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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 13 — Jun. 27, 2005
  • pp: 5013–5023
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Gap random-phase lattice solitons

Robert Pezer, Hrvoje Buljan, JasonW. Fleischer, Guy Bartal, Oren Cohen, and Mordechai Segev  »View Author Affiliations


Optics Express, Vol. 13, Issue 13, pp. 5013-5023 (2005)
http://dx.doi.org/10.1364/OPEX.13.005013


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Abstract

We theoretically study gap random-phase lattice solitons (gap-RPLSs) in nonlinear waveguide arrays with self-defocusing nonlinearity. We find that the intensity structure and statistical (coherence) properties of gap-RPLSs conform to the lattice periodicity, while their Floquet-Bloch power spectrum is multi-humped with peaks in the anomalous diffractions regions. It is shown that a gap-RPLS can be generated when a simple incoherent beam with bell-shaped power spectrum and single-hump intensity is launched at a proper angle into the waveguide array. The input incoherent beam evolves in the lattice while shedding off some radiation, and eventually attains the features of gap-RPLS.

© 2005 Optical Society of America

1. Introduction

The behavior of light in photonic lattices is driven by interference, which crucially depends on the coherence of light. This fact motivates exploring the propagation of partially coherent light in nonlinear photonic lattices. Among the nonlinear phenomena in nonlinear photonic lattices, of particular interest are lattice solitons [1

1. D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424817–823 (2003). [CrossRef]

, 2

2. Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981).

, 3

3. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987). [CrossRef] [PubMed]

, 4

4. D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

, 5

5. Yu. Kivshar, “Self-localization in arrays of de-focusing waveguides,” Opt. Lett. 18, 1147–1149 (1993). [CrossRef] [PubMed]

, 6

6. J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media”, Opt. Lett. 20, 1302–1304 (1993). [CrossRef]

, 7

7. H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998). [CrossRef]

, 8

8. S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity”, JETP 86, 682–686 (1998). [CrossRef]

, 9

9. H.S. Eisenberg, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “Diffraction menagement,” Phys. Rev. Lett. 85, 1863–1866 (2000) [CrossRef] [PubMed]

, 10

10. R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-Focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 863296 (2001). [CrossRef] [PubMed]

, 11

11. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of discrete solitons in optically-induced real time waveguide arrays” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]

, 12

12. 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 (London) 422, 147 (2003). [CrossRef]

, 13

13. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Multiband vector lattice solitons” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

, 14

14. A.A. Sukhorukov and Y.S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]

, 15

15. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings”, Opt. Lett. 28, 710–712 (2003) [CrossRef] [PubMed]

, 16

16. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003). [CrossRef] [PubMed]

, 17

17. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92, 093904 (2004). [CrossRef] [PubMed]

, 18

18. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 4 (2004). [CrossRef]

, 19

19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

, 20

20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

], such as gap solitons [2

2. Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981).

, 3

3. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987). [CrossRef] [PubMed]

, 5

5. Yu. Kivshar, “Self-localization in arrays of de-focusing waveguides,” Opt. Lett. 18, 1147–1149 (1993). [CrossRef] [PubMed]

, 6

6. J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media”, Opt. Lett. 20, 1302–1304 (1993). [CrossRef]

, 10

10. R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-Focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 863296 (2001). [CrossRef] [PubMed]

, 11

11. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of discrete solitons in optically-induced real time waveguide arrays” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]

, 17

17. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92, 093904 (2004). [CrossRef] [PubMed]

, 18

18. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 4 (2004). [CrossRef]

], discrete solitons [4

4. D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

, 7

7. H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998). [CrossRef]

, 12

12. 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 (London) 422, 147 (2003). [CrossRef]

], dipole-like (“twisted”) solitons [8

8. S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity”, JETP 86, 682–686 (1998). [CrossRef]

, 15

15. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings”, Opt. Lett. 28, 710–712 (2003) [CrossRef] [PubMed]

], multi-band vector solitons [13

13. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Multiband vector lattice solitons” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

, 14

14. A.A. Sukhorukov and Y.S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]

], and so forth. Until recently, such self-localized wavepackets were studied solely with coherent light. This situation has been changed by the prediction [19

