## Gap random-phase lattice solitons

Optics Express, Vol. 13, Issue 13, pp. 5013-5023 (2005)

http://dx.doi.org/10.1364/OPEX.13.005013

Acrobat PDF (536 KB)

### 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

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]

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, 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]

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. 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]

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]

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]

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. 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. 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]

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

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]

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]

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]

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]

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]

**433**, 500 (2005). [CrossRef]

**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]

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]

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

## 2. Description of the physical system

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]

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]

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

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]

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]

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]

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

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]

*E*(

*x*,

*z*,

*t*) of such a beam randomly fluctuates. The characteristic time of these fluctuations (the coherence time) τ

_{c}, is much smaller than the response time of the nonlinearity τ

_{m}[24

**77**, 490–493 (1996). [CrossRef] [PubMed]

**79**, 4990–4993 (1997). [CrossRef]

*ψ*

_{m}(

*x*,

*z*), and associated modal weights

*d*

_{m}. The electric field is

*E*(

*x*,

*z*,

*t*)=∑

_{m}

*c*

_{m}(

*t*)

*ψ*

_{m}(

*x*,

*z*) [26

**79**, 4990–4993 (1997). [CrossRef]

*c*

_{m}(

*t*) are randomly fluctuating coefficients such that 〈

*c*

_{m}(

*t*)

*t*)〉=

*d*

_{m}

*δ*

_{mm}

_{′}; brackets 〈…〉 denote the time-average over the response time

*τ*

_{m}. The statistical properties of the incoherent light are contained within the mutual coherence function [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]

*B*(

*x*

_{1},

*x*

_{2},

*z*)=∑

_{m}

*d*

_{m}

*ψ*

_{m}(

*x*

_{1},

*z*)

*ψ**

_{m}(

*x*

_{2},

*z*). The time-averaged intensity is

*I*(

*x*,

*z*)=

*B*(

*x*,

*x*,

*z*)=∑

_{m}

*d*

_{m}|

*ψ*

_{m}(

*x*,

*z*)|

^{2}. The evolution of the coherent waves

*ψ*

_{m}, and hence the evolution of correlation function

*B*(

*x*

_{1},

*x*

_{2},

*z*), is governed by a set of coupled nonlinear wave equations [26

**79**, 4990–4993 (1997). [CrossRef]

*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.

*ψ*

_{m}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:

*u*

_{n,l}(

*x*) are orthonormal (real) eigenfunctions, and

*κ*

_{n,l}are the real eigenvalues of the defect modes [19

**92**, 223901 (2004). [CrossRef] [PubMed]

*V*(

*x*)=

*p*(

*x*)+

*δn*(∑

_{n,l}

*d*

_{n,l}|

*u*

_{n,l}(

*x*)|

^{2}); the index

*n*=1,2,… in the notation for the eigen-modes

*u*

_{n,l}denotes that the propagation constant

*κ*

_{n,l}is in the gap below the

*n*th band, while the index l describes the hierarchy within a single gap: if

*l*<

*l*

^{′}then

*κ*

_{n,l}<

*κ*

_{n,l}0. A gap-RPLS may be formed from several defect modes from the first gap, several modes from the second gap, and so forth. The number of modes that can be excited in different gaps, and the modal weights

*d*

_{n,l}corresponding to these modes, depend on the parameters of the lattice and the nonlinearity. We solve Eq. (2) self-consistently by using the iterative procedure used for calculating multi-gap solitons [13

**91**, 113901 (2003). [CrossRef] [PubMed]

## 3. Results

*δn*(

*I*)=-

*γ*

*I*/(1+

*I*/

*I*

_{S}). The parameters are

*γIS*=0.00015,

*n*

_{0}=2.3,

*k*=2

*πn*

_{0}/

*λ*, where

*λ*=488nm; the lattice is of the form

*p*(

*x*)=

*p*

_{0}∑

_{i}exp(-((

*x*-

*iD*)/

*x*

_{0})

^{8}), where

*x*

_{0}=3.7

*µ*m,

*D*=10

*µ*m, and

*p*

_{0}=6·10

^{-4}. Fig. 1(a) shows the band-gap structure of the periodic system, and the propagation constants of the defect modes that comprise the gap-RPLS. The gap-RPLS consists of 11 coherent waves (defect modes). Eight of these modes originate from the first band, two from the second, and one from the third band. The propagation constants of these (induced) defect modes are (via the self-defocusing nonlinearity) pushed into the gaps below the first, second, and third band, respectively [see Fig. 1(a)]. The modes with larger propagation constants within each gap contain less power. The distribution of power among the modes residing within the first gap is (3.35,4.89,6.78,8.93,11.20,13.36,15.15,16.34)%, among the second-gap modes it is (4.14,13.86)%, and also there is 2% of the total power in the third-gap mode. The total power within the gap-RPLS is 11.0

*I*

_{S}

*D*. In contrast to the RPLSs in the self-focusing medium [19

**92**, 223901 (2004). [CrossRef] [PubMed]

**433**, 500 (2005). [CrossRef]

*u*

_{n,l}is supported only within the nth Brillouin zone [19

**92**, 223901 (2004). [CrossRef] [PubMed]

*u*

_{1,l}are orthogonal to the FB waves from the 2nd band, and so on.

