Unformatted text preview: PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 Observation of HighOrder PolarizationLocked Vector Solitons in a Fiber Laser
D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (Received 13 March 2008; revised manuscript received 30 August 2008; published 10 October 2008) We report on the experimental observation of a new type of polarizationlocked vector soliton in a passively modelocked ﬁber laser. The vector soliton is characterized by the fact that not only are the two orthogonally polarized soliton components phaselocked, but also one of the components has a doublehumped intensity proﬁle. Multiple phaselocked highorder vector solitons with identical soliton parameters and harmonic mode locking of the vector solitons were also obtained in the laser. Numerical simulations conﬁrmed the existence of stable highorder vector solitons in the ﬁber laser.
DOI: 10.1103/PhysRevLett.101.153904 PACS numbers: 42.81.Dp, 05.45.Yv Solitons, as stable localized nonlinear waves, have been observed in various physical systems and have been extensively studied [1,2]. Optical solitons were ﬁrst experimentally observed in single mode ﬁbers (SMFs) by Mollenauer, Stolen, and Gordon in 1980 [3]. It was shown that the dynamics of the solitons could be well described by ¨ the nonlinear Schrodinger equation, a paradigm equation governing optical pulse propagation in ideal SMFs. However, in reality, a SMF always supports two orthogonal polarization modes. Taking ﬁber birefringence into account, it was later found that, depending on the strength of ﬁber birefringence, different types of vector solitons, such as the group velocitylocked vector solitons [4,5], the rotating polarization vector solitons [6,7], and the phaselocked vector solitons [8–10], could also be formed in SMFs. Optical solitons were also observed in modelocked ﬁber lasers. Pulse propagation in a ﬁber laser cavity is different from that in a SMF. Apart from propagating in the ﬁbers that form the laser cavity, a pulse propagating in a laser is also subject to actions of the laser gain and other cavity components. The dynamics of solitons formed in a ﬁber laser are governed by the GinzburgLandau (GL) equation, which takes account not only of the ﬁber dispersion and Kerr nonlinearity but also the laser gain and losses. However, it was shown that, under suitable conditions, solitons formed in ﬁber lasers have analogous features to those of solitons formed in SMFs. Furthermore, vector solitons were also predicted in modelocked ﬁber lasers and recently conﬁrmed experimentally [11,12]. Among the various vector solitons formed in modelocked ﬁber lasers or SMFs, the phaselocked one has attracted considerable attention. Back in 1988, Christodoulides and Joseph ﬁrst theoretically predicted a form of phaselocked vector soliton in birefringent dispersive media [8], which is now known as a highorder phaselocked vector soliton in SMFs. The fundamental form of the phaselocked vector solitons was recently experimentally observed [12–14]. However, to the best of our knowledge, no highorder temporal vector solitons have been 00319007= 08 =101(15)=153904(4) demonstrated. Numerical studies have shown that the highorder phaselocked vector solitons are unstable in SMFs [7]. In this Letter, we report on the experimental observation of a stable phaselocked highorder vector soliton in a modelocked ﬁber laser. Multiple highorder vector solitons with identical soliton parameters coexisting in a laser cavity and harmonic mode locking of the highorder vector solitons were also observed. Moreover, based on a coupled GinzburgLandau equation model, we show numerically that phaselocked highorder vector solitons are stable in modelocked ﬁber lasers. The experimental setup is shown in Fig. 1. The ﬁber laser has a ring cavity consisting of a 4.6 m length of erbiumdoped ﬁber with group velocity dispersion parameter 10 ps=km=nm and a total length of 5.4 m of standard SMF with group velocity dispersion parameter 18 ps=km=nm. Mode locking of the laser is achieved with a semiconductor saturable absorption mirror (SESAM). A polarizationindependent circulator was used to force the unidirectional operation of the ring and simultaneously to incorporate the SESAM in the cavity. FIG. 1. Schematic of the vector soliton ﬁber laser. WDM: Wavelength division multiplexer. EDF: Erbiumdoped ﬁber. 1539041 Ó 2008 The American Physical Society PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 Note that within one cavity roundtrip the pulse propagates twice in the SMF between the circulator and the SESAM. The laser was pumped by a high power ﬁber Raman laser source (BWCFL14801) of wavelength 1480 nm. A 10% ﬁber coupler was used to output the signals. The SESAM used was based on GalnNAs quantum wells. It has a saturable absorption modulation depth of 5%, a saturation ﬂuence of 90 J=cm2 , and a recovery time of 10 ps. The central absorption wavelength of the SESAM is at 1550 nm. A ﬁber polarization controller was inserted in the cavity to ﬁnetune the net cavity birefringence. As no polarizer was used in the cavity, depending on the net cavity linear birefringence, various types of vector solitons such as the group velocitylocked vector soliton, the