19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

] and experimental observation [20

20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

] of random-phase lattice solitons (RPLSs). These studies have shown that the interplay of statistical (coherence) properties of the light, nonlinearity, and lattice structure, determines the properties of RPLSs. More specifically, a partially incoherent wavepacket can excite modes across many Brillouin zones, among multiple bands, whose dynamics is determined by the degree of coherence, the sign and strength of the nonlinearity, and the underlying curvature of the bands. For example, a self-focusing (self-defocusing) nonlinearity can localize modes arising from normal (anomalous) diffraction regions of the Brillouin zones, resulting in a multi-humped Floquet-Bloch power spectrum of RPLS [19

19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

]. The propagation constants of the randomly-excited modes comprising an RPLS are located in the gap(s) of the spectrum of the linear system (the fact that self-focusing (defocusing) nonlinearity can jointly self-trap multiple modes arising from normal (anomalous) diffraction regions was shown in the context of multi-mode multi-band vector lattice solitons [13

13. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Multiband vector lattice solitons” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

]). Particularly interesting is the dynamics of partially coherent light associated with the generation of RPLSs observed in Ref. [20

20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

]. A “simple” incoherent beam with a bell-shaped intensity structure and single-humped power spectrum was launched at a normal angle into the nonlinear waveguide array and dynamically evolved into a RPLS with a multi-humped power spectrum. Thus, the generation of RPLSs did not require engineering of the input beam to match the RPLS properties [20

20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

]. Instead, it occurred due to the energy transfer between the modes of the linear system (the Floquet-Bloch waves) induced by the self-focusing nonlinearity [20

20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

, 21

21. H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, and D.N. Christodoulides, “Partially coherent waves in nonlinear periodic lattices,” Stud. Appl. Math., in print (2005). [CrossRef]

]. Further exploration of incoherent light dynamics in (self-focusing and defocusing) nonlinear photonic lattices have lead to the technique for Brillouin-zone spectroscopy of such lattices [22

22. G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of nonlinear photonic lattices,” Phys. Rev. Lett. 94, 163902 (2005). [CrossRef] [PubMed]

]. Recently, it was numerically shown that incoherent gap solitons can be efficiently generated in a self-defocusing medium by engineering the input excitation [23

23. K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, “Incoherent multi-gap optical solitons in nonlinear photonic lattices,” Optics Express 132916 (2005). [CrossRef] [PubMed]

]. In that scheme, two properly constructed incoherent beams are launched with opposite angles into the waveguide array [23

23. K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, “Incoherent multi-gap optical solitons in nonlinear photonic lattices,” Optics Express 132916 (2005). [CrossRef] [PubMed]

] (equivalent input geometry was suggested by Feng [6

6. J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media”, Opt. Lett. 20, 1302–1304 (1993). [CrossRef]

] for the excitation of coherent gap solitons).

Here we pursuit a different avenue, and investigate the possibility of attaining gap random-phase lattice solitons (gap-RPLSs) in experimental settings where dynamics in a (1+1)D nonlinear waveguide array naturally evolves an incoherent input beam into a beam with gap-RPLS structure. As a first step, we find gap-RPLSs and identify their features: intensity structure and coherence properties (expressed through the complex coherence factor) both conform to the lattice periodicity, while their Floquet-Bloch power spectrum is multi-humped, with humps being located mainly in the anomalous diffraction regions. Finally, we find that a “simple” incoherent beam with a bell-shaped intensity structure and a singly-humped power spectrum, when launched at a proper angle into a nonlinear waveguide array, naturally evolves (under proper self-defocusing conditions) into a beam with gap-RPLS properties.

