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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 16302–16307
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Soliton fission management by dispersion oscillating fiber

Alexej A. Sysoliatin, Andrew K. Senatorov, Andrey I. Konyukhov, Leonid A. Melnikov, and Vladimir A. Stasyuk  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 16302-16307 (2007)
http://dx.doi.org/10.1364/OE.15.016302


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Abstract

We report the experimental observation of the fission of picosecond solitons in a fiber with sine-wave variation of the core diameter along the longitudinal direction of propagation. The experimental pulse dynamics is reproduced by numerical simulations. The fission of high-intensity solitons caused by both the variation of the fiber dispersion and stimulated Raman scattering is demonstrated. The number of output pulses and their frequencies can be managed by periodical modulation of the fiber dispersion even under the strong effect of the Raman scattering.

© 2007 Optical Society of America

1. Introduction

The control of the optical solitons has become important with development of solitonic highspeed optical communication systems and modelocked fiber lasers. It is well known that high-order solitons are unstable against perturbations that lead to the soliton fission. The fission of high-order soliton can be stimulated by self-steepening [1

1. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

], Raman scattering [1

1. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

, 2

2. K. Tai, A. Hasegawa, and N. Bekki“Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988). [CrossRef] [PubMed]

] and cubic dispersion [3

3. P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986). [CrossRef] [PubMed]

]. However, it is difficult to manage the fission process using these effects. For controlling soliton dynamics a variation of the chromatic dispersion along transmission line (dispersion management) was proposed.

First kind of dispersion management implies that the dispersion coefficient periodically alternates between positive and negative values. The dispersion-managed soliton [4

4. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. , 32, 54–55 (1996). [CrossRef]

], split-step soliton [5

5. R. Driben and B. A. Malomed, “Split-step solitons in long fiber links”, Opt. Commun. 185, 439–456 (2000). [CrossRef]

] and stationary rescaled pulse [6

6. T. Inoue, H. Tobioka, and S. Namiki, “Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations” Phys. Rev. E 72, 025601(R) (2005). [CrossRef]

] have been discovered, and they have been analyzed aimed the soliton stability. The soliton can be stable even in transmission line with a not quite periodical change of a fiber dispersion [7

7. M. Böhm and F. Mitschke, “Soliton propagation in a dispersion map with deviation from periodicity” Appl. Phys. B 81, 983–987 (2005). [CrossRef]

]. Shape, power and dynamics of such solitons are somewhat different from their counterparts in constant-dispersion fibers.

For the second kind of dispersion management the sign of the dispersion is preserved. Such kind of the management can be used for the splitting of high-order soliton into few sech-shaped fundamental solitons. A stepwise change of dispersion, a localized loss element or filter will generate pairs of fundamental solitons with shifted central wavelengths [8

8. K. Lee and J. Buck, “Wavelength conversion through higher-order soliton splitting initiated by localized channelperturbations”, J. Opt. Soc. Am. B 20, 514–519 (2003). [CrossRef]

]. In segmented fiber, the multiple breakups of each soliton generate Cantor set fractals [9

9. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons”, Phys. Rev. Lett. 84, 1902–1905 (2000). [CrossRef] [PubMed]

]. Splicing losses and transient processes that arise due to stepwise change of the dispersion restrict the application of multisegmented fibers for soliton splitting. Disadvantages of multisegmented fibers are surmountable in a fiber with longitudinal sine-wave modulation of the core diameter. The soliton can be split both due to the variation of the fiber dispersion [10

10. B.A. Malomed, D.F. Parker, and N.F. Smyth “Resonant shape oscillations and decay of a soliton in a periodicallyinhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993). [CrossRef]

, 11

11. R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–195 (1995). [CrossRef]

] and nonlinearity [12

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. 66, 161–164 (1991). [CrossRef] [PubMed]

, 13

13. H. Sakaguchi and B.A. Malomed “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004). [CrossRef]

]. The splitting of vector soliton can be induced by periodical variation of the fiber birefringence [14

14. B.A. Malomed and N.F. Smyth “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994). [CrossRef]

]. The periodic perturbation in nonlinear medium induces the generation of dispersive waves and/or the splitting of soliton [12

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. 66, 161–164 (1991). [CrossRef] [PubMed]

]. When the period of the modulation of the fiber dispersion or nonlinearity approaches the soliton period z 0 [15

15. G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).

], the soliton splits into pulses propagating with different group velocities [11

11. R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–195 (1995). [CrossRef]

, 12

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. 66, 161–164 (1991). [CrossRef] [PubMed]

]. The nonlinear Schrödinger equation with variable dispersion and nonlinear coefficients with special choice of the variations has analytical solution in a form of nonautonomous solitons [16

16. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva “Nonautonomous solitons in external potentials” Phys. Rev. Lett. 98, 074102 (2007). [CrossRef] [PubMed]

].

