Abstract

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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Appl. Phys. B (2)

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, H. Kuroda, �??Nonlinear refraction in CS2,�?? Appl. Phys. B 78, 433�??438 (2004).
[CrossRef]

M. Falconieri, G. Salvetti, �??Simultaneous measurement of pure-optical and thermo-optical nonlinearities induced by high-repetition-rate, femtosecond laser pulses: application to CS2,�?? Appl. Phys. B. 69, 133-136 (1999).
[CrossRef]

Appl. Phys. Lett. (2)

J. M. Halbout, C. L. Tang, �??�??Femtosecond interferometry for nonlinear optics,�??�?? Appl. Phys. Lett. 40, 765-767 (1982)
[CrossRef]

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, �??Periodic breakup of optical beams due to self-focusing,�?? Appl. Phys. Lett. 23, 628-630 (1973).
[CrossRef]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, A. A. Zozulya, �??�??Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,�??�?? Europhys. Lett. 35, 25-30 (1996);
[CrossRef]

IEEE J. Quantum Electron. (2)

D. McMorrow, W. T. Lotshaw, G. A. Kenneywallace, �??�??Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,�??�?? IEEE J. Quantum Electron. 24, 443-454 (1988)).
[CrossRef]

A. Piekara, �??On self-trapping of a laser beam,�?? IEEE J. Quantum Electron. QE-2, 249-250 (1966).
[CrossRef]

J. Opt. Soc. Am B. (1)

M. Saffman, G. McCarthy and W. Krolikowski, �??Two-dimensional modulational instability in photorefractive media,�?? J. Opt. Soc. Am. B. 6, 387-403 (2004).

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Am. B. (1)

G. Fibich, B. Ilan, �??�??Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation,�??�?? J. Opt. Soc. Am. B 17, 1749-1758 (2000);
[CrossRef]

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, C. Umeton, �??Routing of anisotropic spatial solitons and modulational instability in liquid crystals,�?? Nature 432, 733 (2004).
[CrossRef] [PubMed]

Opt. Commun. (4)

H. Maillotte, J. Monneret, A. Barthelemy, C. Froehly, �??�??Laser beam self-splitting into solitons by optical Kerr nonlinearity,�??�?? Opt. Commun. 109, 265-271 (1994)
[CrossRef]

A. Barthelemy, S. Maneuf, C. Froehly, �??Soliton Propagation and Self-Confinement of Laser-Beams by Kerr Optical Non-Linearity,�?? , Opt. Commun. 55, 201-206 (1985).
[CrossRef]

R.A. Ganeev, A.I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, H. Kuroda, �??�??Two- and three-photon absorption in CS2,�??�?? Opt. Commun. 231, 431-436 (2004).
[CrossRef]

H. Schroeder and S. L. Chin, �??Visualization of the evolution of multiple filaments in methanol,�?? Opt. Commun. 234, 399-406 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Phys. Rep. (1)

Y. Kivshar, D. Pelinovsky, �??Self-focusing and transverse instabilities of solitary waves,�?? Phys. Rep. 331, 118-195 (2000).
[CrossRef]

Phys. Rev. A (3)

M. Centurion, Y. Pu, M. Tsang, D. Psaltis, �??Dynamics of filament formation in a Kerr medium,�?? Phys. Rev. A 71 063811 (2005).
[CrossRef]

A. Vinçotte, L. Berge, �??�?(5) susceptibility stabilizes the propagation of ultrashort laser pulses in air,�?? Phys. Rev. A 70, 061802 (2004)
[CrossRef]

R. Mcleod, K. Wagner and S. Blair, �??(3+1)-Dimensional Optical Soliton Dragging Logic,�?? Phys. Rev. A 52, 3254-3278 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Petrovi, D. Träger, A. Strinic, M. Belic, J. Schroder, C. Denz, �??Solitonic lattices in photorefractive crystals,�?? Phys. Rev. E 68, 055601 (2003)
[CrossRef]

Phys. Rev. Lett. (6)

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, �??Optically Turbulent Femtosecond Light Guide in Air,�?? Phys. Rev. Lett. 83, 2938-2941 (1999).
[CrossRef]

L. Berge, S. Skupin, F. Lederer, G. Mejean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Woste, R. Bourayou, R. Sauerbrey, �??Multiple filamentation of Terrawatt laser pulses in air,�?? Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv and B. Fischer, �??Spatial Solitons in Photorefractive Media,�?? Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Vanstryland, G. I. Stegeman, L. Torner, C. R. Menyuk, �??Observation of 2-Dimensional Spatial Solitary Waves in a Quadratic Medium,�?? Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef]

Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, J. P. Torres, �??�??Eliminating the transverse instabilities of Kerr solitons,�??�?? Phys. Rev. Lett. 85, 4888�??4891 (2000);
[CrossRef] [PubMed]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, S. Wabnitz, �??�??Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,�??�?? Phys. Rev. Lett. 78, 2756-2759 (1997);
[CrossRef]

Science (2)

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, D. N. Christodoulides, �??Modulation instability and pattern formation in spatially incoherent light beams,�?? Science 290, 495-498 (2000).
[CrossRef] [PubMed]

G. I. Stegeman, M. Segev, �??Optical spatial solitons and their interactions: Universality and diversity,�?? Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov, A. M. Rubenchik, �??�??Instability of waveguides and solitons in nonlinear media,�??�?? Sov. Phys. JETP 38, 494-500 (1974).

Other (1)

R Boyd, Nonlinear Optics, Academic Press, 2003.

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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