## Nonlinear similariton tunneling effect in the birefringent fiber |

Optics Express, Vol. 18, Issue 16, pp. 17548-17554 (2010)

http://dx.doi.org/10.1364/OE.18.017548

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### Abstract

We derive analytical bright and dark similaritons of the generalized coupled nonlinear Schrödinger equations with distributed coefficients. An exact balance condition between the dispersion, nonlinearity and the gain/loss has been obtained. Under this condition, we discuss the nonlinear similariton tunneling effect.

© 2010 Optical Society of America

## 1. Introduction

1. A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. **23**, 142–170 (1973). [CrossRef]

2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. **45**, 1095–1098 (1980). [CrossRef]

1. A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. **23**, 142–170 (1973). [CrossRef]

*εσ*> 0) and dark (

*εσ*< 0) solitons. Generally, bright solitons are well localized structures of light while dark solitons appear as localized intensity dips on a finite carrier wave background and are more robust than bright solitons.

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. **57**, 261–272 (2010). [CrossRef]

_{1}(

*z, t*) and Ψ

_{2}(

*z, t*) denote two orthogonal components of an electric field,

*z*is the coordinate along the propagation direction of the carrier wave, and

*t*is the retarded time. The second term represents GVD, the third and fourth terms are SPM and XPM for representing the nonlinear effect, and the term in the right stands for the amplification (

*g*> 0) or the attenuation (

*g*< 0). Bright soliton solutions of Eq. (2) have been investigated [4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. **57**, 261–272 (2010). [CrossRef]

5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E **71**, 056619 (2005). [CrossRef]

8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. **19**, 1126–1133 (1978). [CrossRef]

8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. **19**, 1126–1133 (1978). [CrossRef]

11. R. C. Yang and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express **16**, 17759–17767 (2008). [CrossRef] [PubMed]

## 2. General similarity transformations

5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E **71**, 056619 (2005). [CrossRef]

*θ*and

*ϑ*are constants, the amplitude

*ρ*(

*z*) and the phase

*ϕ*(

*z, t*) are real functions, Φ(

*Z, T*) is a complex function, the effective propagation distance

*Z*(

*z*) and the similarity variable

*T*(

*z, t*) are both to be determined.

*s*

_{0}= 0) expressed as Eq. (8) in [4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. **57**, 261–272 (2010). [CrossRef]

**57**, 261–272 (2010). [CrossRef]

## 3. Bright and dark similaritons

12. L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. **234**, 169–176 (2004). [CrossRef]

*m*= 1, …,

*n, j*= 1, 2, complex spectral parameters

*λ*

^{*}

_{m}is the complex conjugate of

*λ*,

_{m}*T,Z,ρ*(

*z*) and

*ϕ*(

*z, t*) satisfy Eqs. (5) and (6). (

*φ*

_{1,1}(

*λ*

_{1}),

*φ*

_{2,1}(

*λ*

_{1}))

^{T}is the eigenfunction corresponding to

*λ*

_{1}for Ψ

_{0}and

_{0}= 0 into Eq. (8), one can obtain one similariton solution for Eq. (2). Using that one similariton solution as the seed solution in Eq. (8), we can obtain two-similaritons. Thus in recursion, one can generate up to

*n*-similaritons. Here we present bright one- and two-similariton solutions in explicit forms. The one similariton read

*F*

_{1}=

*b*

_{1}cosh(

*δ*

_{1}+

*δ*

_{2}) +

*b*

_{2}cosh(

*δ*

_{1}−

*δ*

_{2}) +

*b*

_{3}cos (

*κ*

_{2}−

*κ*

_{1}),

*a*

_{3}=

*η*

_{1}

*η*

_{2}(

*ξ*

_{1}−

*ξ*

_{2}) and

*b*

_{3}= −

*η*

_{1}

*η*

_{2},

*j*= 1,2.

*δ*

_{j}and

*κ*

_{j}are given by Eq. (10).

