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PRL 98, 053902 (2007)
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PHYSICAL REVIEW LETTERS
Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber Darren Rand,1,* Ivan Glesk,1 Camille-Sophie Bre`s,1 Daniel A. Nolan,2 Xin Chen,2 Joohyun Koh,2 Jason W. Fleischer,1 Ken Steiglitz,3 and Paul R. Prucnal1 1
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA 2 Science and Technology Division, Corning Incorporated, Corning, New York 14831, USA 3 Department of Computer Science, Princeton University, Princeton, New Jersey 08544, USA (Received 15 May 2006; published 1 February 2007)
We report the experimental observation of temporal vector soliton propagation and collision in a linearly birefringent optical fiber. To the best of the authors’ knowledge, this is both the first demonstration of temporal vector solitons with two mutually incoherent component fields, and of vector soliton collisions in a Kerr nonlinear medium. Collisions are characterized by an intensity redistribution between the two components, and the experimental results agree with numerical predictions of the coupled nonlinear Schro¨dinger equation. DOI: 10.1103/PhysRevLett.98.053902
PACS numbers: 42.65.Tg, 42.81.Dp
Because of their multicomponent structure, vector solitons have richer propagation dynamics than their scalar, one-component counterparts. For example, a scalar soliton propagates with an unchanging profile by balancing selfphase modulation (SPM) with dispersion or diffraction, while in the vector case one must also account for crossphase modulation (XPM) between components. For collisions, scalar solitons are characterized by phase and position shifts, while vector solitons have the added dynamics of possible intensity redistributions between the component fields [1,2]. In addition to fundamental interest in such solitons, e.g., optical vector solitons [1], multispecies condensates [3], and plasmas [4], collisions of vector solitons make possible applications including collision-based logic and universal computation [5,6]. Previously, such energyexchanging collisions [7] and information transfer [8] of vector solitons have been demonstrated in the spatial beam case, using photorefractive crystals with a saturable nonlinearity. However, to date, vector soliton collisions have not been observed in the temporal case. This is unfortunate, for optical pulses act as information carriers in modern telecommunications, yielding both ready-made equipment for observation of complex dynamics and a potential area of application for energy-exchanging results. Temporal soliton pulses in optical fiber were first predicted in [9], followed by the first experimental observation in [10]. Later, Menyuk accounted for the birefringence in polarization maintaining fiber (PMF) and predicted that vector solitons, in which two orthogonally polarized components trap each other, are stable under certain operating conditions [11]. Specifically, self-trapping of two orthogonally polarized pulses can occur when XPM-induced nonlinearity compensates the birefringence-induced group velocity difference, causing the pulse in the fiber’s fast axis to slow down and the pulse in the slow axis to speed up. The first demonstration of temporal soliton trapping was performed in the subpicosecond regime [12], in which additional ultrashort pulse effects such as Raman scattering 0031-9007=07=98(5)=053902(4)
are present. This effect results in a redshift that is linearly proportional to the propagation distance, as observed in a later temporal soliton trapping experiment [13]. Recently, soliton trapping in the picosecond regime was observed with equal amplitude pulses [14]. However, vector soliton propagation could not be shown because the pulses propagated for less than 1.5 dispersion lengths. In other work, phase-locked vector solitons in a weakly birefringent fiber laser cavity with nonlinear coherent coupling between components was observed in [15], while vector soliton stability under the combined effects of coherent and incoherent nonlinear coupling was studied numerically in [16]. In this Letter, we demonstrate stable propagation of arbitrary-amplitude vector solitons, in which soliton trapping of picosecond pulses is observed and no ultrashort pulse effects, coherent coupling, or polarization instability are present. In addition, we demonstrate energyexchanging collisions. To the best of the authors’ knowledge, this is both the first reported observation of temporal vector soliton propagation with incoherently coupled components and of vector soliton collisions (temporal or spatial) in a Kerr nonlinear medium. The experimental setup is shown in Fig. 1. We synchronized two actively mode-locked erbium-doped fiber lasers
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EDFA
TDL 2:1
EDFL 2
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EDFA
PLC
LP λ/2
OSA PBS
LB-PMF
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FIG. 1. Experimental setup.
