Abstract

Localized energy states such as two-color gap solitons are theoretically and numerically predicted in a periodic structure in the presence of a frequency doubling nonlinearity. These parametric solitons exhibit appealing features as compared to the Kerr case. Novel effects such as merging and all-optical buffering are envisaged.

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  1. W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
    [CrossRef] [PubMed]
  2. C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.
  3. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Nonlinear pulse propagation in Bragg gratings," J. Opt. Soc. Am. B 14, 2980-2993 (1997).
    [CrossRef]
  4. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, "Nonlinear self- switching and multiple gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 328-330 (1998).
    [CrossRef]
  5. Y. Kivshar, "Gap solitons due to cascading," Phys. Rev. E 51, 1613-1615 (1995).
    [CrossRef]
  6. C. Conti, S. Trillo, and G. Assanto, "Doubly resonant Bragg simultons via second-harmonic generation," Phys. Rev. Lett. 78, 2341-2344 (1997).
    [CrossRef]
  7. H. He and P. D. Drummond, "Ideal soliton environment using parametric band gaps," Phys. Rev. Lett. 78, 4311-4314 (1997).
    [CrossRef]
  8. T. Peschel, U. Peschel, F. Lederer, and B. Malomed, "Solitary waves in Bragg gratings with a quadratic nonlinearity," Phys. Rev. E 55, 4730-4739 (1997).
    [CrossRef]
  9. C. Conti, S. Trillo, and G. Assanto, "Bloch functions approach for parametric gap solitons," Opt. Lett. 22, 445-447 (1997).
    [CrossRef] [PubMed]
  10. C. Conti, S. Trillo, and G. Assanto, "Optical gap solitons via second-harmonic generation: exact solitary solutions," Phys. Rev. E, 57, R1251-R1254 (1998).
    [CrossRef]
  11. C. Conti, S. Trillo, and G. Assanto, "Trapping of slowly moving or stationary two-color gap solitons," Opt. Lett. 23, 334-336 (1998).
    [CrossRef]
  12. C. Conti, G. Assanto, and S. Trillo, "Read/write all-optical buffer by self trapped gap simultons," Electron. Lett. 34, 689-690 (1998).
    [CrossRef]
  13. C. Conti, G. Assanto, and S. Trillo, "Excitation of self-transparency Bragg solitons in quadratic media," Opt. Lett. 22, 1350-1352 (1997).
    [CrossRef]
  14. C. Conti, A. De Rossi, and S. Trillo, "Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media," Opt. Lett. 23, 1265-1267 (1998).
    [CrossRef]
  15. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional spatial solitary waves in a quadratic medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
    [CrossRef] [PubMed]
  16. R. Schiek, Y. Baek, G. I. Stegeman, "One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides," Phys. Rev. E 53, 1138 (1996).
    [CrossRef]
  17. C. M. de Sterke, D. G. Salinas, J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
    [CrossRef]
  18. J. Söchtig et al., Electron. Lett. 31, 551-552 (1995).
    [CrossRef]
  19. Z. Weissman et al., "Second-harmonic generation in Bragg-resonant quasi-phase-matched peri- odically segmented waveguide," Opt. Lett. 20, 674-676 (1995).
    [CrossRef] [PubMed]
  20. P. G. Kazansky and V. Pruneri, "Electric-field poling of quasi-phase-matched optical fibers," J. Opt. Soc. Am. B, 14, 3170-3179 (1997).
    [CrossRef]
  21. Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, "Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating," Opt. Lett. 23, 1195-1197 (1998).
    [CrossRef]
  22. A. V. Buryak and Y. S. Kivshar, "Spatial optical solitons governed by quadratic nonlinearity," Opt. Lett. 19, 1612-1614 (1994).
    [CrossRef] [PubMed]
  23. Guided-Wave Optoelectronic, chapter 2, T. Tamir ed., (Springer, Berlin, 1980).
  24. L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, "Walking solitons in quadratic nonlinear media," J. Opt. Soc. Am. B 15, 1476-1487 (1998).
    [CrossRef]
  25. C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, "Stability of temporal chirped solitary waves in quadratically nonlinear media," Phys. Rev. E 55, 6155-6161 (1997).
    [CrossRef]
  26. C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, "Collision of solitary waves in media with a second-order nonlinearity," Phys. Rev. A 52, R3444-R3447 (1995).
    [CrossRef] [PubMed]

Other (26)

W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Nonlinear pulse propagation in Bragg gratings," J. Opt. Soc. Am. B 14, 2980-2993 (1997).
[CrossRef]

D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, "Nonlinear self- switching and multiple gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 328-330 (1998).
[CrossRef]

