Abstract

Dynamical behavior of counterpropagating (CP) mutually incoherent vector solitons in a 5 × 5 × 23 mm SBN:60Ce photorefractive crystal is investigated. Experimental study is carried out, displaying rich dynamics of three-dimensional CP solitons and higher-order multipole structures, and a theory formulated that is capable of capturing such dynamics. We find that our numerical simulations agree well with the experimental findings for various CP beam structures. Linear stability analysis is also performed, predicting a threshold for the modulational instability of CP beams, and an appropriate control parameter is identified. We attempt at utilizing these results to CP solitons, but find only qualitative agreement with the numerical simulations and experimental findings. However, when broader hyper-Gaussian CP beams are used in simulations, an improved agreement with the theory is obtained.

© 2005 Optical Society of America

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References

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    [CrossRef]
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J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Opt. Phot. News (1)

Special Issue on solitons, Ed. M. Segev, Opt. Phot. News 13, No. 2 (2002).

Phys. Rev. A (2)

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, "Hexagons and squares in a passive nonlinear optical system," Phys. Rev. A 50, 3471 (1994).
[CrossRef] [PubMed]

O. Sandfuchs, F. Kaiser, and M. Belić, "Self-organization and Fourier selection of optical patterns in a nonlinear photorefractive feedback system," Phys. Rev. A 64, 063809 (2001).
[CrossRef]

Phys. Rev. E (3)

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, Ph. Jander, and F. Kaiser, "Counterpropagating beams in biased photorefractive crystals: anisotropic theory," Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in photorefractive media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

M. Petrović, D. Jović, M. Belić, J. Schröder, Ph. Jander, and C. Denz, "Two Dimensional Counterpropagating Spatial Solitons in Photorefractive Crystals," Phys. Rev. Lett. 95, 053901 (2005).
[CrossRef] [PubMed]

Other (2)

C. Denz, M. Schwab, and C. Weilnau, Transverse pattern formation in photorefractive optics (Springer, Berlin, 2003).
[CrossRef]

S. Trillo, and W. Torruellas eds., Spatial Solitons (Springer, New York, 2001).

Supplementary Material (17)

» Media 1: MPG (2031 KB)     
» Media 2: MOV (878 KB)     
» Media 3: MOV (858 KB)     
» Media 4: MPG (2818 KB)     
» Media 5: MOV (592 KB)     
» Media 6: MOV (502 KB)     
» Media 7: MPG (2818 KB)     
» Media 8: MOV (902 KB)     
» Media 9: MOV (425 KB)     
» Media 10: MOV (1999 KB)     
» Media 11: MOV (1762 KB)     
» Media 12: MOV (805 KB)     
» Media 13: MOV (1719 KB)     
» Media 14: MOV (2637 KB)     
» Media 15: MOV (1640 KB)     
» Media 16: MOV (1008 KB)     
» Media 17: MOV (7292 KB)     

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Figures (11)

Fig. 1.
Fig. 1.

Experimental setup for the investigation of CP solitons. BS: beam splitters, PM: vibrating piezo-mirror, SBN: Strontium-Barium-Niobate crystal.

Fig. 2.
Fig. 2.

Isosurface plots of a CP soliton after a splitup transition. Forward propagating component is displayed in the steady state, (a) View along the entrance face of the crystal, (b) View along the exit face of the crystal. Simulation parameters: |F0 | 2 =|BL | 2 =0.6, Γ=7.17, L=5.75LD =23 mm, and initial beam widths (FWHM) are 20 μm.

Fig. 3.
Fig. 3.

Movies of the Gaussian-Gaussian beam interaction: (a) Exit face of the crystal, experimental (2.031 MB). (b) The corresponding numerical simulation of the backward beam (at the exit face of the crystal), in the direct space (878 KB), (c) in the inverse space (858 KB). Parameters are as in Fig. 2, except for |F0 | 2 =|BL | 2 =7.5.

Fig. 4.
Fig. 4.

Movies of the dipole-mode beam interaction. (a) Experiment: The upper frame presents the exit face of the crystal for the dipole component, the lower frame presents the exit face for the fundamental mode component (2.818MB). (b) Movie of the dipole (592 KB), and (c) movie of the fundamental mode component (502 KB), from the corresponding numerical simulation. They are presented at their own exit faces, looking in the forward propagation direction. Parameters are as in Fig. 2, except for |F0 | 2 = |BL | 2 =4. Initial distance between dipole partners is 40 μm.

Fig. 5.
Fig. 5.

