Vector soliton

Among all the soliton considered, the optical soliton draws the most attention as it is possible to generate ultrafast pulses and has wider application. optical soliton can be classified as two groups:temporal soliton and spatial soliton.however, during the propagation of both temporal soliton and spatial soliton, because of the existence of birefringence,the cross phase modulation and coherent energy exchange among the two orthogonal polarizations of the soliton could induce the intensity difference of these two polarizations.In the meantime, the solitons are no longer linearly polarized but elliptically polarized and called as vector (spatial or optical)soliton.

Definition

Menyuk firstly derived the nonlinear pulse propagation equation in SMF under weakly birefringence. Then, he denoted vector soliton as two soliton (more accurately if called as solitary waves) with orthogonal polarizations co-propagated together without dispersing their energy and remaining their shapes. Because of nonlinear interaction among these two polarizations, despite of the existence of birefringence between these two polarization modes, they could still adjust their group velocity and trap together.

Vector solitons can be spatial or temporal and formed by two orthogonally polarized components of a single optical field or two fields of different frequencies but the same polarization.

History

Back to 1987, Menyuk firstly derived the nonlinear pulse propagation equation in SMF under weakly birefringence. It is an initial equation which open a new area to researcher in the area of "scalar" soliton. His equation has considered the nonlinear interaction (cross phase modulation and coherent energy exchange)between the two orthogonal polarization components. From such milestone equation, researchers have obtained both the analytical or numerical solution of such equation under the weakly or moderately or even strongly birefringence.

IN 1988 Christodoulides and Joseph first theoretically predicted a novel form of phase locked vector soliton in birefringent dispersive media, which is now known as a high order phase locked vector soliton in SMFs.It has two orthogonal polarization component with comparable intensity .Despite of the existence of birefringence, these two polarizations could propagate with the same group velocity as they shift their central frequency in order to make their group velocity apaced.

In 2000, Cundiff and Akhmediev found that these two polarizations could not only form the so called group velocity locked vector soliton but also polarization locked vector soliton. They reported that the intensity ratio of these two polarizations about 0.25-1.00.

However, recently,Han Zhang and Dingyuan Tang have pointed another type of vector soliton---- [induced vector soliton] .Such vector soliton is novel in that though the intensity difference between these two orthogonal polarization are extremely large (20dB).It seems that the weak polarization are unable to form a soliton. However, due to the cross polarizaiton modulation between the strong and weak polarization component, the weak soliton could also be formed. It thus demonstrates that the soliton obtained is not "scalar" soliton with linearly polarizatin mode but vector soliton with large ellipticity. Such observation expand the scope of vector soliton. The intensity ratio between the strong and weak component of the vector soliton could not limited only to 0.25-1.0 but extend to 20 dB as well.