2. Description of the physical system

Before analyzing gap-RPLSs, let us recall some concepts of incoherent light propagation in homogeneous nonlinear media, which have been extensively studied since 1996 [24

24. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [CrossRef] [PubMed]

, 25

25. D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997). [CrossRef]

, 26

26. M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997). [CrossRef]

, 27

27. Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998). [CrossRef] [PubMed]

, 28

28. V.V. Shkunov and D. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998). [CrossRef]

, 29

29. A.W. Snyder and D.J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1425 (1998). [CrossRef]

, 30

30. M. Soljačić, M. Segev, T.H. Coskun, D.N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000). [CrossRef] [PubMed]

, 31

31. D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000). [CrossRef] [PubMed]

, 32

32. S.A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001). [CrossRef]

, 33

33. Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” P. Natl. Acad. Sci. USA 99, 5223–5227 (2002). [CrossRef]

, 34

34. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002). [CrossRef]

, 35

35. H. Buljan, M. Segev, M. Soljačić, N.K. Efremidis, and D.N. Christodoulides, “White-light solitons,” Opt. Lett. 28, 1239–1241 (2003). [CrossRef] [PubMed]

, 36

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

, 37

37. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]

, 38

38. M. Segev and D.N. Christodoulides, Incoherent Solitons in Spatial Solitons, S. Trillo and W. Torruellas eds. (Springer, Berlin, 2001) pp. 87–125.

]. Of particular interest are the phenomena of incoherent solitons [24

24. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [CrossRef] [PubMed]

, 25

25. D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997). [CrossRef]

, 26

26. M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997). [CrossRef]

, 27

27. Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998). [CrossRef] [PubMed]

, 28

28. V.V. Shkunov and D. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998). [CrossRef]

, 29

29. A.W. Snyder and D.J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1425 (1998). [CrossRef]

, 32

32. S.A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001). [CrossRef]

, 35

35. H. Buljan, M. Segev, M. Soljačić, N.K. Efremidis, and D.N. Christodoulides, “White-light solitons,” Opt. Lett. 28, 1239–1241 (2003). [CrossRef] [PubMed]

, 37

37. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]

], and modulation instability with incoherent light [30

30. M. Soljačić, M. Segev, T.H. Coskun, D.N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000). [CrossRef] [PubMed]

, 31

31. D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000). [CrossRef] [PubMed]

, 33

33. Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” P. Natl. Acad. Sci. USA 99, 5223–5227 (2002). [CrossRef]

, 34

34. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002). [CrossRef]

, 36

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

]. Incoherent solitons were found [24

24. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [CrossRef] [PubMed]

] in noninstantaneous nonlinear medium, where an incoherent beam with a randomly fluctuating field induces a smooth multi-mode waveguide and populates (on the average) its modes self-consistently [26

26. M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997). [CrossRef]

, 38

38. M. Segev and D.N. Christodoulides, Incoherent Solitons in Spatial Solitons, S. Trillo and W. Torruellas eds. (Springer, Berlin, 2001) pp. 87–125.

].

iψmz+12k2ψmx2+V(x,z)kn0ψm(x,z)=0.
(1)

The potential V(x,z)=p(x)+δn(I(x,z)) contains both the periodic p(x)=p(x+D), and the nonlinear term δn(I(x,z)); the nonlinearity is of the self-defocusing type, i.e., ∂δn(I)=∂I<0.

Spatial solitons occur in our system when the diffraction of a spatially localized incoherent beam is exactly balanced by nonlinear self-defocusing. This exact balance happens when the self-consistency principle holds: the incoherent beam induces (via the nonlinearity) a defect in the photonic lattice, which has several localized defect modes; the coherent waves ψm of the incoherent beam are the defect modes themselves. The eigenvalues of the localized defect modes reside in the gaps of the spectrum of the linear system. For this reason it is convenient to change the notation for the coherent waves: ψmψn,l=un,l(x)eiκn,lz , where u n,l(x) are orthonormal (real) eigenfunctions, and κ n,l are the real eigenvalues of the defect modes [19

19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

],

12kd2un,ldx2+V(x)kn0un,1(x,z)=κn,lun,l,
(2)

3. Results

Fig. 1. The band-gap structure of the lattice, and the power spectra of the first gap-RPLS example. (a) Band-gap structure of the lattice (solid lines), and propagation constants of the RPLS modes (closed circles). (b) Fourier power spectrum of a gap-RPLS in the self-defocusing nonlinear waveguide array (solid line); the humps in the spectrum are located mainly in the anomalous diffraction regions. The Fourier power spectrum of an input incoherent beam launched at an angle is indicated by the dot-dashed line, and spectrum of the output beam is given by the dashed line (the dot-dashed line and the dashed line almost fully coincide). (c) Same as in (b) but with the Floquet-Bloch power spectrum of the beams. See text for details.

