Abstract

Soliton propagation in slow-light states of nonuniform high-index photonic crystal fibers (PCFs) is studied numerically by a recently developed time-propagating 1+1D equation. It is demonstrated that very slow solitons can be highly stable against even short-period roughness. Soliton trapping by longitudinal inhomogeneities is also found as the soliton velocity decreases due to Raman scattering. Practical limitations and opportunities based on the simulation results are briefly discussed.

© 2011 Optical Society of America

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  1. T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
    [CrossRef]
  2. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
    [CrossRef]
  3. J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
    [CrossRef]
  4. A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero—group-velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. S. M. J. Kelly, “Characteristic side-band instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807(1992).
    [CrossRef]
  9. Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
    [CrossRef]
  13. M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81245321 (2010).
    [CrossRef]
  14. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  15. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).
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2011 (1)

2010 (4)

2009 (2)

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero—group-velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009).
[CrossRef] [PubMed]

2008 (2)

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

2007 (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

2006 (1)

2005 (1)

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

2004 (1)

K. Stoychev, M. Primatarowa, and R. Kamburova, “Resonant scattering of nonlinear Schrodinger solitons from potential wells,” Phys. Rev. E 70, 066622 (2004).
[CrossRef]

2002 (1)

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

1992 (2)

S. M. J. Kelly, “Characteristic side-band instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807(1992).
[CrossRef]

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger-equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

1991 (1)

Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

Allegra, N.

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

Anghel-Vasilescu, P.

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

Beggs, D. M.

Dulashko, Y.

Fei, Z.

Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Fink, Y.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Ginovart, F.

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

Gordon, J. P.

Hughes, S.

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81245321 (2010).
[CrossRef]

Hugonin, J. P.

Ibanescu, M.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Jacobs, S.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Joannopoulos, J.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Joannopoulos, J. D.

Johnson, S.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Johnson, S. G.

Kamburova, R.

K. Stoychev, M. Primatarowa, and R. Kamburova, “Resonant scattering of nonlinear Schrodinger solitons from potential wells,” Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Karalis, A.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Kelly, S. M. J.

S. M. J. Kelly, “Characteristic side-band instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807(1992).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Krauss, T. F.

Kuipers, L.

Lægsgaard, J.

Lalanne, P.

Leon, J.

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

Mazoyer, S.

Melloni, A.

Morichetti, F.

Mortensen, N. A.

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

O’Faolain, L.

Oskooi, A. F.

Patterson, M.

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81245321 (2010).
[CrossRef]

Pedersen, J. G.

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Povinelli, M.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Primatarowa, M.

K. Stoychev, M. Primatarowa, and R. Kamburova, “Resonant scattering of nonlinear Schrodinger solitons from potential wells,” Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Schulz, S. A.

Skorobogatiy, M.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Skryabin, D. V.

Soljacic, M.

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Spasenovic, M.

Stoychev, K.

K. Stoychev, M. Primatarowa, and R. Kamburova, “Resonant scattering of nonlinear Schrodinger solitons from potential wells,” Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Sumetsky, M.

Vázquez, L.

Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Weisberg, O.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

White, T. P.

Xiao, S.

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Yulin, A. V.

Appl. Phys. B (1)

S. Johnson, M. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Electron. Lett. (1)

S. M. J. Kelly, “Characteristic side-band instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807(1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. A: Math. Theor. (1)

J. Leon, P. Anghel-Vasilescu, F. Ginovart, and N. Allegra, “Scattering of slow-light gap solitons with charges in a two-level medium,” J. Phys. A: Math. Theor. 42, 055101 (2009).
[CrossRef]

Nat. Photon. (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (2)

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81245321 (2010).
[CrossRef]

Phys. Rev. E (2)

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

K. Stoychev, M. Primatarowa, and R. Kamburova, “Resonant scattering of nonlinear Schrodinger solitons from potential wells,” Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

Y. S. Kivshar, Z. Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

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Figures (6)

Fig. 1
Fig. 1

Fraction of pulse energy in main peak after 500 ps of propagation for various periods of a cosine Δ ( z ) . (a) Results for a Δ ( z ) magnitude of 10 4 , (b) results for a magnitude of 10 3 .

Fig. 2
Fig. 2

Peak intensity of soliton as a function of propagation distance for a cosine Δ ( z ) of magnitude 10 3 . The initial soliton velocity is 0.044 c .

