Unformatted text preview: PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 Observation of High-Order Polarization-Locked 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 polarization-locked vector soliton in a passively mode-locked ﬁber laser. The vector soliton is characterized by the fact that not only are the two orthogonally polarized soliton components phase-locked, but also one of the components has a doublehumped intensity proﬁle. Multiple phase-locked high-order 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 high-order 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 . 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 velocity-locked 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 mode-locked 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 Ginzburg-Landau (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 mode-locked ﬁber lasers and recently conﬁrmed experimentally [11,12]. Among the various vector solitons formed in modelocked ﬁber lasers or SMFs, the phase-locked one has attracted considerable attention. Back in 1988, Christodoulides and Joseph ﬁrst theoretically predicted a form of phase-locked vector soliton in birefringent dispersive media , which is now known as a high-order phaselocked vector soliton in SMFs. The fundamental form of the phase-locked vector solitons was recently experimentally observed [12–14]. However, to the best of our knowledge, no high-order temporal vector solitons have been 0031-9007= 08 =101(15)=153904(4) demonstrated. Numerical studies have shown that the high-order phase-locked vector solitons are unstable in SMFs . In this Letter, we report on the experimental observation of a stable phase-locked high-order vector soliton in a mode-locked ﬁber laser. Multiple high-order 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 Ginzburg-Landau equation model, we show numerically that phase-locked high-order vector solitons are stable in mode-locked 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 erbium-doped 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 polarization-independent 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: Erbium-doped ﬁber. 153904-1 Ó 2008 The American Physical Society PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 Note that within one cavity round-trip 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 (BWC-FL-1480-1) 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 ﬁne-tune 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 velocity-locked vector soliton, the polarization rotating vector soliton, and the fundamental phase-locked 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 Ginzburg-Landau equation model, which also considered the effects of the saturable absorber and the laser cavity . Encouraged by the results, we then further searched for the high-order phase-locked 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 peak-dip sidebands , is also visible in the spectra. However, in contrast to the polarization resolved spectra of the fundamental phase-locked 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 double-humped 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 single-hump 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.  and Christodoulides  for a highorder phase-locked 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 high-order phase-locked 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 high-order phase-locked 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 153904-2 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 mode-locked high-order phase-locked vector soliton state. Lc: Cavity roundtrip time. Eight vector solitons coexist in the cavity. results show that the formation of high-order phase-locked 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 Ginzburg-Landau 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 wave-number difference between the two modes. 2 ¼ 2=2c is the inverse group velocity difference. k00 is the second-order dispersion coefﬁcient; k000 is the third-order 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 erbium-doped 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 split-step Fourier technique to solve the equations and a so-called pulse tracing method to model the effects of laser oscillation . 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 high-order phase-locked 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 high-order phase-locked vector soliton state numerically calculated. (a) Soliton intensity proﬁles of the two orthogonally polarized components. (b) The corresponding optical spectra of (a). 153904-3 PRL 101, 153904 (2008) PHYSICAL REVIEW LETTERS week ending 10 OCTOBER 2008 vector soliton is a single-hump 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 cross-phase 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 phase-locked vector solitons, such as the one with both soliton components having a double-humped structure, were also numerically obtained. We note that similar high-order vector solitons were also predicted for pulse propagation in weakly birefringent ﬁbers, but they are unstable. However, we found that all of the high-order phase-locked vector solitons obtained numerically were stable in the laser. We believe that the differences in stability of the high-order phase-locked 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 high-order 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 high-order phase-locked vector soliton in a passively mode-locked ﬁber laser. The highorder vector soliton is characterized by the fact that its two orthogonal polarization components are phase-locked, and, while the stronger polarization component is a singlehump pulse, the weaker component has a double-humped structure with 180 phase difference between the humps. Our experimental results are the ﬁrst conﬁrmation of the theoretical predictions of high-order phase-locked vector solitons in birefringent dispersive media. 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This note was uploaded on 11/12/2009 for the course EEE optics taught by Professor Zhang during the Spring '09 term at Nanyang Technological University.
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