© 2007 The American Physical Society
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PHYSICAL REVIEW LETTERS
(EDFLs)—EDFL1 at 1.25 GHz repetition rate, and EDFL2 at 5 GHz. EDFL2 was modulated (MOD) to match with the lower repetition rate of EDFL1. Each pulse train, consisting of 2 ps pulses, was amplified in an erbium-doped fiber amplifier (EDFA) and combined in a fiber coupler. To align polarizations, a polarization loop controller was used in one arm, and a tunable delay line (TDL) was employed to temporally align the pulses for collision. Once combined, both pulse trains passed through a linear polarizer (LP) and a half-wave plate to control the input polarization to the PMF. Approximately 2 m of high birefringence (HB) PMF preceded the specially designed 500 m of low birefringence (LB) PMF used to propagate vector solitons. Although this short length of HB-PMF will introduce some pulse splitting (on the order of 2 –3 ps), the birefringent axes of the HB- and LB-PMF were swapped in order to counteract this effect. Each component of the vector soliton was then split at a polarization beam splitter (PBS), followed by an optical spectrum analyzer (OSA) for measurement. The design of the LB-PMF required careful control over three characteristic length scales: the (polarization) beat length, dispersion length Ld , and nonlinear length Lnl . A beat length Lb =n 50 cm was chosen at a wavelength of 1550 nm, where n is the fiber birefringence. According to the approximate stability criterion of [17], this choice allows stable propagation of picosecond vector solitons. By avoiding the subpicosecond regime, ultrashort pulse effects such as intrapulse Raman scattering will not be present. The dispersion D 2c2 =2 16 ps=km nm and Ld 2T02 =j2 j 70 m, where T0 TFWHM =1:763 is a characteristic pulse width related to the full width at half maximum (FWHM) pulse width. Since Ld Lb , degenerate four-wave mixing due to coherent coupling between the two polarization components can be neglected [18]. Furthermore, the total propagation distance is greater than 7 dispersion lengths. Polarization instability, in which the fast axis component is unstable, occurs when Lnl P1 is of the same order of magnitude or smaller than Lb , as observed in [19]. The nonlinearity parameter 2n2 =Aeff 1:3 km W1 , with Kerr nonlinearity coefficient n2 2:6 1020 m2 =W and measured effective mode area Aeff 83 m2 . In the LB-PMF, the fundamental vector soliton power P 14 W, thus Lnl 55 m Lb , mitigating the effect of polarization instability. The theoretical model for our numerical calculations is the coupled nonlinear Schro¨dinger equation (CNLSE): @Ax;y @Ax;y @2 Ax;y 1x;y 2 i @z @t 2 @t2 2 2 2 jAx;y j jAy;x j Ax;y 0; (1) 3 where t is the local time of the pulse, z is propagation distance along the fiber, and Ax;y is the slowly varying pulse envelope for each polarization component. The parameter
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1x;y is the group velocity associated with each fiber axis, and 2 represents the group velocity dispersion, assumed equal for both polarizations. In addition, we neglect higher order dispersion and assume a lossless medium with an instantaneous electronic response, valid for picosecond pulses propagating in optical fiber. The last two terms of Eq. (1) account for the nonlinearity due to SPM and XPM, respectively. In linearly birefringent optical fiber, a ratio of 2=3 exists between these two terms. When this ratio equals unity, the CNLSE becomes the integrable Manakov system, characterized by soliton collisions with no radiation of dispersive waves [1]. There exist several candidates for the physical realization of Manakov or near-Manakov solitons, including photorefractive crystals [7,8,20], semiconductor waveguides [21], quadratic media [22], optical fiber [18], and BoseEinstein condensates [3]. Although solutions of Eq. (1) are not, strictly speaking, solitons, they are solitary waves, and it was found in [23] that the family of symmetric singlehumped solutions, to which the current investigation belongs, are all stable. Furthermore, it was shown in [24] that collisions of solitary waves in Eq. (1) can be described by perturbation theory of the integrable Manakov equations, indicating the similarities between the characteristics of these two systems. We first studied propagation of vector solitons using both lasers independently. Results are shown in Fig. 2. The peak power of the pulses is 14.3 W, very close to the theoretical prediction of 14 W for vector soliton formation. The total loss (including input and output coupling losses, as well as propagation loss) is less than 0.6 dB. The wavelength shift for each component is shown in Fig. 2(a) as a function of the input