In spite of intensive theoretical studies the soliton fission in the fibers with variable but signpreserving dispersion was not yet realized in accordance with our knowledge. In present work we demonstrate experimentally the soliton fission in single-mode highly-nonlinear dispersion oscillating fiber. The management of the soliton fission by the change of modulation parameters is discussed.

2. Soliton fission in dispersion oscillating fiber

For analysis of the soliton fission, the generalized nonlinear Schrödinger equation was solved.

Az+α2A(z,t)=iβ2(z)22At2+β3(z)63At3+i(PNL+i1πv0PNLt),
(1)

where A(z,t) is the complex pulse envelope, α is the loss coefficient, and ν 0 is the carrier frequency of the pulse. Functions β 2(z) and β 3(z) describe dispersion varying along the fiber length. Nonlinear media polarization includes the Kerr effect and delayed Raman scattering PNL(z,t)=γK(z)|A|2 A+γR(z)QA(z,t), where γK(z) and γR(z) are nonlinear coefficients. The Raman delayed response Q(z,t) is approximated by damping oscillations [17

17. V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley “Simplified field wave equations for the nonlinear propagation of extremely short light pulses” Phys. Rev. A 66, 013811 (2002). [CrossRef]

]: ( 2 Q/∂t 2)+2T -1 2 (∂Q/∂ t)+Ω2 Q(z,t)=Ω2|A(z,t)|2, where T 2=32fs, Ω(2π)-1=13.1THz. The equation (1) was solved using standard split-step Fourier algorithm [15

15. G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).

]. Simulations were carried out with hyperbolic secant input pulses having intensity full-width at half maximum duration T FWHM=2.05ps.

The single-mode dispersion oscillating fiber (DOF) was drawn in the Fiber Optics Research Center (Moscow, Russia) from the preform (Fig. 1(a)) by varying the diameter slightly during the draw in accordance with prearranged law. The variation of outer diameter of the fiber is described by sine-wave function d(z)=d 0(1+dm sin(2πz/zm)), where d 0=133µm, zm=0.16km is the modulation period, d m=0.03 is the modulation depth. The linear loss at 1550 nm is 0.69 dB/km that corresponds to α=0.159km-1 in eq.(1). Dispersion measured for three different fibers drawn from the same preform is shown in Fig. 1(b). Using measurements shown in Fig. 1 the coefficients of eq.(1) were defined as follows:

β2,3(z)=β2,3[1+β2,3(m)sin(2πzzm+φm)],
(2)
γK,R(z)=γK,R[1γmsin(2πzzm+φm)],
(3)

where 〈β 2〉=-12.76ps2km-1, 〈β3〉=0.0761ps3km-1, 〈γK〉=8.2W-1km-1, 〈γR〉=1.8W-1km-1, β 2(m)=0.02, β 3(m)=0.095, γm=0.028, φm is the modulation phase. Fourthorder dispersion coefficient β 4 is around -1.4×10-4 ps4km-1 at λ=1550nm. The magnitude of β 4 is not sufficiently high to play any significant role in considered regimes.

Fig. 1. (a) Distribution of the refractive index in cross-section of the fiber preform, Δn is the difference between refractive index of the glass and refractive index of the silica. (b) The dispersion (D) measured for three fibers with the fixed outer diameter: 128µm, 133µm and 138µm.

