Abstract

We report the behavior of spatial optical solitons propagating through inhomogeneous waveguides with Gaussian, single triangular, and double-triangular refractive index profiles. In a given Gaussian profile, as the soliton amplitude decreases below a certain value, its behavior deviates from that of a particlelike soliton. Dependence of the swing period of a spatial soliton in a single triangular index profile on its amplitude, η, is less significant than that in a Gaussian profile. We also report the interacting behavior of two solitons propagating simultaneously through a waveguide with a double-triangular index profile. Furthermore, we present the effects of the solitons’ initial phase factors and amplitude on their behavior.

© 2009 Optical Society of America

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  1. G. P. Agrawal, Fiber Optic Communication Systems (Wiley, 2002).
    [CrossRef]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  3. E. N. Tsoy and C. M. Sterk, “Soliton dynamics in nonuniform fiber Bragg gratings,” J. Opt. Soc. Am. B 18, 1-6 (2001).
    [CrossRef]
  4. N. J. Doran and D. Wood, “Soliton processing element for all-optical switching and logic,” J. Opt. Soc. Am. B 4, 1843-1846 (1987).
    [CrossRef]
  5. L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photonics Technol. Lett. 9, 1364-1366 (1997).
    [CrossRef]
  6. X. D. Cao and D. D. Meyerhofer, “All-optical switching by means of collisions of spatial vector solitons,” Opt. Lett. 19, 1711-1713 (1994).
    [CrossRef] [PubMed]
  7. F. Garzia, C. Sibilia, and M. Bertolotti, “New phase modulation technique based on spatial soliton switching,” J. Lightwave Technol. 19, 1036-1042 (2001).
    [CrossRef]
  8. V. N. Serkin and M. Matsumoto, “Bright and dark solitary nonlinear bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159-171 (2001).
    [CrossRef]
  9. M. Ebnali-Heidari, M. K. Moravvej-Farshi, and A. Zarifkar, “Multi-channel wavelength conversion using fourth order soliton decay,” J. Lightwave Technol. 25, 2571-2578 (2007).
    [CrossRef]
  10. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
    [CrossRef] [PubMed]
  11. F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
    [CrossRef]
  12. A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264-274 (2006).
    [CrossRef]
  13. Q. Chang and E. Jia, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” J. Comput. Phys. 148, 397-415 (1999).
    [CrossRef]
  14. F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
    [CrossRef]
  15. F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
    [CrossRef]

2007 (1)

2006 (1)

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264-274 (2006).
[CrossRef]

2001 (3)

2000 (1)

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
[CrossRef]

1999 (1)

Q. Chang and E. Jia, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” J. Comput. Phys. 148, 397-415 (1999).
[CrossRef]

1997 (2)

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
[CrossRef]

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photonics Technol. Lett. 9, 1364-1366 (1997).
[CrossRef]

1994 (2)

X. D. Cao and D. D. Meyerhofer, “All-optical switching by means of collisions of spatial vector solitons,” Opt. Lett. 19, 1711-1713 (1994).
[CrossRef] [PubMed]

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

1989 (1)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

1987 (1)

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

G. P. Agrawal, Fiber Optic Communication Systems (Wiley, 2002).
[CrossRef]

Bajer, J.

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

Barthelemy, A.

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photonics Technol. Lett. 9, 1364-1366 (1997).
[CrossRef]

Bertolotti, M.

F. Garzia, C. Sibilia, and M. Bertolotti, “New phase modulation technique based on spatial soliton switching,” J. Lightwave Technol. 19, 1036-1042 (2001).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
[CrossRef]

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

Cao, X. D.

Chang, Q.

Q. Chang and E. Jia, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” J. Comput. Phys. 148, 397-415 (1999).
[CrossRef]

Doran, N. J.

Ebnali-Heidari, M.

Garzia, F.

F. Garzia, C. Sibilia, and M. Bertolotti, “New phase modulation technique based on spatial soliton switching,” J. Lightwave Technol. 19, 1036-1042 (2001).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
[CrossRef]

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

Horak, R.

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

Jia, E.

Q. Chang and E. Jia, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” J. Comput. Phys. 148, 397-415 (1999).
[CrossRef]

Lefort, L.

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photonics Technol. Lett. 9, 1364-1366 (1997).
[CrossRef]

Matsumoto, M.