The intensity profile of this gap-RPLS example, and its complex coherence factor μ(x1,x2)=B(x1,x1)B(x1,x2)B(x2,x2) are shown in Fig. 2. The intensity is located mainly on sites, but there is also a non-negligible amount of power in the interstitial regions [solid line in Fig. 2(b)]. From Fig. 2(a), i.e., from the coherence factor µ(x 1,x 2), we read the coherence properties of the gap-RPLS. For RPLSs, µ(x 1,x 2)∊[-1,1], where µ(x 1,x 2)=0 corresponds to zero correlation between the fields at points x 1 and x 2, whereas µ(x 1,x 2)=1,-1 corresponds to full correlation, where the field is in phase and π-out of phase, respectively. From Fig. 2(a) we see that coherence conforms to the periodicity of the lattice. The coherence increases as we move away from the soliton region, which is shown as a (high visibility) black (µ=-1) and white (µ=1) checkerboard. The structure of the black and white squares at the soliton tails depends on the most slowly decaying mode(s). Most of the grey shaded regions (µ≈0) (low correlation) are within the soliton region, where we confirm the validity of the approximate law µ(x 1,x 2)≈µ(x 1+D,x 2+D) [19

19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

]. The stability of our gap-RPLS example is checked numerically by evolving it for many diffraction lengths with small initial noise on top of the gap-RPLS; Figs. 3(a),(b), and (c) show the evolution of the intensity profile and the power spectra of the gap-RPLS, while Fig. 2(b) shows the diffraction-broadening of an input beam whose wavefunction is identical to that of the gap-RPLS, when it evolves in the lattice without the nonlinearity.

Fig. 2. The intensity structure and coherence properties of the first example of a gap-RPLS. (a) The complex coherence factor µ(x,x )∊[-1,1] of the gap-RPLS; black (white) corresponds to a value of -1 (1). (b) The intensity profile I(x)/IS (solid line) of a gap-RPLS. The intensity structure of an incoherent beam launched at angle θ=0.7π/Dk, and propagated for 50 mm (dashed line). Dotted line illustrates the diffraction-broadened beam after z=30 mm propagation in a linear lattice, for an input beam with a wavefunction identical to that of a gap-RPLS. The vertical lines represent the lattice sites.

Fig. 3. The stable evolution of the (a) intensity structure, (b) Fourier power spectrum, and (c) Floquet-Bloch power spectrum of the first gap-RPLS example. The evolution of an incoherent beam launched at an angle θ=0.7π/Dk into the nonlinear waveguide array; (d) intensity structure, (e) Fourier power spectrum, and (f) Floquet-Bloch power spectrum.

ψkx(x,z=0)=I0(x)G(kx)eikxxeiKx,

where I 0(x)=exp[-((x+0.35D)/3D)8] is the intensity structure of the beam; kx corresponds to the transverse momentum of the kx th coherent wave, while G(kx )∝exp[-(kxD/3.5π)2] expresses the relative power within the kx th coherent wave. The beam enters the waveguide array at the angle θ=K/k=0.7π/Dk. Figures 3(d),(e), and (f) show the evolution of the intensity profile, the Fourier, and Floquet-Bloch power spectrum of such a simple beam, and its evolution into a gap-RPLS.

Fig. 4. Same as in Figure 1 for the second example of a gap-RPLS. See text for details.

Fig. 5. Same as Figure 2 for the second example of a gap-RPLS. See text.

4. Discussion

Fig. 6. Same as Figure 3 for the second example of a gap-RPLS. See text for details.