Y. Kivshar, "Gap solitons due to cascading," Phys. Rev. E 51, 1613-1615 (1995).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, "Doubly resonant Bragg simultons via second-harmonic generation," Phys. Rev. Lett. 78, 2341-2344 (1997).
[CrossRef]

H. He and P. D. Drummond, "Ideal soliton environment using parametric band gaps," Phys. Rev. Lett. 78, 4311-4314 (1997).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, "Solitary waves in Bragg gratings with a quadratic nonlinearity," Phys. Rev. E 55, 4730-4739 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, "Bloch functions approach for parametric gap solitons," Opt. Lett. 22, 445-447 (1997).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, "Optical gap solitons via second-harmonic generation: exact solitary solutions," Phys. Rev. E, 57, R1251-R1254 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, "Trapping of slowly moving or stationary two-color gap solitons," Opt. Lett. 23, 334-336 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, "Read/write all-optical buffer by self trapped gap simultons," Electron. Lett. 34, 689-690 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, "Excitation of self-transparency Bragg solitons in quadratic media," Opt. Lett. 22, 1350-1352 (1997).
[CrossRef]

C. Conti, A. De Rossi, and S. Trillo, "Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media," Opt. Lett. 23, 1265-1267 (1998).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional spatial solitary waves in a quadratic medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, G. I. Stegeman, "One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides," Phys. Rev. E 53, 1138 (1996).
[CrossRef]

C. M. de Sterke, D. G. Salinas, J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

J. Söchtig et al., Electron. Lett. 31, 551-552 (1995).
[CrossRef]

Z. Weissman et al., "Second-harmonic generation in Bragg-resonant quasi-phase-matched peri- odically segmented waveguide," Opt. Lett. 20, 674-676 (1995).
[CrossRef] [PubMed]

P. G. Kazansky and V. Pruneri, "Electric-field poling of quasi-phase-matched optical fibers," J. Opt. Soc. Am. B, 14, 3170-3179 (1997).
[CrossRef]

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, "Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating," Opt. Lett. 23, 1195-1197 (1998).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, "Spatial optical solitons governed by quadratic nonlinearity," Opt. Lett. 19, 1612-1614 (1994).
[CrossRef] [PubMed]

Guided-Wave Optoelectronic, chapter 2, T. Tamir ed., (Springer, Berlin, 1980).

L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, "Walking solitons in quadratic nonlinear media," J. Opt. Soc. Am. B 15, 1476-1487 (1998).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, "Stability of temporal chirped solitary waves in quadratically nonlinear media," Phys. Rev. E 55, 6155-6161 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, "Collision of solitary waves in media with a second-order nonlinearity," Phys. Rev. A 52, R3444-R3447 (1995).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Twin bandgap structure with the definition of the Bragg frequencies ωoj , and upper and lower gap edge frequencies ωuj and ωlj , respectively.

Fig. 2.
Fig. 2.

Injection of a FF pulse in a finite nonlinear grating (κ 2 = 1, v 2 = 0.5): Excitation of a two-color GS propagating at low velocity in the LB-LB case; Here we show the contour levels of the total FF (a) and SH (b) intensity, respectively.

Fig. 3.
Fig. 3.

Formation of a stationary gap solitons via inelastic collision of two counterpropagating low-speed solitons: (a) FF; (b) generated SH.

Fig. 4.
Fig. 4.

Interrogation process of the GS formed as in Fig. 3 (same parameters), by means of launching a new moving GS.

Fig. 5.
Fig. 5.

Long range dynamics of excitation of slow GSs in a singly resonant semi-infinite grating from Eqs. (22) with δ 1 = -0.7, δ 1 = 2, κ 2 = 0 and v 2 = 0.5.

Equations (67)