Experimental setup is shown in Fig. 2. For our experiments bandwidth-limited pulses with a pulse width in a range 1.6⋯2.1 ps (FWHM) are generated by actively mode-locked fiber laser “Pritel UOC” at a repetition rate of 10 GHz. The central wavelength of laser pulses is 1550.6 nm. EDFA increase the average power of the pulse train up to 350 mW. After propagation in 0.8 km length of DOF pulses were analyzed by autocorrelator “Femtochrome” with the scan range exceeding 100 ps, wide-bandwidth oscilloscope “Agilent Infinium DCA 86100A” and “Ando” optical spectrum analyzer with a resolution bandwidth of 0.002 nm.

For initial pulse width T FWHM=2.05ps the soliton period [15

15. G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).

] is z 0=0.16π |〈β 2〉|-1 T 2 FWHM=0.166km. When zm is near z 0, the soliton splitting is possible at a short propagation distance (z<0.8km). The soliton with N=1.72 can be split into two pulses (Fig. 3). At the certain range of zm the soliton splitting is not clearly discernible. That is why the curves in Fig. 3(a) have discontinuities. The change between stable and unstable propagation regimes of two bound solitons with variation of modulation parameters was predicted in [12

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. 66, 161–164 (1991). [CrossRef] [PubMed]

]. Our experiments were performed for two values ϕm=0 and ϕm=π. For the last the soliton splitting have higher efficiency. That is agree with simulations (Fig. 3(a)). Due to the Raman scattering the maximum separation between pulses ΔT is achieved at zm that not coincide with z 0=0.166km (Fig. 3(a)). However, with zmz 0 only one modulation period of DOF is sufficient for the soliton splitting (Fig. 3(b)). In Fig. 3(b) the normalized intensities of output pulses are 0.9 and 1.0. Such an asymmetry is caused mainly by the effect of the Raman scattering. For regime shown in Fig. 3 the self-steepening operator (∂PNL/∂t) in (1) can be neglected. While for solitons of higher order the self-steepening leads to considerable changes in amplitudes and group velocities of the pulses.

Fig. 2. Experimental setup: Pritel UOC, picosecond pulse source; EDFA, Er-doped fiber amplifier; DOF, dispersion oscillating fiber; AC, autocorrelator; OSA, optical spectrum analyzer, OSCI, wide-bandwidth oscilloscope.

The splitting of the soliton was detected using both the autocorrelation trace (Fig. 3(c)) and output spectrum (Fig. 3(d)). The intensity spectrum of initial pulses has sech 2 envelope. After the splitting the oscillations appear in the spectrum (Fig. 3(d)). Such a structure of output spectrum arises due to the interference between two pulses with shifted carrier frequencies.

Fig. 3. Soliton splitting into the pulse pair. (a) Temporal distance ΔT between peaks of the pulses at z=0.8km. Simulations were performed for φm=π (red curve) and φm=0 (blue curve). (b) Contour plot of pulse trajectory (bottom) and intensity pulse shape at z=0.8km (top). (c) Autocorrelation traces for output pulses; (d) Output spectrum, dashed curve on the top shows the spectrum of input pulse; In simulations the soliton order N=1.72 was used. T FWHM=2.05 ps. For Fig. (b),(c) and (d) φm=π, zm=0.16km. Plots are normalized.

In Fig. 4 the fission of third-order soliton under the simultaneous effect of the stimulated Raman scattering (SRS) and periodical modulation of the fiber dispersion is shown. The input pulse was split into three fundamental solitons (Fig. 4(a)). Due to the Raman scattering the spectrum is broadened mainly to the direction of longer wavelengths (Fig. 4(b)). The spectral component at 1555nm <λ<1560nm (Fig. 4(b)) corresponds to the pulse with the highest peak intensity (Fig. 4(a), pulse No.3). Spectra of other two pulses are overlapped in the range 1545nm<λ<1555nm. The splitting into three pulses was detected experimentally using the oscilloscope record (Fig. 4(c)).

Fig. 4. Splitting of high-intensity pulse. (a) Contour plot of pulse trajectory (bottom) and intensity pulse shape at z=0.8km (top); (b) output spectrum; (c) oscilloscope record. N=3.02, φm=π, zm=0.16km Other parameters are the same as in Fig. 2.

3. Soliton fission management

The soliton fission can be managed by the change of the parameters of the modulation of the fiber diameter. Using numerical simulations we can study the effect of the change of the modulation depth dm and modulation period zm on the soliton fission. The constants β 2(m), β 3(m) and γm in eq. (2),(3) are practically linearly dependent on the modulation depth dm for considered fiber diameters.