*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

**92**, 223901 (2004). [CrossRef] [PubMed]

*d*

_{n,l}) to obtain different soliton solutions. For example, consider the case where all parameters are identical to those of the previous example, but that the distribution of power within coherent waves is different: the power within the eight modes with propagation constants in the first gap is (9.94,9.96,9.98,10.00,10.01,10.03,10.03,10.04)%; the power distribution within the second gap modes is (8.97,9.03)%, and there is 0.02% of the total power within the third gap mode. The propagation constants and the power spectra of this gap-RPLS example are shown in Fig. 4, while its coherence properties and intensity structure are illustrated in Fig. 5. The Floquet-Bloch power spectrum is again multi-humped with the peaks of the humps in the anomalous diffraction regions, as in the previous example. However, the intensity structure now has a dip in the very center of the soliton, while the peaks of the Fourier power spectrum within the 1st Brillouin zone (BZ) are shifted from the edges of the 1st BZ towards its center. The differences between the two examples stem from a different choice of the modal populations; the modes with Fourier power spectrum closer to

*k*

_{x}=0 have more power in this gap-RPLS example. Figures 6(d),(e), and (f) show the stable stationary propagation of the intensity profile, the Fourier, and Floquet-Bloch power spectrum of the second example of a gap-RPLS.

**433**, 500 (2005). [CrossRef]

*I*

_{0}(

*x*)=exp[-((

*x*+0.35

*D*)/3

*D*)

^{8}] is the intensity structure of the beam;

*k*

_{x}corresponds to the transverse momentum of the

*k*

_{x}th coherent wave, while

*G*(

*k*

_{x})∝exp[-(

*k*

_{x}

*D*/3.5

*π*)

^{2}] expresses the relative power within the

*k*

_{x}th coherent wave. The beam enters the waveguide array at the angle

*θ*=

*K*/

*k*=0.7

*π*/

*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.

*z*=50 mm of propagation the intensity structure has the same width, and the output power spectra match the power spectra of the first gap-RPLS example [see Fig. 2(b) and Figs. 1(b) and (c)]. For different initial conditions, i.e., for an incoherent beam that has all parameters identical as in the previous case, but with a slightly broader initial intensity structure [

*I*

_{0}(

*x*)=exp[-((

*x*+0.35

*D*)/3.5

*D*)

^{8}]] the beam evolves in its structure, becoming similar to our second example of a gap-RPLS. Figures 6(d),(e), and (f) show the evolution of the intensity profile, the Fourier, and Floquet-Bloch power spectra, respectively, of this incoherent beam. Again, we observe some radiation escaping during the initial stages of evolution and reshaping of the beam spectra. The initial beam with such a simple structure evolves into an incoherent beam with multi-humped power spectra, with humps in the anomalous diffraction regions [see Figs. 4(b) and (c), and Figs. 6(e) and (f)].

## 4. Discussion

*d*

_{n,l}of the gap-RPLS. By setting the modal weights

*d*

_{n,l}, one may seek for gap-RPLS solutions with several modes from the 1st gap, some from the 2nd gap, and so forth. It should be emphasized that even though there are many degrees of freedom when calculating gap-RPLSs, random-phase lattice solitons do not exist for any arbitrary choice of modal weights

*d*

_{n,l}. Given the parameters of the lattice and the nonlinearity, all possible sets of values

*d*

_{n,l}for which gap-RPLSs exist define the existence range of gap-RPLSs. The notion of the existence range of incoherent solitons is described in Ref. [26

**79**, 4990–4993 (1997). [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]

## 5. Conclusion

## References and links

1. | D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) |

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. |

3. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. |

4. | D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

5. | Yu. Kivshar, “Self-localization in arrays of de-focusing waveguides,” Opt. Lett. |

6. | J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media”, Opt. Lett. |

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. |

8. | S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity”, JETP |

9. | H.S. Eisenberg, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “Diffraction menagement,” Phys. Rev. Lett. |

10. | R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-Focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. |

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. |

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) |

13. | O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Multiband vector lattice solitons” Phys. Rev. Lett. |

14. | A.A. Sukhorukov and Y.S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. |

15. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings”, Opt. Lett. |

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. |

17. | D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. |

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. |

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. |

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) |

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. |

23. | K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, “Incoherent multi-gap optical solitons in nonlinear photonic lattices,” Optics Express |

24. | M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. |

25. | D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. |

26. | M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. |

27. | Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science |

28. | V.V. Shkunov and D. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. |

29. | A.W. Snyder and D.J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. |

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. |

31. | D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science |

32. | S.A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E |

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 |

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 |

35. | H. Buljan, M. Segev, M. Soljačić, N.K. Efremidis, and D.N. Christodoulides, “White-light solitons,” Opt. Lett. |

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. |

38. | M. Segev and D.N. Christodoulides, |

**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

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- Yu. Kivshar, �??Self-localization in arrays of de-focusing waveguides,�?? Opt. Lett. 18, 1147-1149 (1993). [CrossRef] [PubMed]
- J. Feng, �??Alternative scheme for studying gap solitons in infinite periodic Kerr media�??, Opt. Lett. 20, 1302-1304 (1993). [CrossRef]
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- 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]
- 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]
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- A.A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, andW. Krolikowski, �??Spatial solitons in optically induced gratings�??, Opt. Lett. 28, 710-712 (2003) [CrossRef] [PubMed]
- 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]
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- 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]
- 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]
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- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A.W. Snyder and D.J. Mitchell, �??Big incoherent solitons,�?? Phys. Rev. Lett. 80, 1422-1425 (1998). [CrossRef]
- 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]
- 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]
- S.A. Ponomarenko, �??Twisted Gaussian Schell-model solitons,�?? Phys. Rev. E 64, 036618 (2001). [CrossRef]
- 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]
- 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]
- 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]
- T. Schwartz, T. Carmon, H. Buljan, and M. Segev, �??Spontaneous pattern formation with incoherent white light,�?? Phys. Rev. Lett. 93, (2004). [CrossRef] [PubMed]
- A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, �??Incoherent solitons in instantaneous response nonlinear media,�?? Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]
- 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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