*ρ*= (

*ω*

_{1}−

*μ*) (

*ω*

_{2}−

*μ*),

*ω*= (

_{j}*ξ*−

_{j}*iη*) [

_{j}*ξ*+

_{j}*iη*tanh(

_{j}*δ*)]/

_{j}*μ*,

*δ*=

_{j}*η*[

_{j}*T*−

*T*

_{j0}− 2(Ω + ξ

_{j})

*εZ*],,

*λ*=

_{j}*ξ*+ i

_{j}*η*and

_{j}*μ*= ∣

*λ*∣,

_{j}*j*= 1,2. And the exact dark one similariton read

## 4. Nonlinear similariton tunneling effect

9. V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. **74**, 573–577 (2001). [CrossRef]

10. J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B **25**, 1254–1260 (2008). [CrossRef]

*I*= ∣Ψ

_{1}∣

^{2}of the nonlinear tunneling of the single bright similariton with

*s*

_{0}≠ 0 or soliton with

*s*

_{0}= 0 [Eq. (11)] through both the DB or DW [Eq. (15)] and the NB or NW [Eq. (16)]. As

*β*(

*z*) > 0, we assume

*h*

_{1}> −1, where

*h*

_{1}> 0 indicates the DB, and −1 <

*h*

_{1}< 0 represents the DW. Similarly,

*h*

_{2}> 0 and −1 <

*h*

_{2}< 0 denote the NB and NW, respectively. As shown in Figs. 1(a) and 1(b), it can be seen that when the similariton and soliton pass through the DB, the pulses are amplified and form the peaks, then attenuates and recovers original shape, respectively. When the similaritons pass through the DW and the NB, the pulses’ amplitudes diminish. The pulses form the dips, then attenuates and increases their amplitudes, as shown in Figs. 1(c) and 1(d), respectively. When the similariton goes through the NW, the dynamical behavior is similar to the case in Fig. 1(a) except increasing its amplitudes after passing across the well, and we omit it. From Figs. 1(a) and 1(d), the DB and NB have different effects on the propagation of the similaritons, i.e., one forms a peak, the other a dip.

*z*=

*z*

_{02}and a peak near

*z*=

*z*

_{01}. The peak and channel originating from the NB and DB also influence the level of interaction, as shown in Figs. 3(c) and 3(d). If the locations of the barriers coincides with that of interaction, the DB in Fig. 3(c) can partially counteract the influence of interaction, and the shape in the location of interaction becomes from concave to convex. While the NB in Fig. 3(d) increases the degree of interaction and form a deeper dip in the location of interaction.

*g*

_{0}. When

*g*

_{0}= 0.055, the amplitude of the wave is almost unchanged. When

*g*

_{0}> 0.055 and

*g*

_{0}< 0.055, the amplitudes gradually increases and decreases along

*z*, respectively. This implies that the parameter

*g*

_{0}can control the amplitude of similariton after passing through the barrier.

_{2}are similar to that of Ψ

_{1}, and here we omit it.

## 5. Conclusions

*g*

_{0}can control the amplitude of similariton after passing through the barrier. We expect that these results for similariton tunneling would inaugurate a new and exciting area in the application of optical similaritons.

## Acknowledgements

## References and links

1. | A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. |

2. | L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. |

3. | G. P. Agrawal, |

4. | C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. |

5. | V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E |

6. | V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B |

7. | C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. |

8. | A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. |

9. | V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. |

10. | J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B |

11. | R. C. Yang and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express |

12. | L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 17, 2010

Revised Manuscript: July 22, 2010

Manuscript Accepted: July 24, 2010

Published: July 30, 2010

**Citation**

Chaoqing Dai, Yueyue Wang, and Jiefang Zhang, "Nonlinear similariton tunneling effect in the birefringent fiber," Opt. Express **18**, 17548-17554 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17548

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### References

- A. Hasegawa, and F. Tappet, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–170 (1973). [CrossRef]
- L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).
- C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]
- V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71, 056619 (2005). [CrossRef]
- V. I. Kruglov, and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006). [CrossRef]
- C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010). [CrossRef] [PubMed]
- A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978). [CrossRef]
- V. N. Serkin, and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001). [CrossRef]
- J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008). [CrossRef]
- R. C. Yang, and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008). [CrossRef] [PubMed]
- L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004). [CrossRef]

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