polarization angle , controlled through the half-wave plate. Because of the anomalous dispersion of the fiber at this wavelength, the component in the slow (fast) axis will shift to shorter (longer) wavelengths to compensate the birefringence. The total amount of wavelength shift between components xy 1 =D 0:64 nm, where 1 j1x 1y j 10:3 ps=km is the birefringence-induced group velocity difference. As approaches 0 (90 ), the vector soliton approaches the scalar soliton limit, and the fast (slow) axis does not shift in wavelength. At 45 , a symmetric shift results. For unequal amplitude solitons, the smaller component shifts more in wavelength than the larger component, because the former experiences more XPM. The numerically simulated curves, given by the dashed lines of Fig. 2(a), agree well with the experimental results. Also shown in Fig. 2 are two cases, 45 and 37 , as well as the numerical prediction. The experimental spectra show some oscillatory features at 5 GHz, which are a modulation of the EDFL2 repetition rate on the optical spectrum. The input pulse spectrum from EDFL1 is shown in the inset of Fig. 2, which shows no modulation due to the limited resolution of the OSA. Vector solitons from both lasers
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induced walkoff, and adjusted the delay line in such a way that the collision occurred halfway down the fiber. We define a collision length Lcoll 2TFWHM =D, where is the wavelength separation between the two vector solitons. For our setup, 3 nm, and Lcoll 83:3 m. As a result, the total fiber length is equal to 6 collision lengths, long enough to ensure sufficient separation of solitons before and after collision. To quantify our results, we introduce a quantity R tan2 , defined as the power ratio between the slow and fast components. Manakov theory predicts that the resulting energy exchange will be a function of R1;2 , , and the relative phase 1;2 between the two components of each soliton, where soliton 1 (2) is the shorter (longer) wavelength soliton [1,2,5]. Because only one half-wave plate is used in our experiment, it was not possible to prepare each vector soliton individually with an arbitrary R. In addition, due to the wavelength dependence of the half-wave plate, it was not possible to adjust without affecting R. Beyond Manakov theory, e.g., the current case of birefringent fiber, the unequal (2=3) ratio between SPM and XPM nonlinearities will in general be characterized by a phase difference 1;2 that changes as a function of propagation distance [23]. To bypass this, we ensure that all collisions reported below occur at the same spatial point in the fiber. First, we investigated the phase dependence of the collision. This was done by increasing the length of the HB-
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FIG. 2. Arbitrary-amplitude vector soliton propagation. (a) Wavelength shift vs angle to fast axis , numerical curves given by dashed lines; (b) and (d) experimental results for 45 and 37 with EDFL2, respectively. Inset: input spectrum for EDFL1; (c) and (e) corresponding numerical simulations of 45 and 37 , respectively.
produced similar results. In this and all subsequent plots, the slow and fast axis components are depicted by solid and dashed lines, respectively. As the two component amplitudes become more unequal, satellite peaks become more pronounced in the smaller component. We attribute this to two factors: (i) a nonideal input spectral profile, as shown in the inset of Fig. 2; and (ii) the pulse power, which is calibrated for the 45 case. In this case, the power threshold for vector soliton formation is largest due to the 2=3 factor between SPM and XPM nonlinear terms in the CNLSE. As the input is rotated towards unequal components, there will be extra power in the input. Both factors will result in the radiation of dispersive waves as the vector soliton forms. Because of the nature of this system, the dispersive waves can be nonlinearly trapped, giving rise to the satellite features in the optical spectra. This effect is present in the simulations, though not as prominent because an ideal soliton spectral profile was used. To set up a collision, we operated both lasers simultaneously, detuned in wavelength to allow for dispersion-
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FIG. 3. Demonstration of the phase dependence of energyexchanging collisions. In (a) –(c) the collision phase 2 was determined by the short length of HB-PMF; and (e) –(f) by the long length; (a),(d) experimental result, without collision; (b),(e) experimental result, with collision; (c),(f) simulated collision result with (c) 2 90 and (f) 2 50 . Values of the slow-fast power ratio are given above each soliton.
