Abstract

Quasi-periodic arrays of bright soliton-like beams are obtained experimentally in the picosecond regime as a result of the transverse modulational instability of a noisy continuous background in a planar CS2 waveguide. For a given propagation length, the array is stable from a laser shot to another and for a wide range of input intensities. The experimental period corresponds to the maximum gain of modulational instability only for the intensity just sufficient for soliton formation. On the other hand the mean period increases with the propagation length. We show by a numerical simulation that the leading edge of the pulse governs the dynamical formation of the array owing to the finite relaxation time of the reorientational Kerr nonlinearity in CS2.

© 2002 Optical Society of America

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References

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  1. S. Trillo and W. E. Torruellas, Spatial Solitons (Springer Verlag, Berlin, 2001).
  2. Special Issue on Solitons, Opt. Photon. News13, 17–76 (2002).
  3. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286, 1518–1523 (1999).
    [Crossref] [PubMed]
  4. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
    [Crossref]
  5. R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [Crossref]
  6. A. Bramati, W. Chinaglia, S. Minardi, and P. Di Trapani “Reconstruction of blurred images by controlled formation of spatial solitons,” Opt. Lett. 18, 1409–1411 (2001).
    [Crossref]
  7. P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, “Generation of a periodic array of dark spatial solitons in the regime of effective amplification,” J. Opt. Soc. Am. B 11, 1254–1260 (1994).
    [Crossref]
  8. H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), 167–169.
  9. C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).
  10. R. Malendevich, L. Jankovic, G. I. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett. 26, 1879–1881 (2001).
    [Crossref]
  11. R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
    [Crossref] [PubMed]
  12. C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
    [Crossref]
  13. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
    [Crossref]
  14. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
    [Crossref] [PubMed]

2002 (1)

2001 (4)

R. Malendevich, L. Jankovic, G. I. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett. 26, 1879–1881 (2001).
[Crossref]

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

A. Bramati, W. Chinaglia, S. Minardi, and P. Di Trapani “Reconstruction of blurred images by controlled formation of spatial solitons,” Opt. Lett. 18, 1409–1411 (2001).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

1999 (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

1997 (1)

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

1996 (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[Crossref]

1994 (1)

1980 (1)

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Aitchison, J. S.

Baboiu, D-M.

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

Bosshard, C.

Bramati, A.

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Cambournac, C.

C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
[Crossref]

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

Chauvet, M.

C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
[Crossref]

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

Chinaglia, W.

Di Trapani, P.

Dudley, J. M.

C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
[Crossref]

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

Emplit, Ph.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Fang, H.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

Fuerst, R. A.

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

Grasser, R.

H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), 167–169.

Haelterman, M.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Hasegawa, A.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Jankovic, L.

Lantz, E.

C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
[Crossref]

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

Lawrence, B.

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

Maillotte, H.

C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002).
[Crossref]

H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), 167–169.

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

Malendevich, R.

R. Malendevich, L. Jankovic, G. I. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett. 26, 1879–1881 (2001).
[Crossref]

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[Crossref]

Mamyshev, P. V.

Minardi, S.

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[Crossref]

Schiek, R.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

Segev, M.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

Stegeman, G. I.

R. Malendevich, L. Jankovic, G. I. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett. 26, 1879–1881 (2001).
[Crossref]

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

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

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, “Generation of a periodic array of dark spatial solitons in the regime of effective amplification,” J. Opt. Soc. Am. B 11, 1254–1260 (1994).
[Crossref]

Torruellas, W. E.

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

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer Verlag, Berlin, 2001).

Trillo, S.

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

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer Verlag, Berlin, 2001).

Van Simaeys, G.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Wabnitz, S.

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

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[Crossref]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, and 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. (1)

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Phys. Rev. Lett. (3)

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

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

Science (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

Other (4)

H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), 167–169.

C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer Verlag, Berlin, 2001).

Special Issue on Solitons, Opt. Photon. News13, 17–76 (2002).

Supplementary Material (2)

» Media 1: MOV (531 KB)     
» Media 2: MOV (1225 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Experimental set-up: Excitation of a planar Kerr-like waveguide by an intense, extended, and quasi-plane pulsed TE wave.

Fig. 2.
Fig. 2.

Typical experimental results of noise-initiated spatial modulational instability: (a) Output profiles at low input mean intensity (red), and for nonlinear propagation regime (blue), (b) corresponding Fourier spectra.

Fig. 3.
Fig. 3.

Experimental images for different intensities and two propagation lengths.

Fig. 4.
Fig. 4.

Dashed curve: period of maximum MI gain versus the mean input intensity. Blue circles: measured periods of the spontaneously generated arrays for L = 3 cm and different intensities. Red circles: the same for L = 7 cm. The letters correspond to the images of Fig. 3.

Fig. 5.
Fig. 5.

Simulated time-integrated intensity profiles at the output of the 7-cm waveguide for an input mean intensity of 260 MW/cm2 (blue) and 440 MW/cm2 (red).

Fig. 6.
Fig. 6.

(544 KB) Evolution of the time-integrated intensity with propagation distance for an input mean intensity of 440 MW/cm2.

Fig. 7.
Fig. 7.

(1,255 KB). Spatiotemporal repartition of the intensity for two different lengths of the waveguide and for an input mean intensity of 440 MW/cm2. Movie: evolution of this spatiotemporal repartition with the propagation distance (every 0.5 cm)

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Δ n ( t ) = 1 τ t n 2 I ( t 1 ) exp ( t 1 t τ ) d t 1 ,
δ = Im { Ω Ω 2 4 β 2 n 2 I 0 n 0 } ,
δ max = β n 2 I 0 / n 0 is obtained at Ω max = ± β 2 n 2 I 0 / n 0 = ± Ω c / 2 .
G = exp ( δ max L ) = constant .
p = K L ,
p ( L = 7 cm ) = 125 μm p ( L = 3 cm ) = 84 μm 7 3 .

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