Fig. 3
Fig. 3

Peak intensity of soliton as a function of propagation distance for a cosine Δ ( z ) of magnitude 4 · 10 3 and L p = 2.5 μm . Inset shows the pulse shape after 500 ps propagation. The initial soliton velocity is 0.044 c and initial energy density is 15 pJ / μm .

Fig. 4
Fig. 4

(a) Two examples of random Δ ( z ) distributions with Gaussian disorder. The RMS value of Δ ( z ) is 10 3 . (b) Fraction of pulse energy in main peak after 500 ps of propagation for various values of L c . For each L c value, four different realizations of the random structure were investigated.

Fig. 5
Fig. 5

Ratio of energy in soliton main peak E main to total pulse energy E tot versus propagation time for solitons with different initial velocities propagating in a Gaussian disorder field with L c = 0.5 μm and RMS width 10 4 . Other parameters as in Fig. 4.

Fig. 6
Fig. 6

(a) Soliton position as a function of time, in a Gaussian Δ ( z ) with RMS = 10 3 . Inset shows the spatial intensity of the final soliton on a logarithmic scale. (b) Soliton velocity plotted together with Δ ( z ) , scaled to facilitate comparison.

Equations (38)

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× E = μ 0 H t ,
× H = ε 0 ε ( r ) E t + δ P t .
δ P = ε 0 δ ε ( r , z ) E + P NL .
H ( r , t ) = 1 2 π m d β [ ( A m ( t , β ) + δ m ( t , β ) ) h m ( r , β ) e i ( ω m ( β ) t β z ) + ( A m * ( t , β ) + δ m * ( t , β ) ) h m * ( r , β ) e i ( ω m ( β ) t + β z ) ] ,
E ( r , t ) = 1 2 π m d β [ A m ( t , β ) e m ( r , β ) e i ( ω m ( β ) t β z ) + A m * ( t , β ) e m * ( r , β ) e i ( ω m ( β ) t + β z ) ] ,
h m ( r , t ; β ) = h m ( r , β ) e i ( ω m ( β ) t β z ) , e m ( r , t ; β ) = e m ( r , β ) e i ( ω m ( β ) t β z ) ,
× e m ( r , t ; β ) = μ 0 h m ( r , t ; β ) t , × h m ( r , t ; β ) = ε 0 ε ( r ) e m ( r , t ; β ) t ,
ε 0 d r ε ( r ) e m * ( r , β ) · e n ( r , β ) = μ 0 d r h m * ( r , β ) · h n ( r , β ) = 1 2 δ mn .
A m ( t , β ) t = i ω m ( β ) δ m ( t , β ) δ m ( t , β ) t ,
A m ( t , β ) t i ω m ( β ) δ m ( t , β ) δ m ( t , β ) i ω m ( β ) A m ( t , β ) t .
δ m ( t , β ) t = i ω ( β ) 2 A m ( t , β ) t 2 .
A m ( t , β ) t = 1 2 π d r e m * ( r , t ; β ) · δ P + t .
d r e m * ( r , t ; β ) · ε 0 δ ε ( r , z ) E ( r , t ) t 1 2 π d z e i ( β z ω m ( β ) t ) 1 2 π d β 1 i ω n ( β 1 ) e i ( ω n ( β 1 ) t β 1 z ) A ( t , β 1 ) d r ε 0 δ ε ( r , z ) e m * ( r , β ) · e n ( r , β 1 ) e i ω m ( β ) t d z e i β z B n ( t , z ) Δ mn ( z ) .
Δ mn ( z ) = d r ε 0 δ ε ( r , z ) e m * ( r , β = 0 ) · e n ( r , β = 0 ) .