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In the second set of results (Fig. 4), we changed R while keeping all other parameters constant. More specifically, we used the short HB-PMF, with initial phase difference 2 90 , and changed the power ratio. In agreement with our numerical predictions and Manakov theory [1,2,5], the same direction of energy exchange is observed as in Figs. 3(a)–3(c). Along with the observed energy redistribution, in both Figs. 3 and 4 there is a wavelength shift as expected by the vector soliton propagation dynamics of Fig. 2. For example, comparing Figs. 3(d) and 3(e), we see that soliton 1 (2) shifts to shorter (longer) wavelengths. The observed spectral shift of 0.1 (0.05) nm is in good agreement with the expected 0.08 (0.04) nm shift from Fig. 2. In summary, we demonstrated for the first time the stable propagation and collisions of temporal vector solitons in a linearly birefringent fiber. The control of energyexchanging collisions was realized, which may find applications in all-optical switching, logic, and computation.
FIG. 4. Additional energy-exchanging collisions. In (a) –(c) the collision phase 2 was determined by the short length of HBPMF; and (e) –(f) by the long length; (a),(d) Experiment, without collision; (b),(e) experiment, with collision; (c),(f ) simulated collision result, using 2 90 inferred from the experiment of Fig. 3.
PMF by approximately 0.5 m, while keeping R and constant. As a result, we could change 1;2 due to the birefringence of the HB-PMF. The results are shown in Fig. 3, where the left and right columns correspond to the short and long HB-PMFs, respectively. Figures 3(a) and 3(d) show the two vector solitons, which propagate independently when no collision occurs; as expected, the two results are similar because the OSA measurement does not depend on 1;2 . The result of the collision is depicted in Figs. 3(b) and 3(e), along with the corresponding simulation results in Figs. 3(c) and 3(f). In both of these collisions, an energy exchange between components occurs, and two important relations are satisfied: the total energy in each soliton and in each component is conserved. It can be seen that when one component in a soliton increases as a result of the collision, the other component decreases, with the opposite exchange in the second soliton. The difference between these two collisions is dramatic, in that the energy redistributes in opposite directions. For the simulations, idealized sech pulses for each component were used as initial conditions, and propagation was modeled without accounting for polarization-dependent losses. The experimental power ratio was used, and (without loss of generality [1,2,5]) 2 was varied while 1 0. Best fits gave 2 90 [Fig. 3(c)] and 50 [Fig. 3(f)]. Despite the model approximations, experimental and numerical results all agree to within 15%.
*Electronic address: [email protected] [1] S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 505 (1973) [Sov. Phys. JETP 38, 248 (1974)]. [2] R. Radhakrishnan et al., Phys. Rev. E 56, 2213 (1997); R. Radhakrishnan et al., ibid. 69, 046607 (2004). [3] C. J. Myatt et al., Phys. Rev. Lett. 78, 586 (1997); J. Babarro et al., Phys. Rev. A 71, 043608 (2005). [4] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge University Press, Cambridge, England, 2000), 2nd ed. [5] M. H. Jakubowski et al., Phys. Rev. E 58, 6752 (1998). [6] K. Steiglitz, Phys. Rev. E 63, 016608 (2000); 63, 046607 (2001). [7] C. Anastassiou et al., Phys. Rev. Lett. 83, 2332 (1999). [8] C. Anastassiou et al., Opt. Lett. 26, 1498 (2001). [9] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). [10] L. F. Mollenauer et al., Phys. Rev. Lett. 45, 1095 (1980). [11] C. R. Menyuk, Opt. Lett. 12, 614 (1987); J. Opt. Soc. Am. B 5, 392 (1988). [12] M. N. Islam et al., Opt. Lett. 14, 1011 (1989). [13] N. Nishizawa and T. Goto, Opt. Express 10, 256 (2002). [14] A. E. Korolev et al., Opt. Lett. 30, 132 (2005). [15] S. T. Cundiff et al., Phys. Rev. Lett. 82, 3988 (1999). [16] J. M. Soto-Crespo et al., Phys. Rev. E 51, 3547 (1995). [17] X. D. Cao and C. J. McKinstrie, J. Opt. Soc. Am. B 10, 1202 (1993). [18] C. R. Menyuk, IEEE J. Quantum Electron. 25, 2674 (1989). [19] Y. Barad and Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997). [20] M. Shih and M. Segev, Opt. Lett. 21, 1538 (1996); D. N. Christodoulides et al., Appl. Phys. Lett. 68, 1763 (1996); Z. Chen et al., Opt. Lett. 21, 1436 (1996). [21] J. U. Kang et al., Phys. Rev. Lett. 76, 3699 (1996). [22] V. V. Steblina et al., J. Opt. Soc. Am. B 17, 2026 (2000). [23] J. Yang, Physica (Amsterdam) 108D, 92 (1997). [24] J. Yang, Phys. Rev. E 59, 2393 (1999).
