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Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals

M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and Paul Mandel

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Abstract

We study numerically the counterpropagating vector solitons in SBN:60 photorefractive crystals. A simple theory is provided for explaining the symmetry-breaking transverse instability of these solitons. Phase diagram is produced that depicts the transition from stable counterpropagating solitons to bidirectional waveguides to unstable optical structures. Numerical simulations are performed that predict novel dynamical beam structures, such as the standing-wave and rotating multipole vector solitonic clusters. For larger coupling strengths and/or thicker crystals the beams form unstable self-trapped optical structures that have no counterparts in the copropagating geometry.

©2004 Optical Society of America

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Supplementary Material (14)

Media 1: MOV (726 KB)     
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Figures (6)
Fig. 1.
Fig. 1. Phase diagram in the parameter plane for the symmetry-breaking instabilities of bidirectional solitons in 1D. Below the lower curve CP solitons exist, above the curve stable bidirectional waveguides appear. The insets depict typical beam intensity distributions in the (x, z) plane at the points indicated. At and above the upper critical curve the waveguides loose stability. The points are numerically determined, the curves represent inverse power polynomial fits. L is measured in units of LD , while Γ is dimensionless.

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Fig. 2.
Fig. 2. Movies of Gaussian forward beam undergoing transverse displacement at Γ=9.2 and L=1.8. Beam size 15 µm FWHM. (a) The (x,y) plane (727 KB). (b) The (y,z) plane (971 KB).

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Fig. 3.
Fig. 3. Movies of stable interacting dipoles. (a)-(b) Parallel-parallel geometry. (c)–(d) Parallel-perpendicular geometry. (a) (1.263 MB) and (c) (1.307 MB) Output forward beams. (b) (1.263 MB) and (d) (1.298 MB) Output backward beams. Parameters: Γ=9.2, L=1.8, initial distance between dipole partners 24 µm, initial beam widths 10 µm FWHM.

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Fig. 4.
Fig. 4. Movies of standing multipole waves, as a result of collision of two identical vortices at Γ=19.1, for different values of L. Forward output beam is depicted, backward wave is mirror-image. Initial beam widths 24.5 µm FWHM. (a) L=0.8 (1.391MB), (b) L=1.1 (1.359 MB), (c) L=1.5 (1.307 MB).

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Fig. 5.
Fig. 5. Movies of stable rotating forward beam at the output face, resulting from the collision of two oppositely charged vortices. The backward beam executes a mirror-image rotation. (a) The (x,y) plane (1.322 MB). (b) The (y,z) plane (1.841 MB). Parameters as in Fig. 4 (a).

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Fig. 6.
Fig. 6. Movies of unstable outputs of vortex-vortex collisions, forward output beam. (a) The same beam as in Fig. 5, only L is larger, L=1.5 (2.050 MB). (b) The same beam as in (a), only wider, the initial beam width 30 µm FWHM (1.444 MB). (c) Unstable vortex beam for the parameter values: Γ=16.1, L=1.3, initial beam width 30 µm FWHM (1.286 MB).

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Equations (12)

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1 + I = ( 1 + I 0 ) [ 1 + ε ∣ m ∣ cos ( 2 kz + Δ ϕ ) ] ,
E = E 0 + 1 2 [ E 1 exp ( 2 ikz ) + cc ] ,
i ∂ z F = − Δ F + Γ [ E 0 F + E 1 B ⁄ 2 ] , − i ∂ z B = − Δ B + Γ [ E 0 B + E 1 * F ⁄ 2 ] ,
τ ∂ t E 0 + E 0 = − I 0 1 + I 0 , τ ∂ t E 1 + E 1 = − ε m 1 + I 0 ,
〈 x 〉 ( z ) = 1 I t ∫ − ∞ + ∞ ∫ − ∞ + ∞ x ∣ A ( x , y , z ) ∣ 2 dxdy , 〈 y 〉 ( z ) = 1 I t ∫ − ∞ + ∞ ∫ − ∞ + ∞ y ∣ A ( x , y , z ) ∣ 2 dxdy ,
d 〈 x 〉 dz = p x , d p x dz = − 〈 ∂ V ∂ x 〉 ,
i ( d dz ) A = ( p 2 + V ) A ( ρ , z ) ,
( d dz ) 2 〈 x 〉 = − Γ 〈 ∂ E 0 ∂ x 〉 .
( d dz ) 2 d = ( d dz ) 2 ( 〈 d ∣ ρ ∣ d 〉 − 〈 0 ∣ ρ ∣ 0 〉 ) = − Γ ( 〈 d ∣ ∇ E 0 ∣ d 〉 − 〈 0 ∣ ∇ E 0 ∣ 0 〉 ) .
( d dz ) 2 d = − K d ,
K = − 2 Γ 〈 0 ∣ ( e ̂ · ∇ ) 2 E 0 ∣ 0 〉 = − 2 Γ I t ∬ ( e ̂ · ∇ ) 2 E 0 ∣ A ∣ 2 dxdy ,
( L K ) c = π 2 ,
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