polarization rotating vector soliton, and the fundamental phaselocked vector soliton were obtained in the laser. We found that the experimentally observed features of these vector solitons could be well described by an extended coupled GinzburgLandau equation model, which also considered the effects of the saturable absorber and the laser cavity [15]. Encouraged by the results, we then further searched for the highorder phaselocked vector solitons theoretically predicted. Through splicing a ﬁberpigtailed optical isolator between the output port and the external cavity measurement apparatus, which serves to suppress the inﬂuence of backreﬂections on the operation of the laser, we could indeed obtain one such vector soliton. Figure 2 shows, for example, the optical spectra and autocorrelation traces of the soliton observed. Polarization locking of the soliton is identiﬁed by measuring the polarization evolution frequency (PEF) of the soliton pulse train [13,14]. No PEF could be detected. As the vector soliton has a stationary elliptic polarization, we could use an external polarizer to separate its two orthogonal polarization components. The optical spectra of the components are shown in Fig. 2(a). The spectra have the same central wavelength and about 10 dB peak intensity difference. Both spectra display soliton sidebands, showing that both of the components are solitonic. In addition, coherent energy exchange between the two soliton components, represented by the appearance of spectral peakdip sidebands [15], is also visible in the spectra. However, in contrast to the polarization resolved spectra of the fundamental phaselocked vector solitons, there is a strong spectral dip at the center of the spectrum of the weaker component. No such dip appears in the spectrum of the stronger component. To identify the cause of the spectral dip, we further measured the autocorrelation traces of each of the soliton components. It turned out that the weak component of the vector soliton had a doublehumped intensity proﬁle as shown in Fig. 2(b). The pulse width of the humps is about 719 fs if a sech2 proﬁle is assumed, and the separation between the humps is about 1.5 ps. The strong component of the vector soliton is a singlehump soliton. It has a pulse width of about 1088 fs if a sech2 FIG. 2. Polarization resolved soliton spectra and autocorrelation traces of the vector soliton observed. (a) Soliton spectra. (b) Autocorrelation traces. proﬁle is assumed. The components of the vector soliton have pulse intensity proﬁles exactly like those predicted by Akhmediev et al. [10] and Christodoulides [8] for a highorder phaselocked vector soliton. Based on the autocorrelation traces, obviously the spectral dip is formed due to the spectral interference between the two humps, and the strong dip at the center of the spectrum indicates that the two humps have 180 phase difference, which is also in agreement with the theoretical prediction. Once the laser operation conditions were appropriately selected, the highorder phaselocked vector soliton operation was always obtained in the laser. Experimentally, multiple such vector solitons with identical soliton parameters were also obtained. By carefully changing the pump strength, one could even control the number of vector solitons in a cavity, without changing the structure of the vector solitons. Like the scalar solitons observed in conventional soliton ﬁber lasers, harmonic mode locking of the highorder phaselocked vector solitons was also observed, as shown in Fig. 3, where 8 such vector solitons were equally spaced in the cavity. All of our experimental 1539042 PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 l À l0 juj2 þ jvj2 @ls ¼À s À ls ; Trec Esat @t (3) FIG. 3. Oscilloscope trace of a harmonically modelocked highorder phaselocked vector soliton state. Lc: Cavity roundtrip time. Eight vector solitons coexist in the cavity. results show that the formation of highorder phaselocked vector solitons is a natural consequence of pulse circulation in the ﬁber laser under the current operation conditions. To conﬁrm our experimental observations, we also numerically simulated the operation of the laser. We used the following coupled GinzburgLandau equations to describe the pulse propagation in the weakly birefringent ﬁbers in the cavity: @u @u ik00 @2 u ik000 @3 u ¼ iu À À þ @z @t 2 @t2 6 @t3 2 i g g @2 u þ i juj2 þ jvj2 u þ v2 uÃ þ u þ ; 3 3 2 22 @t2 g @v @v ik00 @2 v ik000 @3 v ¼ Àiv þ À þ @z @t 2 @t2 6 @t3 2 i g g @2 u þ i jvj2 þ juj2 v þ u2 vÃ þ v þ ; 3 3 2 22 @t2 g (1) where u and v are the normalized envelopes of the optical pulses in the two orthogonal polarized modes of the optical ﬁber. 2 ¼ 2Án= is the wavenumber difference between the two modes. 