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PHYSICAL REVIEW LETTERS
Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber Darren Rand,1,* Ivan Glesk,1 Camille-Sophie Bre`s,1 Daniel A. Nolan,2 Xin Chen,2 Joohyun Koh,2 Jason W. Fleischer,1 Ken Steiglitz,3 and Paul R. Prucnal1 1
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA 2 Science and Technology Division, Corning Incorporated, Corning, New York 14831, USA 3 Department of Computer Science, Princeton University, Princeton, New Jersey 08544, USA (Received 15 May 2006; published 1 February 2007)
We report the experimental observation of temporal vector soliton propagation and collision in a linearly birefringent optical fiber. To the best of the authors’ knowledge, this is both the first demonstration of temporal vector solitons with two mutually incoherent component fields, and of vector soliton collisions in a Kerr nonlinear medium. Collisions are characterized by an intensity redistribution between the two components, and the experimental results agree with numerical predictions of the coupled nonlinear Schro¨dinger equation. DOI: 10.1103/PhysRevLett.98.053902
PACS numbers: 42.65.Tg, 42.81.Dp
Because of their multicomponent structure, vector solitons have richer propagation dynamics than their scalar, one-component counterparts. For example, a scalar soliton propagates with an unchanging profile by balancing selfphase modulation (SPM) with dispersion or diffraction, while in the vector case one must also account for crossphase modulation (XPM) between components. For collisions, scalar solitons are characterized by phase and position shifts, while vector solitons have the added dynamics of possible intensity redistributions between the component fields [1,2]. In addition to fundamental interest in such solitons, e.g., optical vector solitons [1], multispecies condensates [3], and plasmas [4], collisions of vector solitons make possible applications including collision-based logic and universal computation [5,6]. Previously, such energyexchanging collisions [7] and information transfer [8] of vector solitons have been demonstrated in the spatial beam case, using photorefractive crystals with a saturable nonlinearity. However, to date, vector soliton collisions have not been observed in the temporal case. This is unfortunate, for optical pulses act as information carriers in modern telecommunications, yielding both ready-made equipment for observation of complex dynamics and a potential area of application for energy-exchanging results. Temporal soliton pulses in optical fiber were first predicted in [9], followed by the first experimental observation in [10]. Later, Menyuk accounted for the birefringence in polarization maintaining fiber (PMF) and predicted that vector solitons, in which two orthogonally polarized components trap each other, are stable under certain operating conditions [11]. Specifically, self-trapping of two orthogonally polarized pulses can occur when XPM-induced nonlinearity compensates the birefringence-induced group velocity difference, causing the pulse in the fiber’s fast axis to slow down and the pulse in the slow axis to speed up. The first demonstration of temporal soliton trapping was performed in the subpicosecond regime [12], in which additional ultrashort pulse effects such as Raman scattering 0031-9007=07=98(5)=053902(4)
are present. This effect results in a redshift that is linearly proportional to the propagation distance, as observed in a later temporal soliton trapping experiment [13]. Recently, soliton trapping in the picosecond regime was observed with equal amplitude pulses [14]. However, vector soliton propagation could not be shown because the pulses propagated for less than 1.5 dispersion lengths. In other work, phase-locked vector solitons in a weakly birefringent fiber laser cavity with nonlinear coherent coupling between components was observed in [15], while vector soliton stability under the combined effects of coherent and incoherent nonlinear coupling was studied numerically in [16]. In this Letter, we demonstrate stable propagation of arbitrary-amplitude vector solitons, in which soliton trapping of picosecond pulses is observed and no ultrashort pulse effects, coherent coupling, or polarization instability are present. In addition, we demonstrate energyexchanging collisions. To the best of the authors’ knowledge, this is both the first reported observation of temporal vector soliton propagation with incoherently coupled components and of vector soliton collisions (temporal or spatial) in a Kerr nonlinear medium. The experimental setup is shown in Fig. 1. We synchronized two actively mode-locked erbium-doped fiber lasers
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EDFL 1
EDFA
TDL 2:1
EDFL 2
MOD
EDFA
PLC
LP λ/2
OSA PBS
LB-PMF
HB-PMF
FIG. 1. Experimental setup.
