term in Eq. (3) is well justified.
Fig. 5. The laser emission over 4 consecutive cavity roundtrips (a) the overall laser emission in 200 ns; (b) Zoom-in of a small segment in the four roundtrips.
To check the stability of the dark pulses we also recorded the laser emission over 4 consecutive cavity roundtrips, as shown in Fig. 5(a). For the purpose of comparison, we also plotted the noise level of our detection system. Due to the threshold-less property of the dark soliton formation, experimentally it was difficult to control the number of dark solitons formed in a laser cavity. In our experiment many dark pulses were simultaneously formed, they had different darkness and were randomly distributed in the cavity. At the first sight the laser emission looked very “noise”. However, when zoomed in to a time scale where the individual dark pulses were well resolved, as shown in Fig. 5(b), it was to see that the same dark pulse pattern actually repeated itself with the cavity roundtrip time. The result clearly shows that the dark pulses are stable in the cavity. They are indeed dark solitons rather than random intensity fluctuations. 4. Conclusion
In conclusion, we have studied the dark soliton formation in fiber lasers without an antisaturable absorber in cavity. It is shown that under suitable conditions the NLSE type dark solitons, characterized by the simultaneous formation of dark solitons with different darkness and pulse width, could be formed in the fiber lasers. Dark soliton formation in an erbium#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19836
doped fiber ring laser without an anti-saturable absorber in cavity was first experimentally demonstrated, and the features of the observed dark solitons are well in agreement with those of the numerical simulations. Our experimental result once again confirmed the formation of dark solitons in all-normal dispersion cavity fiber lasers. Acknowledgments
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.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19837
2
School of Physics and Electronic Engineering, Jiangsu Normal University, China Jiangsu Key Laboratory of Advanced Laser Materials and Devices, Jiangsu Normal University, China 3 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 4 College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China * [email protected]
Abstract: Dark soliton formation and soliton dynamics in all-normal dispersion cavity fiber ring lasers without an anti-saturable absorber in cavity is studied both theoretically and numerically. It is shown that under suitable conditions the dark solitons formed could be described by the nonlinear Schrödinger equation. The dark soliton formation in an allnormal-dispersion cavity erbium-doped fiber ring laser without an antisaturable absorber in cavity is first experimentally demonstrated. Individual dark solitons are experimentally identified. Excellent agreement between theory and experiment is observed. ©2014 Optical Society of America OCIS codes: (060.4370) Nonlinear optics, fibers; (060.5530) Pulse propagation and temporal solitons.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Y. S. Kivishar and G. Agrawal, Optical Solitons: From fiber to photonic crystals, Academic Press, 2003. Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012). L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). P. Emplit, J. P. Hamaide, F. Reynaud, C. Groehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62(6), 374–379 (1987). S. M. J. Kelly, K. Smith, K. J. Blow, and N. J. Doran, “Average soliton dynamics of a high-gain erbium fiber laser,” Opt. Lett. 16(17), 1337–1339 (1991). C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17(6), 417–419 (1992). I. N. Iii, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16(8), 539–541 (1991). D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005). H. Zhang, D. Y. Tang, L. M. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A 80(4), 045803 (2009). D. Y. Tang, L. Li, Y. F. Song, H. Zhang, and D. Y. Shen, “Evidence of dark solitons in all-normal dispersion fiber lasers,” Phys. Rev. A 88(1), 013849 (2013). M. J. Ablowitz, T. P. Horikis, S. D. Nixon, and D. J. Frantzeskakis, “Dark solitons in mode-locked lasers,” Opt. Lett. 36(6), 793–795 (2011). V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823– 828 (1973). S. A. Gredeskul and Y. S. Kivshar, “Generation of dark solitons in optical fibers,” Phys. Rev. Lett. 62(8), 977 (1989). S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49(19), 1224–1226 (1986).