FWM spectral sideband in Vector soliton

Passive mode-locking of erbium-doped fiber lasers with a semiconductor saturable absorber mirror (SESAM) has been extensively investigated [1, 2] . In contrast to the nonlinear polarization rotation mode-locking, mode-locking incorporating a SESAM does not require any polarization element inside the laser cavity, thereby under suitable condition of the cavity birefringence, vector solitons could be formed in the lasers [3] . Recently, it was reported that even the polarization-locked vector solitons (PLVSs) could be formed in the mode-locked fiber lasers [4, 5] . Formation of a PLVS requires not only that the group velocities of the two orthogonal polarization components of a vector soliton are locked but also that their phase velocities are also locked. It is well known that through the self-phase modulation (SPM) and cross-phase modulation (XPM) between the two polarization-modes of a fiber, the group velocity locked vector solitons (GVLVSs) could be formed [6] . Although it was also pointed out that the four-wave-mixing (also called coherent energy exchange) coupling between the polarization components of a vector soliton could have contributed to the formation of the PLVSs [4, 5] , so far no experimental evidence on the soliton internal FWM has been shown. In this Letter we report on the experimental observation of FWM between the two orthogonal polarization components of a vector soliton formed in a fiber laser passively mode locked with a SESAM. A new type of spectral sidebands was first experimentally observed on the polarization resolved soliton spectra of the PLVSs of the fiber lasers. The new spectral sidebands are characterized by that their positions on the soliton spectrum vary with the strength of the linear cavity birefringence, and while on one vector soliton polarization component the sideband appears as a spectral peak, then on the orthogonal polarization component it is a spectral dip, indicating the energy exchange between the two orthogonal polarization components of the vector solitons. Numerically we confirmed that the formation of the new type of spectral sidebands was formed by the FWM between the two polarization components of the vector solitons. 2. Experimental setupThe fiber laser is illustrated in Fig.1. It has a ring cavity consisting of a piece of 4.6 m Erbium-doped fiber (EDF) with group velocity dispersion parameter of 10 ps/km/nm and a total length of 5.4 m standard single mode fiber (SMF) with group velocity dispersion parameter of 18 ps/km/nm. The cavity has a length of 4.6EDF+5.4SMF=10m. Note that within one cavity round-trip the signal propagates twice in the SMF between the circulator and the SESAM. A circulator is used to force the unidirectional operation of the ring and simultaneously to incorporate the SESAM in the cavity. An intra cavity polarization controller is used to change the cavity’s linear birefringence. The laser is pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. A 10% fiber coupler is used to output the signals. The laser operation is monitored with an optical spectrum analyzer (Ando AQ-6315B), a 26.5 GHz RF spectrum analyzer (Agilent E4407BESA-E SERIES) and a 350 MHz oscilloscope (Agilent 54641A) together with a 5 GHz photodetector. A commercial autocorrelator (Femtochrome FR-103MN) is used to measure the pulse width of the soliton pulses. The SESAM used is made based on a GalnNAs quantum well. The SESAM has a saturable absorption modulation depth of 5%, a saturation fluence of 90 µJ/cm2 and 10 ps relaxation time. The central absorption wavelength of the SESAM is at 1550nm.

Fig. 1. Schematic of the fiber laser. SESAM: semiconductor saturable absorber mirror; PC: polarization controller; WDM: wavelength-division multiplexer; EDF: erbium-doped fiber.

3. Experimental resultsExperimentally, it was noticed that after mode-locking multiple mode locked pulses were always initially formed in the cavity. Depending on the net cavity birefringence, they were either the GVLVSs, characterized by the rotation of soliton polarization state along the cavity, or the PLVSs, characterized by the fixed polarization at the laser output. With multiple vector solitons in cavity, as a result of mutual soliton interaction complicated relative soliton movement or vector soliton bunches with random fixed soliton separations were observed. To exclude the complications caused by soliton interactions, we have always reduced the number of solitons in cavity through carefully decreasing pump power so that only one or a few widely separated solitons exist in cavity. Fig. 2 shows typical measured optical spectra of the PLVSs of the laser. The soliton feature of the mode-locked pulses is confirmed by the existence of soliton sidebands. However, apart from the existence of the conventional Kelly soliton sidebands, on the vector soliton spectrum there are also extra sets of spectral sidebands. Experimentally it was noticed that different from the Kelly sidebands whose positions are almost independent of the laser operation conditions, such as the pump strength and polarization controller orientation change (linear cavity birefringence change), the positions of the new spectral sidebands varied sensitively with the linear cavity birefringence. We note that Cundiff et al have reported a new type of spectral sidebands on the GVLVS spectrum and interpreted their formation as caused by the vector soliton polarization evolution in the cavity [7] . However, in our experiment the sidebands were observed on the PLVSs, whose polarization remains unchanged as they propagate along the laser cavity. To determine the physical origin of the extra sideband formation, we then conducted polarization resolved measurement of the vector soliton spectrum. To this end the laser output was first passed through a rotatable external cavity linear polarizer. To separate the two orthogonal polarization components of a vector soliton, we always first located the orientation of the polarizer to the maximum soliton transmission, which sets the long axis of an elliptically polarized vector soliton; we then rotated the polarizer by 90 degree to determine the soliton polarization component along the short axis of the polarization ellipse. Through separating the two orthogonal polarization components of the vector solitons, it turned out that the formation of the extra spectral sidebands was due to the coherent energy exchange between the two soliton polarization components. As can be clearly seen from the polarization resolved spectra, at the positions of extra spectral sidebands, while the spectral intensity of one soliton polarization component has a spectral peak, the orthogonal polarization component then has a spectral dip, indicting coherent energy exchange between them. We note that the energy flow between the two polarization components is not necessarily from the strong one to the weak one. Energy flow from the weak component to the strong component was also observed. In addition, it is to see from the polarization resolved spectra that the extra sidebands are symmetric with respect to the soliton peak frequency and at different wavelength positions the peak-dip can also alternate, suggesting that the energy exchange is the relative phase of the coupled components dependent. Fig. 2 (a) and (b) were obtained from the same laser but under different intra cavity polarization controller orientations. Obviously, the positions of the extra sidebands are the cavity birefringence dependent.