Movies of the dipole-dipole interaction. (a) Experiment: the forward beam (upper) and backward beam (lower), at the exit face of the crystal (2.818 MB). The corresponding numerical simulations of the backward beam: (b) with an extra noise of 5% added to the input beam intensity (902 KB), (c) without noise (426 KB). Parameters are as in Fig. 4, except for |F0 | 2 =|BL | 2 =1.3.

Fig. 6.
Fig. 6.

Threshold curves obtained from Eq. (11). Inset provides an extended view. Two clusters of open-circled points, obtained numerically, represent jumps in k2L of the peaked soliton-like structures in the inverse space. Dashed lines are the average values of k2L for the two sets of points. Filled circles represent points where the broader hyper-Gaussian CP beams undergo MI.

Fig. 7.
Fig. 7.

(a) Threshold intensity |F0 | 2 versus the square of the transverse wave vector k2 , for Γ=4 and L=5LD . Blue color represents the intensity region with stable solitons, yellow represents the intensity region with splitup(s).(b) The square of the transverse wave vector(kmax2) corresponding to the maximum value of the far-field intensity versus time, for |F0 | 2 =2.5. Steady state is reached.

Fig. 8.
Fig. 8.

(a) Threshold intensity |F0 | 2 versus the square of the transverse wave vector k2 , for Γ=6 and L=5LD . Colours have the same meaning as in Fig. 7(a), red represents the intensity region of unstable behavior, where the steady state is never reached. (b) The square of the transverse wave vector corresponding to the maximum value of the far-field intensity (kmax2) versus time, for |F0 | 2 =4. After a transient, a limit cycle is reached.

Fig. 9.
Fig. 9.

(a) Threshold intensity |F0 | 2 versus the square of the transverse wave vector k2 , for Γ = 7.17 and L=5.75LD . Colours have the same meaning as in Fig. 8(a). (b) The square of the transverse wave vector (kmax2) corresponding to the maximum value of the far-field intensity versus time, |F0 | 2 =7.5. Irregular behavior is observed.

Fig. 10.
Fig. 10.

Movies of modulational instabilities of a broad hyper-Gaussian beam, for the backward component. (a) (1999 KB) Direct space (7293 KB version), (b) (1763 KB) Inverse space. The order of the hyper-Gaussian is 4, FWHM=150 μm, other parameters: Γ=27.6, L=0.5LD , |F0 | 2 = |BL | 2 =3.

Fig. 11.
Fig. 11.

Movies of the modulational instabilities of hyper-Gaussians, resulting in a rotating hexagon, (a) (805 KB) direct space, (b) (1719 KB) inverse space intensity distributions (2637 KB), and in a steady-state octagonal pattern, (c) (1640 KB) direct space, (d) (1008 KB) inverse space intensity distributions. The parameters for both cases are the same, Γ=16.14, L=3LD , |F0 | 2 = |BL | 2 =5, the only difference is the beam widths: 100 μm in the first, and 150 μm in the second case.

Equations (17)

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i z F = Δ F + Γ EF ,
i z B = Δ B + Γ EB ,
τ t E + E = I 1 + I ,
F 0 ( z ) = F 0 ( 0 ) exp ( i Γ E 0 z ) ,
B 0 ( z ) = B 0 ( L ) exp ( i Γ E 0 ( z L ) ) ,
F = F 0 ( 1 + f ) ,
B = B 0 ( 1 + b ) ,
E = E 0 ( 1 + e ) ,
f ( 0 ) = b ( L ) = 0 .
1 + cos ( k 2 L ) cos ( k 4 L 2 4 A Γ k 2 L 2 ) +
( k 2 L 2 A Γ L ) k 4 L 2 4 A Γ k 2 L 2 sin ( k 2 L ) sin ( k 4 L 2 4 A Γ k 2 L 2 ) = 0
2 + 2 cos ( Ψ 1 ) cos ( Ψ 2 ) + ( Ψ 1 Ψ 2 + Ψ 2 Ψ 1 ) sin ( Ψ 1 ) sin ( Ψ 2 ) = 0 ,
H 1 H 2 = 0 ,
H 1 ( Ψ 1 , Ψ 2 ) = Ψ 1 Ψ 2 sin ( Ψ 1 2 ) sin ( Ψ 2 2 ) + cos ( Ψ 1 2 ) cos ( Ψ 2 2 )
A Γ L = ( m 2 n 2 ) π 2 4 k 2 L ,
k 2 L = .
k 2 < Γ 2 ,

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