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H t = 1 μ 0 E z ; E t = 1 ε 0 n 2 ( z ) ( H z + P N L t ) ,
A ± ( z , t ) = 1 2 n ( z ) n 0 [ E ( z , t ) ± μ 0 ε 0 H ( z , t ) n ( z ) ] ,
i n ¯ ( z ) · A t = M ¯ · A + B ,
M ¯ [ i c z i c n ' ( z ) 2 n ( z ) i c n ' ( z ) 2 n ( z ) i c z ] ,
B ± = i 2 ε 0 n 0 n ( z ) P N L t .
P N L ( z , t ) = ε 0 χ E 2 ( z , t ) .
M ¯ · Ψ μ = ω μ n ¯ · Ψ μ .
Ψ j μ · n ¯ · Ψ j ' μ ' = 0 L Ψ j μ · n ¯ · Ψ j ' μ ' d z = N δ j j ' δ μ μ ' .
A = A 1 e i ω o 1 t + A 2 e i ω o 2 t + c . c . ,
A j = f j u Ψ j u + f j l Ψ j l + p u , l f j p Ψ j p f j h Ψ j h ,
ω o j n ¯ · A j ¯ + i η n ¯ · A ¯ j t 1 = M ¯ ( 0 ) · A ¯ j i η V ¯ · A ¯ j z 1 + B ¯ j
i f j u t Δ j 2 f j u V j f j l z + ϑ j u = 0 ,
i f j l t Δ j 2 f j l V j f j u z + ϑ j l = 0 ,
α r p q n 0 χ 0 d ϕ r , 1 ( z ) ϕ p , 1 ( z ) ϕ q , 2 ( z ) d z ,
α p q r n 0 χ 0 d ϕ p , 1 ( z ) ϕ q , 1 ( z ) ϕ r , 2 ( z ) d z .
E j + = 1 2 ( f j u i f j l ) ; E j = 1 2 ( f j u i f j l ) ,
f i = Θ E j [ f j u f j l ] = [ f j 1 f j 2 ] = [ 1 1 i i ] [ E j 1 E j 2 ] ,
i ( E j + t + V j E j + z ) + V j Γ j E j + τ j + = 0 ,
i ( E j t - V j E j z ) + V j Γ j E j + + τ j = 0 ,
[ τ j + τ j ] [ τ j 1 τ j 2 ] 1 2 [ 1 i 1 i ] [ ϑ j 1 ϑ j 2 ] ,
τ 1 j = ω o 1 2 exp ( iδωt ) H j h k ( E 1 h ) * E 2 k ; τ 2 j = ω o 1 2 exp ( + iδωt ) H h k j E 1 h E 1 k ;
H j h k ( Θ j m ) ( Θ p h ) * Θ q k α m p q ; H h k j ( Θ j m ) Θ p h Θ q k α p q m .
H j h k = { ( 1 ) j [ ( 1 ) h + k α 11 1 + α 12 2 ] + ( 1 ) h α 21 2 ( 1 ) k α 22 1 } +
+ i { ( 1 ) h + k α 21 1 α 22 2 + ( 1 ) j [ ( 1 ) h α 11 2 ( 1 ) k α 12 2 ] } ,
H h k j = { ( 1 ) j [ ( 1 ) h + k α 11 1 α 22 1 ] + ( 1 ) h α 21 2 + ( 1 ) h α 12 2 } +
+ i { ( 1 ) h + k α 11 2 + α 22 2 + ( 1 ) j [ ( 1 ) k α 21 1 + ( 1 ) h α 12 1 ] } .
n ( z ) = n B + n 1 sin ( 2 π d z ) + n 2 sin ( 4 π d z )
ϕ u j ( z ) = N j cos ( j π z d ) ,
ϕ l j ( z ) = N j sin ( j π z d ) ,
α u l l = α l u l = α l l u = α u u u = 0 ,
α l l l = α u l u = α l u u = α u u l = d 4 n 0 χ N L N 1 2 N 2 α .
H 1 11 = H 2 22 = 4 i α ; H 11 1 = H 22 1 = 4 ,
i ( E 1 + t + V 1 E 1 + z ) + V 1 Γ 1 E 1 2 i α ω o 1 ( E 1 + ) * E 2 + e i δ ω t = 0 ,
i ( E 1 t V 1 E 1 z ) + V 1 Γ 1 E 1 + 2 i α ω o 1 ( E 1 ) * E 2 e i δ ω t = 0 ,
i ( E 2 + t + V 2 E 2 + z ) + V 2 Γ 2 E 2 + 2 i α ω o 1 ( E 1 + ) 2 e i δ ω t = 0 ,
i ( E 2 t V 2 E 2 z ) + V 2 Γ 2 E 2 + + 2 i α ω o 1 ( E 1 ) 2 e i δ ω t = 0 .