We may think of gap-RPLSs as solitons obtain by localizing (self-consistently) a number of FB waves by pushing their propagation constants into the gaps (through the action of the self-defocusing nonlinearity). The FB waves that are most easily pushed into the gap are those whose propagation constants are the closest to the gap, i.e., at the bottom of a band; these modes are from the edges of the BZs at the anomalous diffraction regions. That is, the modes that are most easily localized in the gap have their FB power spectrum mainly in the anomalous diffraction regions. This is underpinned by the fact that anomalous diffraction is balanced by the self-defocusing nonlinearity, and it reflects onto the FB power spectra of gap-RPLSs [see Figs. 4(c) and 4(c)]

Let us discuss the existence range(s) and features of gap-RPLSs. For lattices with narrower gaps (e.g., for lattices that are more shallow), intuition suggests that the existence range of gap-RPLSs is smaller than for lattices with broader gaps. Namely, one may excite (localize) more defect modes within a broader gap. Following the intuition that modes with FB power spectrum closer to the edges of the BZs in the anomalous diffraction regions are more easily localized with self-defocusing nonlinearity, it follows that for lattices with narrower gaps (smaller existence range), the multi-humped feature of the FB spectrum is expected to be more pronounced. For lattices with broad gaps (e.g. for deep lattices), the existence range is large. As more and more modes are being localized in a gap, the width of the FB power spectrum of the gap-RPLS within a single BZ broadens. Eventually, the FB power spectrum may fill the entire BZ [see Fig. 4(c)].

Before closing, let us add that a similar line of reasoning can be developed for the self-focusing type of nonlinearity, where modes can also be localized in the semi-infinite gap. The discussion here is restricted to (1+1)D systems, while (2+1)D systems certainly offer more possibilities and will definitely lead to exciting results. All of the above discussion points at the fact that partially incoherent light in nonlinear photonic lattices is a rich dynamical system worthy of further exploration. One example of this is a recently developed technique for Brillouin-zone spectroscopy of nonlinear photonic lattices [22

22. G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of nonlinear photonic lattices,” Phys. Rev. Lett. 94, 163902 (2005). [CrossRef] [PubMed]

], which utilizes partially coherent light.

5. Conclusion

In conclusion, we have theoretically studied gap random-phase solitons in self-defocusing nonlinear waveguide arrays (gap-RPLSs). We have found gap-RPLS solutions and identified their features, showing that both their intensity structure and their statistical (coherence) properties conform to the lattice periodicity, while their Floquet-Bloch power spectrum is multi-humped with peaks in the anomalous diffractions regions. We have shown that a self-trapped beam with such gap-RPLS power spectrum may be obtained naturally by launching an incoherent beam with a bell-shaped power spectrum and single-hump intensity, at a proper angle into a nonlinear waveguide array. The structure of this incoherent beam evolves into a gap-RPLS structure partially due to the energy exchange between the Floquet-Bloch waves induced by the self-defocusing nonlinearity, and partially by shedding off some radiation.

This work was supported by the German-Israeli DIP Project.

References and links

1.

D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424817–823 (2003). [CrossRef]

2.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981).

3.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987). [CrossRef] [PubMed]

4.

D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

5.

Yu. Kivshar, “Self-localization in arrays of de-focusing waveguides,” Opt. Lett. 18, 1147–1149 (1993). [CrossRef] [PubMed]

6.

J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media”, Opt. Lett. 20, 1302–1304 (1993). [CrossRef]

7.

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998). [CrossRef]

8.

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity”, JETP 86, 682–686 (1998). [CrossRef]

9.

H.S. Eisenberg, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “Diffraction menagement,” Phys. Rev. Lett. 85, 1863–1866 (2000) [CrossRef] [PubMed]

10.

R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-Focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 863296 (2001). [CrossRef] [PubMed]

11.

J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of discrete solitons in optically-induced real time waveguide arrays” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]

12.

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 (London) 422, 147 (2003). [CrossRef]

13.

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Multiband vector lattice solitons” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

14.

A.A. Sukhorukov and Y.S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]

15.

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings”, Opt. Lett. 28, 710–712 (2003) [CrossRef] [PubMed]

16.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003). [CrossRef] [PubMed]

17.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92, 093904 (2004). [CrossRef] [PubMed]

18.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 4 (2004). [CrossRef]

19.