1. Introduction Soliton formation is an attractive topic that has been extensively investigated in diverse branches of physics such as nonlinear optics, condensed matter physics, fluid mechanics, and particle physics. In nonlinear optics the study on optical soliton formation and soliton dynamics have attracted great attention over the past two decades [1, 2]. Optical solitons can be formed either in the time or spatial domains. While the optical spatial solitons have been
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19831
realized under a diversity of nonlinear effects, the optical temporal solitons are mostly observed in the light propagation in the single mode fibers (SMFs). It is now well known that the optical temporal soliton formation in SMFs is governed by the nonlinear Schrödinger equation (NLSE). In particular, while the bright solitons are formed in the anomalous dispersion SMFs, the dark solitons, these are the intensity dips embedded in a CW background, are formed in the normal dispersion SMFs. Experimentally, both the bright and dark soliton formation in SMFs has been demonstrated [3,4]. Moreover, bright soliton formation in mode locked fiber lasers has also been theoretically studied and experimentally demonstrated [5–8]. Although the existence of laser gain and cavity output affect the detailed parameters of the formed solitons in a fiber laser, it was found that the essential dynamics of the solitons are still governed by the NLSE. In view of the dark soliton formation in normal dispersion SMFs, one would also expect that the dark solitons could be formed in the normal dispersion cavity fiber lasers. However, so far the research on the dark soliton fiber lasers was less addressed. In a recent paper we reported the experimental observation of dark pulse emission of a normal dispersion cavity fiber laser [9]. Based on the features of the observed dark pulses, it was postulated that the observed dark pulses could be related to the dark solitons in the laser. This postulation was confirmed by further experimental study on the dark pulses [10]. Using an ultrahigh speed real-time electronic detection system the individual dark solitons formed in the fiber lasers could be clearly identified. The high speed real-time measurement also clarified why associated with a broad optical spectral profile a broad dark pulse was formed in the fiber lasers. It was actually a bunch of the dark solitons. The previous dark soliton fiber lasers have a cavity design that an artificial anti-saturable absorber is automatically incorporated. Numerical simulations have shown that with an antisaturable absorber in cavity, stable black solitons will always be formed in a fiber laser [9– 11]. In addition, an anti-saturable absorber in cavity also introduces detrimental effects. Not only the black solitons have the tendency of bunching, it also causes strong CW background intensity fluctuation. In this letter we show theoretically that dark solitons could also be formed in normal dispersion cavity fiber lasers without an anti-saturable absorber in cavity. Especially, under suitable conditions dark solitons formed in such a fiber laser could be described by the nonlinear Schrödinger equation. The dark soliton formation in such a fiber laser was numerically simulated. It was found that an arbitrary intensity dip on the CW laser emission could simultaneously generate multiple dark soliton pairs with different darkness and pulse widths. An all-normal-dispersion cavity erbium-doped fiber laser without an antisaturable absorber in cavity is also constructed. Formation of individual dark solitons in the laser is experimentally identified. The observed dark soliton formation and soliton dynamics are well in agreement of the theoretical and numerical predictions. 2. Simulations Under the condition that the cavity length is far shorter than the dispersion and nonlinear length, and the optical field is resonant with the cavity, the operation of a fiber ring laser is described by the extended NLS equation. i
∂u β 2 ∂ 2 u g −α g ∂ 2u 2 − +γ u u −i u −i =0 2 ∂z 2 ∂t 2 2Ω 2g ∂t 2
(1)
where u is the slowly varying amplitude of the light field, β2 is the average cavity dispersion parameter, γ is the nonlinearity of the fiber, α is the cavity loss coefficient, g is the effective laser gain coefficient, and Ωg is the effective bandwidth of the laser gain. The gain saturation of the laser is described by g=
g0 1 + u dt Es 2
(2)
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19832
where g0 is a small signal gain and Es is the saturation energy. Eq. (1) can be further normalized with LNL = 1 γ P0 , Ld = T02 β 2 , U P0 = u , η = sgn ( β 2 ) , ξ = z / Ld , τ = t / T0 , N 2 = Ld LNL = 1 to a dimensionless form:
i where G = gLd = g 0 Ld
∂U η ∂ 2U (G − L) G ∂ 2U 2 − + U U − i U − i =0 ∂ξ 2 ∂τ 2 2 2Ω 2 ∂τ 2
(1 + U
2
)
(3)
dt I s , L = α L , Ω = Ω g T0 I s = Es P0 T0 are the normalized gain d