i ( ± w 1 , ξ ± + w 1 , τ ± v 1 ) + κ 1 w 1 + ( w 1 ± ) * w 2 ± e i ω τ = 0 ,
i ( ± w 2 , ξ ± + w 2 , τ ± v 2 ) + κ 2 w 2 + ( w 1 ± ) 2 2 e i ω τ = 0 ,
i ( ± u 1 , ξ ± + u 1 , τ ± v 1 ) + δ 1 u 1 ± + κ 1 u 1 + ( u 1 ± ) * u 2 ± = 0 ,
i ( ± u 2 , ξ ± + u 2 , τ ± v 2 ) + δ 2 u 2 ± + κ 2 u 2 + ( u 1 ± ) 2 2 = 0 ,
σ 0 = [ 1 0 0 1 ] σ 1 = [ 0 i i 0 ] σ 3 = [ 1 0 0 1 ] ,
L m ϕ m + κ m σ 1 ϕ m + N m = 0 ,
N 1 [ ( w 1 + ) * w 2 + ( w 1 ) * w 2 ] e i ω τ ; N 2 [ ( w 1 + ) 2 2 ( w 1 ) 2 2 ] e i ω τ .
ϕ m ( ξ j , τ j ; j 0 ) = n = 0 η n ϕ m ( n ) ( ξ j , τ j ; j 0 ) =
= η 2 ϕ m ( 2 ) ( ξ j , τ j ; j 0 ) + η 3 ϕ m ( 3 ) ( ξ j , τ j ; j 0 ) =
= [ η 2 a m ( ξ j , τ j ; j 1 ) f m ( ± ) + η 3 b m ( ξ j , τ j ; j 1 ) f m ( ) ] exp ( i Q m ξ 0 i Ω m ( ± ) τ 0 ) ,
f m ( + ) ( Q m ) = 1 2 κ m 2 + Q m 2 [ κ m κ m 2 + Q m 2 Q m κ m 2 + Q m 2 Q m ] ,
f m ( ) ( Q m ) = 1 2 κ m 2 + Q m 2 [ κ m 2 + Q m 2 Q m κ m κ m 2 + Q m 2 Q m ] .
b m = i 2 κ m κ m 2 + Q m 2 a m ξ 1 ,
i a 1 τ 2 + i Ω 1 a 1 ξ 2 + Ω 1 2 2 a 1 ξ 1 2 + v 1 χ a 1 * a 2 e i ( Δ Q ¯ ξ 1 ΔΩ ¯ τ 1 ) = 0 ,
i a 2 τ 2 + i Ω 2 a 2 ξ 2 + Ω 2 2 2 a 2 ξ 1 2 + v 2 χ a 1 2 e i ( Δ Q ¯ ξ 1 ΔΩ ¯ τ 1 ) = 0 ,
i a 1 τ 1 + i Ω 2 a 1 ξ 1 = 0 ; i a 2 τ 1 + i Ω 2 a 2 ξ 1 = 0 ,
χ = χ ( Q 1,2 ; κ 1,2 ) [ f 1 + ( ± ) ] 2 f 2 + ( ± ) + [ f 1 ( ± ) ] 2 f 2 ( ± ) ,
Ω m = d Ω m ± d Q m = ± v m Q m κ m 2 + Q m 2 ; Ω m = d 2 Ω m ± d Q m 2
[ u m + u m ] = [ 𝜜 m f m ( ± ) + i 2 ρ m κ m κ m 2 + Q m 2 𝜜 m ξ f m ( ) ] exp ( i Q m ξ i Ω m ( ± ) τ + i δ m v m τ ) ,
i 𝜜 1 τ + i Ω 1 𝜜 1 ξ + Ω 1 2 2 𝜜 1 ξ 2 + v 1 χ 𝜜 1 * 𝜜 2 e i ( Δ Q ξ ΔΩ τ ) = 0 ,
i 𝜜 2 τ + i Ω 2 𝜜 2 ξ + Ω 2 2 2 𝜜 2 ξ 2 + v 2 χ 𝜜 1 2 e i ( Δ Q ξ ΔΩ τ ) = 0 .
𝜜 1 ( ξ , τ ) = Ω 1 2 v 1 v 2 u 1 [ ( ξ Ω 1 τ ) , τ Ω 1 ] ,
𝜜 2 ( ξ , τ ) = Ω 1 v 1 u 2 [ ( ξ Ω 1 τ ) , τ Ω 1 ] exp { i [ Δ Q ξ ( Ω 2 Δ Q 2 / 2 Δ Q Ω 2 ) τ ] } .
i u 1 σ ρ 1 2 2 u 1 s 2 + χ u 1 * u 2 e i β τ = 0 ,
i u 2 σ ρ 2 2 γ 2 u 2 s 2 i δ W u 2 s + χ u 1 2 2 e i β τ = 0 ,
i u 1 τ ρ 1 2 2 u 1 ξ 2 + v 1 2 1 + ρ 2 2 u 1 * u 2 e i β τ = 0 ,
i u 2 τ ρ 2 2 v 2 κ 2 2 u 2 ξ 2 + v 2 2 1 + ρ 2 2 u 1 2 2 e i β τ = 0 ,
[ u m + u m ] = A m { 1 ρ m U m ( ζ ) + 1 2 ρ m 1 [ m V U m ( ζ ) + i ζ U m ( ζ ) ] } e i m V ξ
( i τ ± i ξ ) w 1 ± + w 1 + σ w ± 2 w ± = 0
i τ a + sgn ( δ 1 ) 2 ξ 2 a + σ a 2 a = 0 ,
w 1 + w 1 = 1 2 1 sgn ( δ 1 ) a ( ξ , τ ) exp [ i sgn ( δ 1 ) τ ] .

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