Vector soliton

Among all the solitons, the optical soliton draws the most attention as it is possible to generate ultrafast pulses and has wider application. optical soliton can be divided into :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 termed as vector (spatial or temporal)soliton.

Definition

Menyuk firstly derived the nonlinear pulse propagation equation in SMF under weakly birefringence. Then, Menyuk 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, another type of vector soliton----[induced vector soliton] have been observed recently.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

A new type of spectral sidebands was first experimentally observed on the polarization resolved soliton spectra of the polarization locked vector solitons 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 simulations also 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.

Bound vector soliton

Two adjacent vector solitons could form a bound state. Compared with scalar bound soliton, the polarization state of this soliton is more complex. Through the cross interactions, the bound vector solitons could have much stronger interacion forces.

Vector dark soliton

Dark soliton is characterized as a localized intensity dip on a continuous wave background.Scalar dark soliton,actually linearly polarized dark soliton,could be formed in all normal dispersion fiber laser moded locked by nonlinear polarizaiton rotation method.It could be rather stable.Regarding Vector dark solitons,they are much less stable due to the cross interaction between the two dark polarization components.Therefore,it is very interesting to investigate how the polarization state of these two polarization component evlove.

Vector dark bright soliton

Bright soliton is characterized as a localized intensity peak on a continuous wave (CW) background while dark soliton is featured as a localized intensity peak below a continuous wave (CW) background.Vector dark bright soliton means that one polarization state is a bright soliton while the other polarization is a dark soliton.It has been reported in incoherently coupled spatial DBVSs in a self-defocusing medium and matter wave DBVS in two-species condensates with repulsive scattering interactions. But never verified in the field of optical fiber.

Induced vector soliton

Using a birefringence cavity fiber laser, Induced vector soliton could be formed 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.

Vector Dissipative soliton

Vector dissipative soliton could be formed in a laser cavity with net positive dispersion and its formation mechamism is a natural result of the mutual nonlinear interaction among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, laser gain saturation and gain bandwidth filtering.For a conventional soliton, it is a blance between the dispersion and nonlinearity.Different from a conventional soliton, Vector dissipative soliton is strongly frequency chirped. It would be interesting to find out whether a phase-locked gain-guided vector soliton could be formed in a fiber laser or not:either the polarization rotating or the phase-locked dissipative vector soliton can be formed in a fiber laser with large net normal cavity group velocity dispersion. In addition, multiple Vector dissipative solitons with identical soliton parameters and harmonic mode locking of the Dissipative vector soliton can also be formed in a passively mode-locked fiber laser with a SESAM.

Polarization rotation of vector soliton

Different from scalar soliton, the output polarization is always linearly polarized due to the existence of in cavity polarizer. But for vector soliton, its polarization state could be rotating arbitrary but still locked to the cavity roundtrip time or multiple of it under appropriate conditions.

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Higher-order vector soliton

It is characterized by that not only the two orthogonally polarized soliton components are phase locked, but also one of the components has a double-humped intensity profile. Multiple such 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 fiber lasers.

References

1. O. G. Okhotnikov, T. Jouhti, J. Konttinen, S. Karirnne, and M. Pessa, “1.5-µm monolithic GaInNAs semiconductor saturable-absorber mode locking of an erbium fiber laser,” Opt. Lett. 28, 364-366 (2003).

2. M. Jiang, G. Sucha, M. E. Fermann, J. Jimenez, D. Harter, M. Dagenais, S. Fox, and Y. Hu, “Nonlinearly limited saturable-absorber mode-locking of an erbium fiber laser,” Opt. Lett. 24, 1074-1076 (1999).

3. D. N. Christodoulides and R. I. Joseph, “Vector solitons in birefringent nonlinear dispersive media,” Opt. Lett. 13, 53-55 (1988).

4. B. C. Collings, S. T. Cundiff, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17, 354-365 (2000).

5. S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber, ” Phys. Rev. Lett. 82, 3988-3991 (1999).

6. C. R. Menyuk, “Nonlinear Pulse-Propagation in Birefringent Optical Fibers,” IEEE J. Quantum Electron. QE-23, 174-176 (1987).

7. S. T. Cundiff, B. C. Collings, and W. H. Knox, “Polarization locking in an isotropic, mode locked soliton Er/Yb fiber laser,” Opt. Express 1, 12-21 (1997).

8. 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, 043816 (2005).

9. N. N. Akhmediev, A. Ankiewicz, M. J. Lederer, and B. Luther-Davies, “Ultrashort pulses generated by mode-locked lasers with either a slow or a fast saturable-absorber response,” Opt. Lett. 23, 280-282 (1998).