© 2007 The American Physical Society
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PHYSICAL REVIEW LETTERS
(EDFLs)—EDFL1 at 1.25 GHz repetition rate, and EDFL2 at 5 GHz. EDFL2 was modulated (MOD) to match with the lower repetition rate of EDFL1. Each pulse train, consisting of 2 ps pulses, was amplified in an erbium-doped fiber amplifier (EDFA) and combined in a fiber coupler. To align polarizations, a polarization loop controller was used in one arm, and a tunable delay line (TDL) was employed to temporally align the pulses for collision. Once combined, both pulse trains passed through a linear polarizer (LP) and a half-wave plate to control the input polarization to the PMF. Approximately 2 m of high birefringence (HB) PMF preceded the specially designed 500 m of low birefringence (LB) PMF used to propagate vector solitons. Although this short length of HB-PMF will introduce some pulse splitting (on the order of 2 –3 ps), the birefringent axes of the HB- and LB-PMF were swapped in order to counteract this effect. Each component of the vector soliton was then split at a polarization beam splitter (PBS), followed by an optical spectrum analyzer (OSA) for measurement. The design of the LB-PMF required careful control over three characteristic length scales: the (polarization) beat length, dispersion length Ld , and nonlinear length Lnl . A beat length Lb =n 50 cm was chosen at a wavelength of 1550 nm, where n is the fiber birefringence. According to the approximate stability criterion of [17], this choice allows stable propagation of picosecond vector solitons. By avoiding the subpicosecond regime, ultrashort pulse effects such as intrapulse Raman scattering will not be present. The dispersion D 2c2 =2 16 ps=km nm and Ld 2T02 =j2 j 70 m, where T0 TFWHM =1:763 is a characteristic pulse width related to the full width at half maximum (FWHM) pulse width. Since Ld Lb , degenerate four-wave mixing due to coherent coupling between the two polarization components can be neglected [18]. Furthermore, the total propagation distance is greater than 7 dispersion lengths. Polarization instability, in which the fast axis component is unstable, occurs when Lnl P1 is of the same order of magnitude or smaller than Lb , as observed in [19]. The nonlinearity parameter 2n2 =Aeff 1:3 km W1 , with Kerr nonlinearity coefficient n2 2:6 1020 m2 =W and measured effective mode area Aeff 83 m2 . In the LB-PMF, the fundamental vector soliton power P 14 W, thus Lnl 55 m Lb , mitigating the effect of polarization instability. The theoretical model for our numerical calculations is the coupled nonlinear Schro¨dinger equation (CNLSE): @Ax;y @Ax;y @2 Ax;y 1x;y 2 i @z @t 2 @t2 2 2 2 jAx;y j jAy;x j Ax;y 0; (1) 3 where t is the local time of the pulse, z is propagation distance along the fiber, and Ax;y is the slowly varying pulse envelope for each polarization component. The parameter
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1x;y is the group velocity associated with each fiber axis, and 2 represents the group velocity dispersion, assumed equal for both polarizations. In addition, we neglect higher order dispersion and assume a lossless medium with an instantaneous electronic response, valid for picosecond pulses propagating in optical fiber. The last two terms of Eq. (1) account for the nonlinearity due to SPM and XPM, respectively. In linearly birefringent optical fiber, a ratio of 2=3 exists between these two terms. When this ratio equals unity, the CNLSE becomes the integrable Manakov system, characterized by soliton collisions with no radiation of dispersive waves [1]. There exist several candidates for the physical realization of Manakov or near-Manakov solitons, including photorefractive crystals [7,8,20], semiconductor waveguides [21], quadratic media [22], optical fiber [18], and BoseEinstein condensates [3]. Although solutions of Eq. (1) are not, strictly speaking, solitons, they are solitary waves, and