When N=1.72, the input pulse splits into two pulses (Fig. 3(b)) with different carrier frequencies. For dm=0.03 (Fig. 3) the difference between carrier frequencies of output pulses is Δν=0.214THz. By varying the magnitude of the perturbation, one can vary the difference Δν and temporal separation of output pulses accordingly. From numerical simulations the differences Δν=0.276THz (dm=0.039) and Δν=0.103THz (dm=0.021) were obtained for zm=0.160km.

With increase of peak power of input pulse (N=2.33) the number of output pulses becomes dependent on the modulation period zm (Fig. 5). Without modulation (zm=∞) SRS leads to the soliton fission into two pulses (Fig. 5(a)). Periodical modulation of the fiber dispersion with modulation period (zm=0.16)km) allows to obtain three output pulses (Fig. 5(b)). DOF with reduced value of the modulation period (zm=0.08km) splits the initial pulse into two pulses with nearly identical amplitudes (Fig. 5(c)).

In the fiber without modulation SRS leads to the fission of third-order soliton (N=3.02) into three pulses (Fig. 6(a), red curve). The modulation of the fiber diameter with zm=0.16km only increase the temporal separation between these pulses (Fig. 4(a)). At the modulation period zm=0.08km the pulse dynamics becomes quite different. The pulse splits into several lowintensity pulses (Fig. 6(b), red curve). As a result the output spectrum does not reach the infrared region λ>1557nm (Fig. 6(b), blue curve). This example demonstrates that the DOF can be used for the management of the fission of high-order soliton even under the strong effect of SRS.

4. Conclusion

The soliton fission initiated by DOF was demonstrated experimentally. Good qualitative agreement between numerical modelling and experiment was obtained. The model includes the Raman self-frequency shift, third-order dispersion, and nonlinear dispersion as well as modulation of fiber parameters. For high-order solitons (N>2) the stimulated Raman scattering becomes important for the fission of 2-picosecond solitons. Change of the modulation period and modulation depth of the DOF allows to manage the number of output pulses and their frequencies even in presence of the strong effect of the Raman scattering.

Fig. 5. Effect of the modulation of the fiber diameter on the soliton fission in presence of the stimulated Raman scattering. (a) Fiber diameter is constant (zm=∞); (b) zm=0.16km; (c) zm=0.08km; Top inserts (red curves) show output pulses (z=0.8km). N=2.33, φm=π. Other parameters are the same as in Fig. 3.
Fig. 6. Fission of third-order soliton under the strong effect of the stimulated Raman scattering. Red curves show output pulse shapes. Blue curves show output spectra. Propagation distance is z=0.8km. (a) Soliton fission due to the Raman scattering. Fiber diameter is constant (zm=∞); (b) Simultaneous effect of dispersion oscillation (zm=0.08km) and Raman scattering on the soliton fission. N=3.02, φm=π. Pulses and spectra are normalized to the peak power of input radiation. Other parameters are the same as in Fig. 3.

Presented paper gives the proof of concept of the soliton splitting by DOF. This work can be extended in various directions. It would be interesting to investigate the soliton splitting in fibers with complex variation of core diameter. For practical applications a detailed analysis of fiber parameters suitable for symmetrical splitting of second order and high-order solitons is required. We hope the results of present work may be of interest in the field of optical timedivision multiplexed networks and nonlinear optical signal processing.

Acknowledgments

This work was supported in part by BRHE REC-006 grant. Authors are grateful to Prof. E.M. Dianov and Dr. K.V. Reddy for fruitful discussions and contribution to this work.

References and links

1.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

2.

K. Tai, A. Hasegawa, and N. Bekki“Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988). [CrossRef] [PubMed]

3.

P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986). [CrossRef] [PubMed]

4.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. , 32, 54–55 (1996). [CrossRef]

5.

R. Driben and B. A. Malomed, “Split-step solitons in long fiber links”, Opt. Commun. 185, 439–456 (2000). [CrossRef]

6.

T. Inoue, H. Tobioka, and S. Namiki, “Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations” Phys. Rev. E 72, 025601(R) (2005). [CrossRef]

7.