2 ¼ 2=2c is the inverse group velocity difference. k00 is the secondorder dispersion coefﬁcient; k000 is the thirdorder dispersion coefﬁcient and represents the nonlinearity of the ﬁber. g is the saturable gain coefﬁcient of the ﬁber, and g is the bandwidth of the laser gain. For undoped ﬁbers, g ¼ 0; for erbiumdoped ﬁber, we considered its gain saturation as R 2 ðjuj þ jvj2 Þdt g ¼ G exp À ; Psat (2) where Trec is the absorption recovery time, l0 is the initial absorption of the absorber, and Esat is the absorber saturation energy. We used the following parameters for our simulations for possibly matching the experimental conditions: ¼ 3 WÀ1 kmÀ1 , g ¼ 24 nm, Psat ¼ 50 pJ, k00 SMF ¼ À23 ps2 =km, k00 EDF ¼ À13 ps2 =km, k000 ¼ À0:13 ps3 =km, Esat ¼ 10 pJ, l0 ¼ 0:3, Trec ¼ 2 ps, and cavity length L ¼ 10 m. We used the standard splitstep Fourier technique to solve the equations and a socalled pulse tracing method to model the effects of laser oscillation [16]. We have always started our simulations with an arbitrary weak light input. Figure 4 shows one of the typical results obtained. With a cavity linear birefringence of Lb ¼ 3L, a stable highorder phaselocked vector soliton state was obtained. The weak polarization component of the vector soliton consists of two bound solitons with pulse separation of $1 ps, while the strong polarization component of the where G is the small signal gain coefﬁcient and Psat is the normalized saturation energy. The saturable absorption of the SESAM is described by the rate equation: FIG. 4. A stable highorder phaselocked vector soliton state numerically calculated. (a) Soliton intensity proﬁles of the two orthogonally polarized components. (b) The corresponding optical spectra of (a). 1539043 PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 vector soliton is a singlehump soliton. It is interesting to see that the pulse of the strong component is only temporally overlapped with one of the two pulses of the weak component. Because of the strong crossphase coupling between the temporally overlapped pulses, the two pulses of the weak components have different pulse widths and intensities. Propagating within the cavity, obvious coherent energy exchange between the two temporally overlapped solitons is visible. Figure 4(b) further gives the calculated spectra of the vector soliton components, which also show that the phase difference between the two bound solitons of the weak component is 180. Depending on the laser parameters selected, other highorder phaselocked vector solitons, such as the one with both soliton components having a doublehumped structure, were also numerically obtained. We note that similar highorder vector solitons were also predicted for pulse propagation in weakly birefringent ﬁbers, but they are unstable. However, we found that all of the highorder phaselocked vector solitons obtained numerically were stable in the laser. We believe that the differences in stability of the highorder phaselocked vector solitons between ﬁber and ﬁber lasers could be traced back to their different soliton natures. While the soliton formed in a SMF is essentially a Hamiltonian soliton, the one formed in a ﬁber laser is a dissipative soliton, which is in fact a strange attractor of the laser system. The formation of multiple identical highorder vector solitons in the ﬁber laser clearly shows the dissipative nature of the vector solitons formed. In conclusion, we have reported the ﬁrst experimental observation of a type of highorder phaselocked vector soliton in a passively modelocked ﬁber laser. The highorder vector soliton is characterized by the fact that its two orthogonal polarization components are phaselocked, and, while the stronger polarization component is a singlehump pulse, the weaker component has a doublehumped structure with 180 phase difference between the humps. Our experimental results are the ﬁrst conﬁrmation of the theoretical predictions of highorder phaselocked vector solitons in birefringent dispersive media. Finally we note that, although the current experiment was conducted on a modelocked ﬁber laser whose dynamics is governed by the coupled GL equations, due to the wide applicability of the coupled GL equations in describing the dynamics of real physical systems, it is expected that similar results could also be observed in other systems. [1] L. A. Dickey, Soliton Equations and Hamiltonian Systems (World Scientiﬁc, New York, 2003). [2] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003). [3] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). [4] C. R. Menyuk, Opt. Lett. 12, 614 (1987); J. Opt. Soc. Am. B 5, 392 (1988). [5] M. N. Islam et al., Opt. Lett. 14, 1011 (1989). [6] V. V. Afanasjev, Opt. Lett. 20, 270 (1995). [7] K. J. Blow, N. J. Doran, and D. Wood, Opt. Lett. 12, 202 (1987). [8] D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988). [9] N. Akhmediev, A. Buryak, and J. M. SotoCrespo, Opt. Commun. 112, 278 (1994). [10] N. N. Akhmediev, A. V. Buryak, and J. M. SotoCrespo, and D. R. Andersen, J. Opt. Soc. Am. B 12, 434 (1995). [11] N. N. Akhmediev, J. M. SotoCrespo, S. T. Cundiff, B. C. Collings, and W. H. Knox, Opt. Lett. 23, 852 (1998). [12] S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. SotoCrespo, K. Bergman, and W. H. Knox, Phys. Rev. Lett. 82, 3988 (1999). [13] B. C. Collings, S. T. Cundiff, N. N. Akhmediev, J. M. SotoCrespo, K. Bergman, and W. H. Knox, J. Opt. Soc. Am. B 17, 354 (2000). [14] S. T. Cundiff, B. C. Collings, and W. H. Knox, Opt. Express 1, 12 (1997). [15] H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, Opt. Express 16, 12 618 (2008). [16] D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005). 1539044 ...
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