B m ( t , z ) = 1 2 π d β e i β z i ω m ( β ) A ˜ m ( t , β ) = 1 2 π d β e i β z i ω m ( β ) A m ( t , β ) e i ω m ( β ) t
e n ( r , β ) e n ( r , β = 0 ) + β e n ( 1 ) ( r )
d r e m * ( r , t ; β ) · ε 0 δ ε ( r , z ) E ( r , t ) t e i ω m ( β ) t d z e i β z [ B n ( t , z ) ( Δ mn ( z ) β Δ nm * ( z ) ) Δ mn ( z ) B n ( t , z ) z ] ,
Δ mn ( z ) = d r ε 0 δ ε ( r , z ) e m * ( r , β = 0 ) · e n ( 1 ) ( r ) .
A ( t , β ) t = e i ω ( β ) t 1 2 π d z e i β z [ N 2 A eff { ( 1 f R ) ( 2 A ˜ ( t , z ) 2 B ( t , z ) + A ˜ 2 ( t , z ) B * ( t , z ) ) + f R ( A ˜ ( t , z ) G ( t , z ) + B ( t , z ) F ( t , z ) ) } + B ( t , z ) Δ ( z ) ] .
F ( t , z ) = t d t 1 R ( t t 1 ) A ˜ ( t 1 , z ) 2 , G ( t , z ) = t d t 1 R ( t t 1 ) A ˜ ( t 1 , z ) 2 .
N 2 = 3 χ s ( 3 ) 4 ε 0 ε m 2 ; χ s ( 3 ) = χ K x x x x ( 3 ) + 2 3 χ R x x x x ( 3 ) ; f R = 2 χ R x x x x ( 3 ) 3 χ s ( 3 ) .
R ( t ) = τ 1 2 + τ 2 2 τ 1 τ 2 2 sin ( t τ 1 ) e t τ 2 .
i A ˜ ( t , z ) t = ω 0 [ 1 Δ ( z ) ] A ˜ ( t , z ) ω 2 2 2 A ˜ ( t , z ) z 2 ,
δ ω ω = d r ε 0 δ ε ( r , z ) e ( r , β = 0 ) 2 = Δ ( z )
δ ω ω = Δ ( z ) > ω ( β ) ω 0 ω = ω 2 2 β 2 ω 0 + ω 2 2 β 2 ω 2 2 ω 0 β 2 .
ω ( β ) = ω 0 + ω 2 β 2 , ω 0 = 2 π c λ 0 ,
A ( t , β ) = A s ( t , β ) + δ A ( t , β ) ,
A s ( t , β ) = 1 2 π d z exp i β z A s ( t , z ) ,
A s ( t , z ) = ξ 0 sech ( z v g t z 0 ) e i t ( ω ( β s ) 1 / T NL ) e i β s z .
z 0 2 = ω 2 Γ ξ 0 ; Γ = ω 0 N 2 A eff ; T NL = 1 Γ ξ 0 .
δ A ( t , β d ) t = i 2 d z { e i t ( ω ( β s ) 1 / 2 T N L ω ( β d ) ) e i z ( β d β s ) δ ω 2 ( e i K z + e i K z ) ξ 0 sech ( z v g t z 0 ) + Γ e i ( β d z ω ( β d ) t ) [ 2 A ˜ s ( t , z ) 2 δ A ˜ ( t , z ) + A ˜ s 2 δ A ˜ * ( t , z ) ] } .
i 2 e i t ( ω ( β s ) 1 / 2 T NL ω ( β d ) ) d z e i z ( β d β s ) δ ω 2 ( e i K z + e i K z ) ξ 0 sech ( z v g t z 0 ) = i 2 e i t ( ω ( β s ) 1 / 2 T NL ω ( β d ) + v g ( β d β s ) ) d u e i u ( β d β s ) δ ω 2 ( e i K v g t e i K u + e i K v g t e i K u ) ξ 0 sech ( u z 0 ) = i 2 e i t ( ω ( β s ) 1 / 2 T NL ω ( β d ) + v g ( β d β s ) ) ξ 0 × [ e i K v g t sech ( π 2 ( β d β s + K ) z 0 ) + e i K v g t sech ( π 2 ( β d β s K ) z 0 ) ] .
ω ( β s ) 1 2 T NL ω ( β d ) + v g ( β d β s ) ± K v g = 0.
1 2 ω 2 ( β s 2 β d 2 ) + ω 2 β s ( β d β s ) ± K v g 1 2 T NL = 1 2 ω 2 ( β s β d ) 2 ± K v g 1 2 T NL = 0.
( β d β s ) 2 = ± 2 K β s z 0 2 = z 0 2 ( 4 π v g T NL L p 1 ) ,
Δ ( z ) = Δ 0 cos ( K z ) , K = 2 π L p
z 0 4 π v g T NL z 0 4 π v g z 0 2 ω 2 4 π β s z 0 1.
Δ ( z ) = Δ 0 m Φ m exp ( 1 2 k m 2 ( L c 2 ) 2 ) exp ( i k m z ) , k m = 2 π m L ,

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