Fig. 2. Optical spectra of the phase locked vector solitons of the laser measured without passing and passing through a polarizer: (a) and (b) were measured under different linear cavity birefringence.

Bound vector soliton

Induced vector soliton

Optical solitons were first experimentally observed in single mode fibers (SMF) by Maulenauer et al. in 1980 [1] . The formation of the solitons was a result of the balanced interaction between the effects of anomalous fiber dispersion and the pulse self-phase modulation (SPM). To observe the solitons it is necessary that the intensity of a pulse be above a threshold where the nonlinear length of the pulse becomes comparable with the dispersion length. Apart from SPM, theoretical studies have also shown that cross-phase modulation (XPM) could lead to soliton formation. Soliton formation through XPM was known as the induced soliton formation. An important potential application of the effect is the light controlling light. Various cases of soliton formation in SMF caused by XPM were predicted, these include the formation of a bright soliton in the normal fiber dispersion regime supported by a dark soliton in the anomalous dispersion regime [2,3] , and bright solitons formation in anomalous fiber dispersion regime supported by each other through XPM [4,5] . Spatial soliton formation through XPM has been experimentally observed [6] . However, to the best of our knowledge, no induced temporal soliton formation experiments have been reported. In this letter, we report on the experimental observation of induced solitons in a passively mode-locked fiber laser. Using a birefringence cavity fiber laser, we observed that due to the cross coupling between the two orthogonal polarization components, if a strong soliton is formed along one principal polarization axis, a weak soliton can always be induced along the orthogonal polarization axis. Especially, the intensity of the weak soliton could be so weak that it alone cannot form a soliton by the SPM. Numerical simulations have well supported the experimental observations.

Our fiber laser is schematically shown in Fig. 1. It is an erbium-doped fiber (EDF) ring laser mode-locked with a semiconductor saturable absorber mirror (SESAM). The ring cavity consists of a piece of 4.6 m EDF with group velocity dispersion (GVD) parameter 10 ps/km/nm, and 5.4 m standard SMF with GVD parameter 18 ps/km/nm. A polarization independent circulator was used in the cavity to force the unidirectional operation of the ring and simultaneously incorporate the SESAM in the cavity. Note that within one cavity round-trip the pulse propagates twice in the SMF between the circulator and the SESAM. A 10% fiber coupler was used to output the signals, and the laser was pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. The SESAM used was made based on GalnNAs quantum wells. It has a saturable absorption modulation depth of 5%, a saturation fluence of 90µJ/cm2 and 10 ps relaxation time. The central absorption wavelength of the SESAM is at 1550nm. An intra cavity polarization controller was used to change the cavity linear birefringence.

As no polarizer was used in the cavity, due to the weak birefringence of the fibers the cavity exhibited obvious birefringence features, e. g. varying the linear cavity birefringence we could observe the various types of vector solitons in the laser [7, 8] . In order to identify features of the vector solitons, we explicitly investigated their polarization resolved spectra under various experimental conditions. To measure their polarization resolved spectra, we let the laser output first pass through a rotatable external cavity polarizer, based on the measured soliton intensity change with the orientation of the polarizer we then identify the long and short polarization ellipse axes of the vector solitons. In our measurements we found that apart from vector solitons with comparable coupled orthogonal polarization components, vector solitons with very asymmetric component intensity also exist. Fig. 2 shows for example two cases experimentally observed. Fig. 2a shows a case that was measured under laser operation with a relatively large cavity birefringence. In this case the spectral intensity difference between the two orthogonal polarization directions at the center soliton wavelength is more than 30 dB. The soliton nature of the strong polarization component is obvious as characterized by the existence of the Kelly sidebands. Kelly sidebands have also appeared on the weak polarization component. We emphasize the different locations of the Kelly sidebands along different polarization directions. It excludes the possibility that the sidebands on the spectrum of the weak component were produced due to an experimental artifact. The first order Kelly sidebands of the strong component are located at 1547.4 nm and 1569.5nm, respectively; those of the weak component are at 1546.8nm and 1568.9nm. The separations of both sets of sidebands are the same. The appearance of Kelly sidebands on spectrum of the weak component suggests that it is also a soliton. In particular, due to the large cavity birefringence the solitons formed along the two orthogonal polarization directions have different center wavelengths. Therefore, their Kelly sidebands have different locations.

