Abstract

We study, theoretically and experimentally, the evolution of optical waves with randomly-fluctuating phases in a spatially chirped nonlinear directional coupler. As the system crosses its linear resonance, we observe collective self-phase-locking (autoresonance) of all mutually-incoherent waves, each with its own pump, and simultaneous amplification until the pumps are exhausted. We show that the autoresonant transition in this system exhibits a sharp threshold, common to all mutually-incoherent waves comprising the light beam.

© 2010 OSA

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  1. Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987).
    [CrossRef]
  2. O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
    [CrossRef] [PubMed]
  3. G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
    [CrossRef] [PubMed]
  4. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  5. S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
    [CrossRef]
  6. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
    [CrossRef]
  7. S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
    [CrossRef]
  8. Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
    [CrossRef] [PubMed]
  9. F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34(16), 2405–2407 (2009).
    [CrossRef] [PubMed]
  10. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
    [CrossRef] [PubMed]
  11. L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998).
    [CrossRef]
  12. J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001).
    [CrossRef]
  13. M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
    [CrossRef]
  14. M. S. Livingston, High-Energy Particle Accelerators (Interscience, New York, 1954).
  15. L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000).
    [CrossRef] [PubMed]
  16. L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998).
    [CrossRef]
  17. L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001).
    [CrossRef]
  18. A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
    [CrossRef] [PubMed]
  19. O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
    [CrossRef] [PubMed]
  20. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
    [CrossRef]
  21. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
    [CrossRef] [PubMed]
  22. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
    [CrossRef] [PubMed]
  23. M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996).
    [CrossRef]
  24. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
    [CrossRef] [PubMed]
  25. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
    [CrossRef] [PubMed]
  26. H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).
  27. T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
    [CrossRef] [PubMed]
  28. T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000).
    [CrossRef]
  29. A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001).
    [CrossRef] [PubMed]
  30. A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
    [CrossRef]
  31. A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005).
    [CrossRef] [PubMed]
  32. H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
    [CrossRef] [PubMed]
  33. W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
    [CrossRef] [PubMed]

2010

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

2009

2008

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

2007

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
[CrossRef] [PubMed]

2006

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
[CrossRef] [PubMed]

2005

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005).
[CrossRef] [PubMed]

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
[CrossRef] [PubMed]

2004

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

2003

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

2002

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
[CrossRef]

2001

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001).
[CrossRef] [PubMed]

L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001).
[CrossRef]

J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001).
[CrossRef]

2000

1998

L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998).
[CrossRef]

L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998).
[CrossRef]

1997

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

1996

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996).
[CrossRef]

1994

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

1991

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
[CrossRef]

1987

Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987).
[CrossRef]

1972

S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Arie, A.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

Aschieri, P.

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005).
[CrossRef] [PubMed]

Aumentado, J.

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

Barak, A.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
[CrossRef] [PubMed]

Bartal, G.

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
[CrossRef] [PubMed]

Buljan, H.

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
[CrossRef] [PubMed]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

Carmon, T.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

Carr, L. D.

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

Carretero-González, R.

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
[CrossRef] [PubMed]

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

Christodoulides, D. N.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

Cohen, O.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Coskun, T. H.

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

Della Valle, G.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

Deutsch, M.

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
[CrossRef]

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

Dreisow, F.

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

Fajans, J.

J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

Friedland, L.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
[CrossRef] [PubMed]

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001).
[CrossRef]

J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001).
[CrossRef]

L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000).
[CrossRef] [PubMed]

L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998).
[CrossRef]

L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998).
[CrossRef]

Golub, J. E.

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
[CrossRef]

Grandpierre, A. G.

Haelterman, M.

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001).
[CrossRef] [PubMed]

Heinrich, M.

Jensen, M. H.

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
[CrossRef] [PubMed]

Kalinikos, B. A.

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

Kapteyn, H. C.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Keil, R.

Lahini, Y.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

Lamhot, Y.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
[CrossRef] [PubMed]

Laporta, P.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

Longhi, S.

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34(16), 2405–2407 (2009).
[CrossRef] [PubMed]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

Lytle, A. L.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Manela, O.

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
[CrossRef] [PubMed]

Meerson, B.

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
[CrossRef]

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

Montes, C.

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
[CrossRef]

Morandotti, R.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

Murnane, M. M.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Naaman, O.

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

Nicolin, A. I.

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
[CrossRef] [PubMed]

Nolte, S.

Ornigotti, M.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

Oron, D.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

Picozzi, A.

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001).
[CrossRef] [PubMed]

Popmintchev, T.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Pozzi, F.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

Schwartz, T.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

Segev, M.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
[CrossRef] [PubMed]

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
[CrossRef] [PubMed]

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
[CrossRef] [PubMed]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

Shagalov, A. G.

L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000).
[CrossRef] [PubMed]

L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998).
[CrossRef]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

Shih, M.-

Siber, A.

