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Self-organization of spatial solitons

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We present experimental results on the transverse modulation instability of an elliptical beam propagating in a bulk nonlinear Kerr medium, and the formation and self-organization of spatial solitons. We have observed the emergence of order, self organization and a transition to an unstable state. Order emerges through the formation of spatial solitons in a periodic array. If the initial period of the array is unstable the solitons will tend to self-organize into a larger (more stable) period. Finally the system transitions to a disordered state where most of the solitons disappear and the beam profile becomes unstable to small changes in the input energy.

©2005 Optical Society of America

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Supplementary Material (3)

Media 1: AVI (79 KB)     
Media 2: AVI (113 KB)     
Media 3: AVI (259 KB)     

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Figures (8)

Fig. 1.
Fig. 1. FTOP Setup. The pump pulse is focused in the material with a cylindrical lens to generate a single column of filaments. The beam profile at the output is imaged on CCD 1. The probe pulse goes through a variable delay line, a polarizer and analyzer and is imaged on CCD.
Fig. 2.
Fig. 2. Beam profile of the pump pulse at the output of the CS2 cell. The power increases form left to right: a) P = 12Pcr , b) 40Pcr , c) 80Pcr , d) 170Pcr , e) 250Pcr , f) 390Pcr , g) 530Pcr , h) 1200Pcr .
Fig. 3.
Fig. 3. Video clip of changes in the beam profile as a result of fluctuations in the pulse energy for P = 170 Pcr (78.5 KB). The image area is 0.36 mm (h) × 0.89 mm (v).
Fig. 4.
Fig. 4. Video clip of changes in the beam profile as a result of fluctuations in the pulse energy for P = 390 Pcr (113 KB). The image area is 0.36 mm (h) × 0.89 mm (v).
Fig. 5.
Fig. 5. Video clip of pulse propagation inside CS2 from 2 mm to 4 mm from the cell entrance for a pulse power of 390 Pcr. An initially uniform beam breaks up into stable filaments (258 KB). The image size is 2.4 mm (h) × 1.6 mm (v).
Fig. 6.
Fig. 6. Pulse trajectories and 1-D Fourier transforms. (a,c): The trajectory of the pulse is reconstructed by digitally adding up the FTOP frames for different positions of the pulse. Each separate image corresponds to frames taken for a fixed position of CCD camera. The camera was moved laterally to capture the beam profile further along inside the cell. The pulse power is 390Pcr in (a) and 1200Pcr in (c). (b,d) Show the 1-D Fourier transforms of the filamentation patterns in (a) and (c), respectively. The central component is blocked to visualize higher frequencies.
Fig. 7.
Fig. 7. Interactions between filaments from 3.5 mm to 4.2 mm from the cell entrance for an input pulse power of 1200Pcr . Some filaments propagate undisturbed (a-c). We have observed fusion of two filaments (b), divergence of a filament (d) and the generation of a new filament (e).
Fig. 8.
Fig. 8. 1-D Fourier transforms for numerically calculated beam propagation. The beam propagation is numerically calculated for four different power levels a) P = 250 Pcr , b) 390 Pcr , c) 530 Pcr , d) 1200 Pcr . A 1-D Fourier transform on the side view of the beam profile is calculated for each along the propagation direction. The total distance is 10 mm. The central peak (DC component) in the Fourier transform is blocked to improve the contrast in the image.

Equations (2)

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P cr = π ( 0.61 ) 2 λ 2 8 n 0 n 2 .
dA dz = i 2 k n 0 ( 2 x 2 + 2 y 2 ) A + ik ( n 2 A 2 ) A ik ( n 4 A 4 ) A β A 2 A


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