Using a commercial autocorrelator we measured the soliton pulse width. Assuming a Sech2 pulse profile it is about 1 ps. If only the SPM is considered, we estimate that the peak power of the fundamental solitons in our laser is about 24W. This is well in agreement of the experimentally measured strong component soliton peak power of about 25W. The experimentally measured weak component soliton peak power is only 0.5W. Obviously with the intensity of the weak pulse it is impossible to form a soliton. The weak soliton should be an induced soliton.

Through adjusting the intra cavity polarization controller the net cavity birefringence could be changed. Another situation as shown in Fig. 2b where both solitons have the same center wavelength was also obtained. Even in the case the two solitons have different Kelly sidebands. In particular, the first order Kelly sidebands of the weak soliton have slightly larger separation than that of the strong soliton, indicating that the induced soliton has a narrower pulse width than that of the strong soliton [9] .

Gain-guide vector soliton

Polarization rotation of vector soliton

Higher-order vector soliton

ee also

* Clapotis
* Freak waves may be a related phenomenon.
* Oscillons
* Q-ball a non-topological soliton
* Soliton (topological).
* Soliton (optics)
* Soliton model of nerve impulse propagation
* Spatial soliton
* Solitary waves in discrete media [http://www.livescience.com/technology/050614_baby_waves.html]
* Topological quantum number

References

* N. J. Zabusky and M. D. Kruskal (1965). "Interaction of 'Solitons' in a Collisionless Plasma and the Recurrence of Initial States." Phys Rev Lett 15, 240

* A. Hasegawa and F. Tappert (1973). "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion." Appl. Phys. Lett. Volume 23, Issue 3, pp. 142-144.

* P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy (1987) "Picosecond steps and dark pulses through nonlinear single mode fibers." Optics. Comm. 62, 374

* P. G. Drazin and R. S. Johnson (1989). "Solitons: an introduction." Cambridge University Press.

* N. Manton and P. Sutcliffe (2004). "Topological solitons." Cambridge University Press.

* Linn F. Mollenauer and James P. Gordon (2006). "Solitons in optical fibers." Elsevier Academic Press.

* http://www3.ntu.edu.sg/home2006/zhan0174/

External links

* [http://www.sciencedaily.com/releases/2005/05/050506141331.htm Solitons, solitary waves and secondary or baby solitary waves in discrete media]
* [http://www.ma.hw.ac.uk/solitons/ Heriot-Watt University soliton page]
* [http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/index-e.html The many faces of solitons]
* [http://www.usf.uni-osnabrueck.de/~kbrauer/solitons.html Klaus Brauer's soliton page]
* [http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions]
* [http://www.ma.hw.ac.uk/~chris/scott_russell.html John Scott Russell and the solitary wave]
* [http://www.severn-bore.co.uk/default.htm Severn Bore web site]
* [http://www.ma.hw.ac.uk/solitons/LocalHeroes/sr.html John Scott Russell biography]
* [http://unic.ece.cornell.edu Soliton in Electrical Engineering]
* [http://web.njit.edu/~miura/ Miura's home page]
* [http://www.ma.hw.ac.uk/solitons/soliton1b.html Photograph of Soliton on the Scott Russell Aqueduct]
* [http://www.pnas.org/cgi/reprint/102/28/9790 Solitons possible agent of nerve transmission (PDF)(pnas.org)]
* [http://www.youtube.com/watch?v=H4rN3Wr4ctw Three Solitons Solution of KdV Equation]
* [http://www.youtube.com/watch?v=5z5SylS2QHE Three Solitons (unstable) Solution of KdV Equation]
* [http://lie.math.brocku.ca/~sanco/solitons/index.html Solitons & Nonlinear Wave Equations]