it was found in [23] that the family of symmetric singlehumped solutions, to which the current investigation belongs, are all stable. Furthermore, it was shown in [24] that collisions of solitary waves in Eq. (1) can be described by perturbation theory of the integrable Manakov equations, indicating the similarities between the characteristics of these two systems. We first studied propagation of vector solitons using both lasers independently. Results are shown in Fig. 2. The peak power of the pulses is 14.3 W, very close to the theoretical prediction of 14 W for vector soliton formation. The total loss (including input and output coupling losses, as well as propagation loss) is less than 0.6 dB. The wavelength shift for each component is shown in Fig. 2(a) as a function of the input polarization angle , controlled through the half-wave plate. Because of the anomalous dispersion of the fiber at this wavelength, the component in the slow (fast) axis will shift to shorter (longer) wavelengths to compensate the birefringence. The total amount of wavelength shift between components xy 1 =D 0:64 nm, where 1 j1x 1y j 10:3 ps=km is the birefringence-induced group velocity difference. As approaches 0 (90 ), the vector soliton approaches the scalar soliton limit, and the fast (slow) axis does not shift in wavelength. At 45 , a symmetric shift results. For unequal amplitude solitons, the smaller component shifts more in wavelength than the larger component, because the former experiences more XPM. The numerically simulated curves, given by the dashed lines of Fig. 2(a), agree well with the experimental results. Also shown in Fig. 2 are two cases, 45 and 37 , as well as the numerical prediction. The experimental spectra show some oscillatory features at 5 GHz, which are a modulation of the EDFL2 repetition rate on the optical spectrum. The input pulse spectrum from EDFL1 is shown in the inset of Fig. 2, which shows no modulation due to the limited resolution of the OSA. Vector solitons from both lasers
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induced walkoff, and adjusted the delay line in such a way that the collision occurred halfway down the fiber. We define a collision length Lcoll 2TFWHM =D, where is the wavelength separation between the two vector solitons. For our setup, 3 nm, and Lcoll 83:3 m. As a result, the total fiber length is equal to 6 collision lengths, long enough to ensure sufficient separation of solitons before and after collision. To quantify our results, we introduce a quantity R tan2 , defined as the power ratio between the slow and fast components. Manakov theory predicts that the resulting energy exchange will be a function of R1;2 , , and the relative phase 1;2 between the two components of each soliton, where soliton 1 (2) is the shorter (longer) wavelength soliton [1,2,5]. Because only one half-wave plate is used in our experiment, it was not possible to prepare each vector soliton individually with an arbitrary R. In addition, due to the wavelength dependence of the half-wave plate, it was not possible to adjust without affecting R. Beyond Manakov theory, e.g., the current case of birefringent fiber, the unequal (2=3) ratio between SPM and XPM nonlinearities will in general be characterized by a phase difference 1;2 that changes as a function of propagation distance [23]. To bypass this, we ensure that all collisions reported below occur at the same spatial point in the fiber. First, we investigated the phase dependence of the collision. This was done by increasing the length of the HB-
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FIG. 2. Arbitrary-amplitude vector soliton propagation. (a) Wavelength shift vs angle to fast axis , numerical curves given by dashed lines; (b) and (d) experimental results for 45 and 37 with EDFL2, respectively. Inset: input spectrum for EDFL1; (c) and (e) corresponding numerical simulations of 45 and 37 , respectively.