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

Siddiqi, I.

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

Silberberg, Y.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987).
[CrossRef]

Soljacic, M.

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

Somekh, S.

S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Sorel, M.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

Stegeman, G. I.

Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987).
[CrossRef]

Suchowski, H.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

Szameit, A.

Tong, W.

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

Tünnermann, A.

Valley, G. C.

M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

Vardi, A.

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
[CrossRef] [PubMed]

Wu, M.

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

Wurtele, J. S.

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Zhang, X.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Am. J. Phys.

J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001).
[CrossRef]

Appl. Phys. Lett.

Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987).
[CrossRef]

S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Astrophys. J.

L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Nature

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Fluids B

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991).
[CrossRef]

Phys. Rev.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007).
[CrossRef]

Phys. Rev. A

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002).
[CrossRef]

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998).
[CrossRef]

Phys. Rev. E. Rapid Communication

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

Phys. Rev. Lett.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001).
[CrossRef] [PubMed]

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005).
[CrossRef] [PubMed]

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010).
[CrossRef] [PubMed]

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994).
[CrossRef] [PubMed]

L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000).
[CrossRef] [PubMed]

L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998).
[CrossRef]

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008).
[CrossRef] [PubMed]

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009).
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006).
[CrossRef] [PubMed]

Other

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

M. S. Livingston, High-Energy Particle Accelerators (Interscience, New York, 1954).

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Figures (4)

Fig. 1
Fig. 1

(a) Linear refractive index profile. (b) Transverse profile of the refractive index at the input face [blue dashed line], along with the intensity profiles of five mutually-incoherent waves (solid lines), at the input face of the directional coupler. Each wave has different initial intensity. All the waves are launched into the right waveguide, where they form the incoherent beam.

Fig. 2
Fig. 2

(a) Evolution of the sum of the squares of the absolute amplitudes in the left [blue solid line] and in the right [red dashed line] waveguides. The inset shows the absolute value of the amplitudes in the right and left waveguides during propagation. As the system crosses the linear resonance [marked in vertical black dashed line], the wave amplitudes in the left waveguide increase, at the expense of the amplitudes in the right waveguide. (b) Propagation of the sum of the population differences [blue solid line], and the theoretical curve, m | c R , m ( 0 ) | 2 R m Λ 0 z / χ , [red circles]. The efficient amplification results in the flip of the population difference from −1 to 1. (c) Evolution of the population difference for each wave. (d) Evolution of the phase mismatch along z, for each wave. In (c) and (d) the plots are slightly diverted, to demonstrate that the dynamics of all waves is identical.

Fig. 3
Fig. 3

(a) Propagation of the total intensity of the incoherent beam below the threshold for autoresonant phase locking. (b) Evolution of the intensity and (c) the phase mismatch of each wave below the threshold. As the system crosses the linear resonance [marked in vertical black dashed line], the phases do not lock and the power transfer is inefficient. (c) Evolution of the population difference below threshold. Since the coupling is inefficient, the population differences do not flip from −1 to 1. (e) Propagation dynamics of the total intensity of the incoherent beam, above the threshold for autoresonant phase locking. Now, as the system crosses the linear resonance [vertical white dashed line], the phases lock and the waveguides efficiently exchange power.

Fig. 4
Fig. 4

Experimental results, displaying beam profiles taken at the exit face of the directional coupler. (a) Total intensity below [blue solid line] and above [red dashed line] the threshold. The threshold parameter is controlled by only one of the (mutually-uncorrelated) waves. Above the threshold, sharply, all the power transfers to left waveguide. (b) Each wave below the threshold. (c) Each wave above the threshold. (b) and (c) were obtained by decreasing the intensity of the medium intensity wave [green dashed line] only. The same result is obtained by varying the intensity of the other waves. This shows that the dynamics is indeed collective.

Equations (8)

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

    i ψ / z = 2 ψ / x 2 + [ Δ n L ( x , z ) + Δ n N L ( x , z ) ]     ψ     .
    i ϕ n / z = 2 ϕ n / x 2 + [ Δ n L ( x , z ) + Δ n N L ( x , z ) ]     ϕ n     .
i d c R , n d z = κ c L , n + χ ( m | c R , m | 2 ) c R , n + Λ 0 c R , n z ,
i d c L , n d z = κ c R , n + χ ( m | c L , m | 2 ) c L , n ,
d R n d z = 2 κ 1 R n 2 sin ( Φ n ) ,
d Φ n d z = Λ 0 z χ ( m | c R , m ( 0 ) | 2 R m ) + 2 κ R n 1 R n 2 cos ( Φ n ) .
d R d z = 2 κ 1 R 2 sin ( Φ ) ,
d Φ d z = Λ 0 z χ I 0 R + 2 κ R 1 R 2 cos ( Φ ) .

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