V. N. Serkin and M. Matsumoto, “Bright and dark solitary nonlinear bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159-171 (2001).
[CrossRef]

Meyerhofer, D. D.

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

Moravvej-Farshi, M. K.

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

Serkin, V. N.

V. N. Serkin and M. Matsumoto, “Bright and dark solitary nonlinear bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159-171 (2001).
[CrossRef]

Sibilia, C.

F. Garzia, C. Sibilia, and M. Bertolotti, “New phase modulation technique based on spatial soliton switching,” J. Lightwave Technol. 19, 1036-1042 (2001).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
[CrossRef]

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

Sterk, C. M.

Suryanto, A.

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264-274 (2006).
[CrossRef]

Tsoy, E. N.

van Groesen, E.

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264-274 (2006).
[CrossRef]

Wood, D.

Zarifkar, A.

IEEE Photonics Technol. Lett. (1)

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photonics Technol. Lett. 9, 1364-1366 (1997).
[CrossRef]

J. Comput. Phys. (1)

Q. Chang and E. Jia, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” J. Comput. Phys. 148, 397-415 (1999).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (4)

V. N. Serkin and M. Matsumoto, “Bright and dark solitary nonlinear bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159-171 (2001).
[CrossRef]

F. Garzia, C. Sibilia, M. Bertolotti, R. Horak, and J. Bajer, “Phase properties of a two-soliton system in a nonlinear planar waveguide,” Opt. Commun. 108, 47-54 (1994).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of Spatial Soliton,” Opt. Commun. 139, 193-198 (1997).
[CrossRef]

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264-274 (2006).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781-798 (2000).
[CrossRef]

Phys. Rev. A (1)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Fiber Optic Communication Systems (Wiley, 2002).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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Figures (12)

Fig. 1
Fig. 1

Swing effect of a spatial soliton propagating through a single Gaussian waveguide with Δ n 0 = 0.05 , and b = 0.1 for input pulse parameters (a)  η = 1 , X 0 = 3.5 , and β = 0 ; (b)  η = 2 , X 0 = 3.5 , and β = 0 . The profile shown on the top belongs to the equivalent perturbing potential, V, induced by the Gaussian index profile.

Fig. 2
Fig. 2

(a) Transverse acceleration and (b) potential energy U for solitons of various amplitudes η as functions of the average transverse coordinate propagating through a Gaussian index profile with Δ n 0 = 0.05 and b = 0.1 .

Fig. 3
Fig. 3

3D view of a spatial soliton with amplitude η = 0.75 and initial average position X 0 = 3.5 propagating through a Gaussian index profile with Δ n 0 = 0.05 and b = 0.1 . Profile of V is shown on the top.

Fig. 4
Fig. 4

Swing effect of a spatial soliton propagating through a single triangular waveguide with Δ n 0 = 0.1 and b = 5 for input pulse parameters (a)  η = 1 , X 0 = 3.5 , and β = 0 ; (b)  η = 2 , X 0 = 3.5 , and β = 0 . Profile of the induced equivalent potential for the triangular index profile is illustrated on the top.

Fig. 5
Fig. 5

(a) Transverse acceleration a and (b) potential energy U for solitons of various amplitudes η, as functions of the average transverse coordinate, propagating in a triangular index profile with Δ n 0 = 0.1 and b = 5 .

Fig. 6
Fig. 6

Swing effect of a spatial soliton propagating through a double-triangular waveguide of parameters Δ n 0 = 0.1 , b 1 = 0 , b 2 = 1.25 , and b 3 = 2.5 , and input pulses of (a)  η 1 = η 2 = 3 and (b)  η 1 = η 2 = 4 . Other pulse parameters for both pulses are fixed: X 01 = 3.5 , X 02 = 3.5 , and β 1 = β 2 = 0 . The induced equivalent potential, V, for a two-triangular index profile is illustrated on the top.

Fig. 7
Fig. 7

(a) Transverse acceleration a and (b) potential energy U for solitons of various amplitudes η, as functions of the average transverse coordinate, propagating in the index profile of Fig. 6.

Fig. 8
Fig. 8

Effect of the input pulse phase on the swing behavior of two solitons, with initial parameters η 1 = η 2 = 4 , X 01 = 3.5 , X 02 = 3.5 , β 1 = 2 , and β 2 = 0 , propagating through a waveguide with the index profile of Fig. 6.