produced similar results. In this and all subsequent plots, the slow and fast axis components are depicted by solid and dashed lines, respectively. As the two component amplitudes become more unequal, satellite peaks become more pronounced in the smaller component. We attribute this to two factors: (i) a nonideal input spectral profile, as shown in the inset of Fig. 2; and (ii) the pulse power, which is calibrated for the 45 case. In this case, the power threshold for vector soliton formation is largest due to the 2=3 factor between SPM and XPM nonlinear terms in the CNLSE. As the input is rotated towards unequal components, there will be extra power in the input. Both factors will result in the radiation of dispersive waves as the vector soliton forms. Because of the nature of this system, the dispersive waves can be nonlinearly trapped, giving rise to the satellite features in the optical spectra. This effect is present in the simulations, though not as prominent because an ideal soliton spectral profile was used. To set up a collision, we operated both lasers simultaneously, detuned in wavelength to allow for dispersion-
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1.47
(e)
0.86
2.23
(c)
1.41
1.58
(f)
0.96
2.57
FIG. 3. Demonstration of the phase dependence of energyexchanging collisions. In (a) –(c) the collision phase 2 was determined by the short length of HB-PMF; and (e) –(f) by the long length; (a),(d) experimental result, without collision; (b),(e) experimental result, with collision; (c),(f) simulated collision result with (c) 2 90 and (f) 2 50 . Values of the slow-fast power ratio are given above each soliton.
053902-3
PHYSICAL REVIEW LETTERS
PRL 98, 053902 (2007) (a)
0.97
1.47
(d)
2.14
2.56
(b)
1.18
1.12
(e)
2.78
1.92
1.15
1.24
2.73
2.29
(c)
(f)
week ending 2 FEBRUARY 2007
In the second set of results (Fig. 4), we changed R while keeping all other parameters constant. More specifically, we used the short HB-PMF, with initial phase difference 2 90 , and changed the power ratio. In agreement with our numerical predictions and Manakov theory [1,2,5], the same direction of energy exchange is observed as in Figs. 3(a)–3(c). Along with the observed energy redistribution, in both Figs. 3 and 4 there is a wavelength shift as expected by the vector soliton propagation dynamics of Fig. 2. For example, comparing Figs. 3(d) and 3(e), we see that soliton 1 (2) shifts to shorter (longer) wavelengths. The observed spectral shift of 0.1 (0.05) nm is in good agreement with the expected 0.08 (0.04) nm shift from Fig. 2. In summary, we demonstrated for the first time the stable propagation and collisions of temporal vector solitons in a linearly birefringent fiber. The control of energyexchanging collisions was realized, which may find applications in all-optical switching, logic, and computation.
FIG. 4. Additional energy-exchanging collisions. In (a) –(c) the collision phase 2 was determined by the short length of HBPMF; and (e) –(f) by the long length; (a),(d) Experiment, without collision; (b),(e) experiment, with collision; (c),(f ) simulated collision result, using 2 90 inferred from the experiment of Fig. 3.
PMF by approximately 0.5 m, while keeping R and constant. As a result, we could change 1;2 due to the birefringence of the HB-PMF. The results are shown in Fig. 3, where the left and right columns correspond to the short and long HB-PMFs, respectively. Figures 3(a) and 3(d) show the two vector solitons, which propagate independently when no collision occurs; as expected, the two results are similar because the OSA measurement does not depend on 1;2 . The result of the collision is depicted in Figs. 3(b) and 3(e), along with the corresponding simulation results in Figs. 3(c) and 3(f). In both of these collisions, an energy exchange between components occurs, and two important relations are satisfied: the total energy in each soliton and in each component is conserved. It can be seen that when one component in a soliton increases as a result of the collision, the other component decreases, with the opposite exchange in the second soliton. The difference between these two collisions is dramatic, in that the energy redistributes in opposite directions. For the simulations, idealized sech pulses for each component were used as initial conditions, and propagation was modeled without accounting for polarization-dependent losses. The experimental power ratio was used, and (without loss of generality [1,2,5]) 2 was varied while 1 0. Best fits gave 2 90 [Fig. 3(c)] and 50 [Fig. 3(f)]. Despite the model approximations, experimental and numerical results all agree to within 15%.
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