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

20.

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef]

21.

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, and D.N. Christodoulides, “Partially coherent waves in nonlinear periodic lattices,” Stud. Appl. Math., in print (2005). [CrossRef]

22.

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of nonlinear photonic lattices,” Phys. Rev. Lett. 94, 163902 (2005). [CrossRef] [PubMed]

23.

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, “Incoherent multi-gap optical solitons in nonlinear photonic lattices,” Optics Express 132916 (2005). [CrossRef] [PubMed]

24.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [CrossRef] [PubMed]

25.

D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997). [CrossRef]

26.

M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997). [CrossRef]

27.

Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998). [CrossRef] [PubMed]

28.

V.V. Shkunov and D. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998). [CrossRef]

29.

A.W. Snyder and D.J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1425 (1998). [CrossRef]

30.

M. Soljačić, M. Segev, T.H. Coskun, D.N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000). [CrossRef] [PubMed]

31.

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000). [CrossRef] [PubMed]

32.

S.A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001). [CrossRef]

33.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” P. Natl. Acad. Sci. USA 99, 5223–5227 (2002). [CrossRef]

34.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002). [CrossRef]

35.

H. Buljan, M. Segev, M. Soljačić, N.K. Efremidis, and D.N. Christodoulides, “White-light solitons,” Opt. Lett. 28, 1239–1241 (2003). [CrossRef] [PubMed]

36.

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

37.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]

38.

M. Segev and D.N. Christodoulides, Incoherent Solitons in Spatial Solitons, S. Trillo and W. Torruellas eds. (Springer, Berlin, 2001) pp. 87–125.

OCIS Codes
(030.6600) Coherence and statistical optics : Statistical optics
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5940) Nonlinear optics : Self-action effects

ToC Category:
Research Papers

History
Original Manuscript: May 3, 2005
Revised Manuscript: June 15, 2005
Published: June 27, 2005

Citation
Robert Pezer, Hrvoje Buljan, Jason Fleischer, Guy Bartal, Oren Cohen, and Mordechai Segev, "Gap random-phase lattice solitons," Opt. Express 13, 5013-5023 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-5013