M. Böhm and F. Mitschke, “Soliton propagation in a dispersion map with deviation from periodicity” Appl. Phys. B 81, 983–987 (2005). [CrossRef]

8.

K. Lee and J. Buck, “Wavelength conversion through higher-order soliton splitting initiated by localized channelperturbations”, J. Opt. Soc. Am. B 20, 514–519 (2003). [CrossRef]

9.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons”, Phys. Rev. Lett. 84, 1902–1905 (2000). [CrossRef] [PubMed]

10.

B.A. Malomed, D.F. Parker, and N.F. Smyth “Resonant shape oscillations and decay of a soliton in a periodicallyinhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993). [CrossRef]

11.

R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–195 (1995). [CrossRef]

12.

A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. 66, 161–164 (1991). [CrossRef] [PubMed]

13.

H. Sakaguchi and B.A. Malomed “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004). [CrossRef]

14.

B.A. Malomed and N.F. Smyth “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994). [CrossRef]

15.

G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).

16.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva “Nonautonomous solitons in external potentials” Phys. Rev. Lett. 98, 074102 (2007). [CrossRef] [PubMed]

17.

V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley “Simplified field wave equations for the nonlinear propagation of extremely short light pulses” Phys. Rev. A 66, 013811 (2002). [CrossRef]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 11, 2007
Revised Manuscript: November 15, 2007
Manuscript Accepted: November 16, 2007
Published: November 26, 2007

Citation
Alexej A. Sysoliatin, Andrew K. Senatorov, Andrey I. Konyukhov, Leonid A. Melnikov, and Vladimir A. Stasyuk, "Soliton fission management by dispersion oscillating fiber," Opt. Express 15, 16302-16307 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16302


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References

  1. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, "Decay of optical solitons," JETP Lett. 42, 87-91 (1985).
  2. K. Tai, A. Hasegawa, N. Bekki "Fission of optical solitons induced by stimulated Raman effect," Opt. Lett. 13, 392-394 (1988). [CrossRef] [PubMed]
  3. P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, "Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers," Opt. Lett. 11, 464-466 (1986). [CrossRef] [PubMed]
  4. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, "Enhanced power solitons in optical fibers with periodic dispersion management," Electron. Lett.,  32, 54-55 (1996). [CrossRef]
  5. R. Driben and B. A. Malomed, "Split-step solitons in long fiber links", Opt. Commun. 185, 439-456 (2000). [CrossRef]
  6. T. Inoue, H. Tobioka, S. Namiki, "Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations" Phys. Rev. E 72, 025601(R) (2005). [CrossRef]
  7. M. Böhm, F. Mitschke, "Soliton propagation in a dispersion map with deviation from periodicity" Appl. Phys. B 81, 983-987 (2005). [CrossRef]
  8. K. Lee, J. Buck, "Wavelength conversion through higher-order soliton splitting initiated by localized channel perturbations", J. Opt. Soc. Am. B 20, 514-519 (2003). [CrossRef]
  9. S. Sears, M. Soljacic, M. Segev, D. Krylov, K. Bergman, "Cantor set fractals from solitons", Phys. Rev. Lett. 84, 1902-1905 (2000). [CrossRef] [PubMed]
  10. B.A. Malomed, D.F. Parker, N.F. Smyth "Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber," Phys. Rev. E 48, 1418-1425 (1993). [CrossRef]
  11. R.G. Bauer, L.A. Melnikov "Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter," Opt. Commun. 115, 190-195 (1995). [CrossRef]
  12. A. Hasegawa, Y. Kodama, "Guiding center solitons," Phys. Rev. Lett. 66, 161-164 (1991). [CrossRef] [PubMed]
  13. H. Sakaguchi, B.A. Malomed "Resonant nonlinearity management for nonlinear Schr¨odinger solitons," Phys. Rev. E 70, 066613 (2004). [CrossRef]
  14. B.A. Malomed, N.F. Smyth "Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber," Phys. Rev. E 50, 1535-1542 (1994). [CrossRef]
  15. G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).
  16. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous solitons in external potentials" Phys. Rev. Lett. 98, 074102 (2007). [CrossRef] [PubMed]
  17. V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley "Simplified field wave equations for the nonlinear propagation of extremely short light pulses" Phys. Rev. A 66, 013811 (2002). [CrossRef]

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