coefficient, loss coefficient, bandwidth and saturation energy, respectively. We note that for a laser under steady state operation the cavity losses are always balanced by the saturated gain. Hence, the forth term of Eq. (3) is practically zero. Moreover, if the spectral bandwidth of the laser emission is far narrower than the effective laser gain bandwidth, one could also ignore the last term of the equation. Consequently, Eq. (3) reduces to the normalized NLSE. The result suggests that under appropriate conditions a fiber laser could exhibit the NLSE dynamics. However, it is to bear in mind that a laser is in nature a dissipative system. The physics is that the laser would behave effectively like a NLSE system when the cavity losses are balanced by the saturated laser gain. The soliton solutions admitted by the NLSE were theoretically extensively studied. In particular, using the inverse scattering technique Zakharov and Shabat have found the following analytical solution [12] U (ξ ,τ ) = u0 [cos φ tan h ( u0 cos φ (τ − u0ξ sin φ ) ) + i sin φ ]exp ( iu02ξ )
(4)
where u0 is a CW background field strength, and φ (0 ≤ φ ≤ π/2) is an arbitrary phase factor. According to Eq. (4) it is obvious that under a fixed CW level, depending on the φ value, dark solitons with different darkness and pulse width could be formed in a NLSE system. To illustrate the feature, we have shown in Fig. 1 the dark solitons calculated from Eq. (4) with different φ values. The one with φ = 0 has the narrowest pulse width, and its peak also reaches the zero intensity. The dark soliton is known as the black soliton. All the other solitons with φ ≠ 0 have a shallower depth. They are known as the grey solitons. The larger the φ, the shallower the depth and the broader the dark pulse width.
Fig. 1. Normalized intensity of the dark solitons calculated from Eq. (4) with different values ofφ.
A striking property of the dark solitons which is different from the bright solitons is that their formation has no threshold [13]. Any a small intensity dip on the CW background could evolve into one or multiple dark solitons. To highlight the property we have used Eq. (1) to
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19833
simulate the dark soliton formation in a fiber laser initialed by an arbitrarily weak intensity dip, as shown in Fig. 2. The dip is defined as u = A(1-B sech2(Ct)), in which A is the amplitude of an arbitrary small CW background, B is the depth of the initial dip, and C is the width of the dip. We considered a fiber laser with the following cavity parameters: averaged cavity dispersion β2 = 22 ps2/km, γ = 3 W−1km−1, cavity length l = 9 m, cavity output loss α = 10%, and small signal gain coefficient g0 = 0.8 m−1. Through appropriately setting the saturation energy Es which is controlled experimentally by the pump power, a stable operation state of the fiber laser can always be obtained.
Fig. 2. Numerically simulated dark soliton formation initialed by a dip of A2 = 0.02W, B = 1, C = 0.25. (a) Contour plot under Es = 0.9pJ. (b) Intensity and phase profiles of the dark solitons at the round trip of 1200.
From Fig. 2(a) it is to see that after a short distance of propagation, the intensity dip split into two pairs of dark pulses with different depths. The two dark pulses within each pair moved to opposite directions, and dark pulses of different pairs moved with different velocities. The pulses can collide with each other. However, after collisions they still remain their pulse profiles, suggesting that they are dark solitons. In Fig. 2(b) we have also shown the intensity and phase profiles of the formed dark pulses at the round trip of 1200. It is obvious that the dark solitons with different depth have different width and phase jumps. We emphasize the simultaneous formation of multiple dark solitons with different pulse widths and darkness from one initial dip. Numerically it is found that the number of soliton pairs formed and the darkness of the formed solitons varied with the initial dip properties and the saturation energy Es. However, once they are formed, all the dark pulses are stable in the cavity. The feature of the dark solitons is distinguished from those of the dark solitons formed in fiber lasers with an anti-saturable absorber in cavity [9–11], where only the black solitons are stable. 3. Experimental results
To experimentally study the NLSE type dark soliton formation and soliton dynamics, we constructed an all-normal dispersion erbium-doped fiber ring laser with a configuration as shown in Fig. 3. The fiber ring was made of 3 m erbium-doped fiber (EDF) with a group velocity dispersion (GVD) parameter of −48 ps/nm/km and 6 m dispersion shifted fiber with a GVD parameter of −4 ps/nm/km. In order to remove the nonlinear polarization rotation induced artificial anti-saturable absorption effect from the cavity, a polarization independent isolator was used, in addition, all the intracavity components were specially made with the dispersion shifted fiber and carefully selected so that they have ignorable polarization dependent losses.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19834
Fig. 3. Schematic configuration of the fiber laser. WDM: wavelength division multiplexer. DSF: dispersion shifted fiber. EDF: erbium doped fiber. PC: polarization controller.