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References

  1. D.N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behaviour in linear and nonlinear waveguide lattices,�?? Nature (London) 424 817-823 (2003). [CrossRef]
  2. Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, �??Stationary waves in nonlinear, periodically modulated media with higher group retardation,�?? Zh. Tekh. Fiz. 51, 902-907 (1981).
  3. W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical-response of superlattices,�?? Phys. Rev. Lett. 58, 160-163 (1987). [CrossRef] [PubMed]
  4. D.N. Christodoulides, and R.I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
  5. Yu. Kivshar, �??Self-localization in arrays of de-focusing waveguides,�?? Opt. Lett. 18, 1147-1149 (1993). [CrossRef] [PubMed]
  6. J. Feng, �??Alternative scheme for studying gap solitons in infinite periodic Kerr media�??, Opt. Lett. 20, 1302-1304 (1993). [CrossRef]
  7. H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383-3386 (1998). [CrossRef]
  8. S. Darmanyan, A. Kobyakov, and F. Lederer, �??Stability of strongly localized excitations in discrete media with cubic nonlinearity�??, JETP 86, 682-686 (1998). [CrossRef]
  9. H.S. Eisenberg, Y. Silberberg, R. Morandotti, and J.S. Aitchison, �??Diffraction menagement,�?? Phys. Rev. Lett. 85, 1863-1866 (2000) [CrossRef] [PubMed]
  10. R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, �??Self-Focusing and defocusing in waveguide arrays,�?? Phys. Rev. Lett. 86 3296 (2001). [CrossRef] [PubMed]
  11. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, �??Observation of discrete solitons in optically-induced real time waveguide arrays�?? Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
  12. 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 (London) 422, 147 (2003). [CrossRef]
  13. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, �??Multiband vector lattice solitons�?? Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]
  14. A.A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]
  15. D. Neshev, E. Ostrovskaya, Y. Kivshar, andW. Krolikowski, �??Spatial solitons in optically induced gratings�??, Opt. Lett. 28, 710-712 (2003) [CrossRef] [PubMed]
  16. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,�?? Phys. Rev. Lett. 90, 053902 (2003). [CrossRef] [PubMed]
  17. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904 (2004). [CrossRef] [PubMed]
  18. D. Neshev, A. A. Sukhorukov, B. Hanna,W. Krolikowski, and Yu. S. Kivshar, �??Controlled generation and steering of spatial gap solitons,�?? Phys. Rev. Lett. 93, 0839054 (2004). [CrossRef]
  19. H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, �??Random-phase solitons in nonlinear periodic lattices,�?? Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]
  20. O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, �??Observation of random-phase lattice solitons,�?? Nature (London) 433, 500 (2005). [CrossRef]
  21. H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, and D.N. Christodoulides, �??Partially coherent waves in nonlinear periodic lattices,�?? Stud. Appl. Math., in print (2005). [CrossRef]
  22. G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, �??Brillouin zone spectroscopy of nonlinear photonic lattices,�?? Phys. Rev. Lett. 94, 163902 (2005). [CrossRef] [PubMed]
  23. K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, �??Incoherent multi-gap optical solitons in nonlinear photonic lattices,�?? Optics Express 13 2916 (2005). [CrossRef] [PubMed]
  24. M. Mitchell, Z. Chen, M. Shih, and M. Segev, �??Self-trapping of partially spatially incoherent light,�?? Phys. Rev. Lett. 77, 490-493 (1996). [CrossRef] [PubMed]
  25. D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, �??Theory of incoherent self-focusing in biased photorefractive media,�?? Phys. Rev. Lett. 78, 646-649 (1997). [CrossRef]
  26. M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, �??Theory of self-trapped spatially incoherent light beams,�?? Phys. Rev. Lett. 79, 4990-4993 (1997). [CrossRef]
  27. Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, �??Self-trapping of dark incoherent light beams,�?? Science 280, 889-892 (1998). [CrossRef] [PubMed]
  28. V.V. Shkunov and D. Anderson, �??Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,�?? Phys. Rev. Lett. 81, 2683-2686 (1998). [CrossRef]
  29. A.W. Snyder and D.J. Mitchell, �??Big incoherent solitons,�?? Phys. Rev. Lett. 80, 1422-1425 (1998). [CrossRef]
  30. M. Solja¡�?i�?, M. Segev, T.H. Coskun, D.N. Christodoulides, and A. Vishwanath, �??Modulation instability of incoherent beams in noninstantaneous nonlinear media,�?? Phys. Rev. Lett. 84, 467-470 (2000). [CrossRef] [PubMed]
  31. D. Kip, M. Solja¡�?i�?, M. Segev, E. Eugenieva, and D. N. Christodoulides, �??Modulation instability and pattern formation in spatially incoherent light beams,�?? Science 290, 495-498 (2000). [CrossRef] [PubMed]
  32. S.A. Ponomarenko, �??Twisted Gaussian Schell-model solitons,�?? Phys. Rev. E 64, 036618 (2001). [CrossRef]
  33. Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, �??Clustering of solitons in weakly correlated wavefronts,�?? P. Natl. Acad. Sci. USA 99, 5223-5227 (2002). [CrossRef]
  34. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, �??Statistical theory for incoherent light propagation in nonlinear media,�?? Phys. Rev. E 65, 035602 (2002). [CrossRef]
  35. H. Buljan, M. Segev, M. Solja¡�?i�?, N.K. Efremidis, and D.N. Christodoulides, �??White-light solitons,�?? Opt. Lett. 28, 1239-1241 (2003). [CrossRef] [PubMed]
  36. T. Schwartz, T. Carmon, H. Buljan, and M. Segev, �??Spontaneous pattern formation with incoherent white light,�?? Phys. Rev. Lett. 93, (2004). [CrossRef] [PubMed]
  37. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, �??Incoherent solitons in instantaneous response nonlinear media,�?? Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]
  38. M. Segev and D.N. Christodoulides, Incoherent Solitons in Spatial Solitons, S. Trillo and W. Torruellas eds. (Springer, Berlin, 2001) pp. 87-125.

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