Fig. 9
Fig. 9

3D view of soliton walk-out from a double-triangular waveguide of Fig. 6. Parameters for the initial solitons are chosen to be η 1 = η 2 = 4 , β 1 = 2.5 , β 2 = 0 , X 01 = 3.5 , and X 02 = 3.5 .

Fig. 10
Fig. 10

3D view of two solitons with different amplitudes: (a)  η 1 = 4 and η 2 = 2 , and (b)  η 1 = 4 and η 2 = 3 interacting in the double-triangular waveguide of Fig. 6. β 1 = 2.5 > β c , | β 2 | = 0 < β c , X 01 = 3.5 , and X 02 = 3.5 .

Fig. 11
Fig. 11

3D view of two solitons with initial phases (a)  β 1 = 3.5 > β c and | β 2 | = 2 < β c and (b)  β 1 = 3.5 > β c and | β 2 | = 2.5 < β c interacting in a double-triangular waveguide of Fig. 6. η 1 = η 2 = 4 , X 01 = 3.5 , and X 02 = 3.5 .

Fig. 12
Fig. 12

3D view of two solitons with equal initial amplitudes η 1 = η 2 = 4 and phases (a)  β 1 = | β 2 | = 2.5 and (b)  β 1 = | β 2 | = 2.75 interacting in a double-triangular waveguide of Fig. 6. X 01 = 3.5 and X 02 = 3.5 .

Equations (21)

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E y ( x , z ) = A ( x , z ) exp ( i k z i ω t ) + c . c . ,
i k A z + 1 2 k 2 2 A x 2 + n 2 n 0 | A | 2 A 0 ,
n ( x ) = n 0 [ 1 + Δ n ( x ) ] ,
i B / Z + 2 B / 2 X 2 + | B | 2 B = V B ,
a ( X ¯ ) = d 2 X ¯ / d Z 2 = U ( X ¯ ) / X ¯ = V ( X ) / X · | B | 2 d X / | B | 2 d X ,
X ¯ ( Z ) = X | B | 2 d X / | B | 2 d X .
a M W 1 0 W a ( X ¯ ) d X ¯ .
Z 0 = 2 { 2 W / a M } 1 / 2 .
B ( X , Z = 0 ) = η sech [ η ( X X 0 ) ] exp ( i β X ) ,
B ( X , Z ) = η sech { η [ X X ¯ ( Z ) ] } exp { i [ υ ( Z ) X + σ ( Z ) ] } ,
υ ( Z ) = d X ¯ ( Z ) d Z ,
d σ ( Z ) d Z = ( η 2 υ 2 ( Z ) ) / 2.
V ( X ) = Δ n 0 exp ( b X 2 ) .
a ( X ¯ ) η b Δ n 0 X ¯ ( 1 + 2 b X ¯ 2 ) exp ( b X ¯ 2 ) .
U ( X ¯ ) η Δ n 0 2 [ 3 + 2 b X ¯ 2 ] exp ( b X ¯ 2 ) .
a M η Δ n 0 2 W [ ( 3 + 2 b W 2 ) exp ( b W 2 ) 3 ] .
V ( X ) = { 0 X < b Δ n 0 ( 1 | X | b ) b X < b 0 X > b .
a ( X ¯ ) = Δ n 0 η b ( tanh ( η ( b X ¯ ) + 2 tanh ( η X ¯ ) tanh ( η ( b + X ¯ ) ) ,
U ( X ¯ ) = Δ n 0 η 2 b ( log ( cosh ( η ( b X ¯ ) ) ) 2 log ( cosh ( η X ¯ ) ) + log ( cosh ( η ( b + X ¯ ) ) ) .
V ( X ) = { 0 X < b 3 Δ n 0 { 1 + X + b 2 b 3 b 2 } b 3 X < b 2 Δ n 0 { 1 X + b 2 b 2 b 1 } b 2 X < b 1 0 b 1 X < b 1 Δ n 0 { 1 + X b 2 b 2 b 1 } b 1 X < b 2 Δ n 0 { 1 X b 2 b 3 b 2 } b 2 X < b 3 0 X b 3 ,
B ( X , Z = 0 ) = η 1 sech [ η 1 ( X + X 01 ) ] exp ( i β 1 X ) + η 2 sech [ η 2 ( X X 02 ) ] exp ( i β 2 X ) .

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