Fig. 4. Dark pulse emission of the fiber laser experimentally observed. (a) A measured oscilloscope trace; (b) the corresponding optical spectrum.
Although there is no polarization selective component in the cavity, it was experimentally found that under strong pumping, the laser could still operate in a linear polarization state. We attribute the feature of the fiber laser operation to the effect of polarization instability of nonlinear light propagation in weakly birefringent fibers [14]. Under the linear polarization operation dark solitons were always observed in the fiber laser. Fig. 4 shows for example a typical case of the dark soliton emission of the laser. Fig. 4(a) is the measured oscilloscope trace of the laser emission, and Fig. 4(b) is the corresponding optical spectrum. We note that with and without the dark pulse formation the spectral bandwidth of the laser emission is obviously different. Different from the laser emission with an anti-saturable absorber in cavity, as reported in [9] where the CW background intensity exhibited strong fluctuations, the CW background of the current laser emission is stable. In addition, the dark solitons formed are well scattered in the cavity. There is no tendency of soliton bunching.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19835
In obtaining Fig. 4 we had carefully controlled the laser operation power so that the individual dark pulses could be well resolved by our detection system. Obviously, the dark pulses formed in the cavity had different darkness and pulse width, which is exactly as predicted by the NLSE and the numerical simulations. Based on the measured 3 dB spectral bandwidth of ~0.16 nm, if a transform-limited sech2 pulse profile is assumed, we estimate that the narrowest dark pulse should have a pulse width of ~17 ps. We note that taking into account the influence by the limited bandwidth of the detection system, the result shown in Fig. 4(a) is well in agreement of the estimation. Here it is worth mentioning that for the erbium-doped fiber lasers the effective gain bandwidth is normally larger than 10nm. From the experimental result shown in Fig. 4(b) the 3dB spectral bandwidth of the dark solitons formed is only about 0.16nm, which is far smaller than the laser gain bandwidth. Hence our above treatment on omitting the last term in Eq. (3) is well justified.
Fig. 5. The laser emission over 4 consecutive cavity roundtrips (a) the overall laser emission in 200 ns; (b) Zoom-in of a small segment in the four roundtrips.
To check the stability of the dark pulses we also recorded the laser emission over 4 consecutive cavity roundtrips, as shown in Fig. 5(a). For the purpose of comparison, we also plotted the noise level of our detection system. Due to the threshold-less property of the dark soliton formation, experimentally it was difficult to control the number of dark solitons formed in a laser cavity. In our experiment many dark pulses were simultaneously formed, they had different darkness and were randomly distributed in the cavity. At the first sight the laser emission looked very “noise”. However, when zoomed in to a time scale where the individual dark pulses were well resolved, as shown in Fig. 5(b), it was to see that the same dark pulse pattern actually repeated itself with the cavity roundtrip time. The result clearly shows that the dark pulses are stable in the cavity. They are indeed dark solitons rather than random intensity fluctuations. 4. Conclusion
In conclusion, we have studied the dark soliton formation in fiber lasers without an antisaturable absorber in cavity. It is shown that under suitable conditions the NLSE type dark solitons, characterized by the simultaneous formation of dark solitons with different darkness and pulse width, could be formed in the fiber lasers. Dark soliton formation in an erbium#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19836
doped fiber ring laser without an anti-saturable absorber in cavity was first experimentally demonstrated, and the features of the observed dark solitons are well in agreement with those of the numerical simulations. Our experimental result once again confirmed the formation of dark solitons in all-normal dispersion cavity fiber lasers. Acknowledgments
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.
#211778 - $15.00 USD Received 9 May 2014; revised 7 Jul 2014; accepted 15 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119831 | OPTICS EXPRESS 19837
