OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 17 — Aug. 26, 2013
  • pp: 19690–19700
« Show journal navigation

Self-trapped leaky waves in lattices: discrete and Bragg soleakons

Maxim Kozlov, Ofer Kfir, and Oren Cohen  »View Author Affiliations


Optics Express, Vol. 21, Issue 17, pp. 19690-19700 (2013)
http://dx.doi.org/10.1364/OE.21.019690


View Full Text Article

Acrobat PDF (1690 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We propose lattice soleakons: self-trapped waves that self-consistently populate slowly-attenuating leaky modes of their self-induced defects in periodic potentials. Two types, discrete and Bragg, lattice soleakons are predicted. Discrete soleakons that are supported by combination of self-focusing and self-defocusing nonlinearities propagate robustly for long propagation distances. They eventually abruptly disintegrate because they emit power to infinity at an increasing pace. In contrast, Bragg soleakons self-trap by only self-focusing nonlinearity. Also, they do not disintegrate because they emit power at a decreasing rate.

© 2013 OSA

1. Introduction

Self-trapped states in periodic systems (lattices) are ubiquitous in nature and play a fundamental role in many branches of science, such as solid state physics (localized modes in crystals and conducting polymer chains) [1

1. A. A. Ovchinnikov, “Localized long-lived vibrational states in molecular crystals,” Zh. Exp. Theor. Phys. 57, 263–270 (1969).

3

3. A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61(8), 970–973 (1988). [CrossRef] [PubMed]

], biology (energy transfer in protein α-helices) [4

4. A. S. Davydov, “The theory of contraction of proteins under their excitation,” J. Theor. Biol. 38(3), 559–569 (1973). [CrossRef] [PubMed]

], nonlinear optics (self-trapped beams and pulses of light in optical lattices) [5

5. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58(2), 160–163 (1987). [CrossRef] [PubMed]

10

10. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]

], mechanics (energy localization in oscillator arrays) [11

11. M. Sato, B. E. Hubbard, and A. J. Sievers, “Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays,” Rev. Mod. Phys. 78(1), 137–157 (2006). [CrossRef]

, 12

12. E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, “Intrinsic localized modes in parametrically driven arrays of nonlinear resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4), 046202 (2009). [CrossRef] [PubMed]

] and quantum mechanics (self-confined excitations in Josephson junction arrays and localized atomic Bose-Einstein condensates) [13

13. E. Trías, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in a josephson array,” Phys. Rev. Lett. 84(4), 741–744 (2000). [CrossRef] [PubMed]

15

15. B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. 92(23), 230401 (2004). [CrossRef] [PubMed]

]. Two types of self-trapped lattice states have been investigated: lattice solitons and lattice breathers. During evolution, the shape of lattice solitons is preserved while it oscillates in lattice breathers. Still, the wave-packets of both lattice solitons and lattice breathers exhibit exponential decay in the trapped directions. As a result, coherent interactions between lattice solitons or breathers are fundamentally short-range.

Self-trapped lattice waves can also be divided according to the location of their energies (or propagation constants) in the band structure. The linear modes of lattices are Floquet-Bloch waves, with their spectra divided into bands that are separated by gaps in which propagating modes do not exist [16

16. F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).

]. The eigenvalues of a self-trapped lattice state can reside in the semi-infinite gap, in which case it is often termed discrete soliton [2

2. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]

, 4

4. A. S. Davydov, “The theory of contraction of proteins under their excitation,” J. Theor. Biol. 38(3), 559–569 (1973). [CrossRef] [PubMed]

, 6

6. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]

] or discrete breather [1

1. A. A. Ovchinnikov, “Localized long-lived vibrational states in molecular crystals,” Zh. Exp. Theor. Phys. 57, 263–270 (1969).

, 3

3. A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61(8), 970–973 (1988). [CrossRef] [PubMed]

, 13

13. E. Trías, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in a josephson array,” Phys. Rev. Lett. 84(4), 741–744 (2000). [CrossRef] [PubMed]

], or in a gap between two bands, hence termed gap [5

5. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58(2), 160–163 (1987). [CrossRef] [PubMed]

, 8

8. J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18(16), 1302–1304 (1993). [CrossRef] [PubMed]

, 15

15. B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. 92(23), 230401 (2004). [CrossRef] [PubMed]

] or Bragg soliton [7

7. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62(15), 1746–1749 (1989). [CrossRef] [PubMed]

]. Notably, discrete and gap solitons often exhibit different properties because discrete solitons are trapped through total internal reflections whereas gap solitons are localized by Bragg reflections [10

10. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]

]. A prime example for a system in which self-localized lattice waves have been investigated experimentally is optical nonlinear waveguide arrays [17

17. 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(16), 3383–3386 (1998). [CrossRef]

24

24. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]

]. Discrete solitons [17

17. 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(16), 3383–3386 (1998). [CrossRef]

22

22. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29(13), 1530–1532 (2004). [CrossRef] [PubMed]

, 24

24. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]

], discrete breathers [25

25. S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5777–5782 (2000). [CrossRef] [PubMed]

], gap solitons [23

23. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92(9), 093904 (2004). [CrossRef] [PubMed]

, 24

24. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]

] and gap breathers [9

9. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90(25), 253902 (2003). [CrossRef] [PubMed]

], as well as more complicated structures such as vector lattice solitons [26

26. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003). [CrossRef] [PubMed]

, 27

27. A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91(11), 113902 (2003). [CrossRef] [PubMed]

] and incoherent lattice solitons [28

28. 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(22), 223901 (2004). [CrossRef] [PubMed]

, 29

29. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature 433(7025), 500–503 (2005). [CrossRef] [PubMed]

], have been explored in one and two dimensional arrays of waveguides.

Lattice solitons and lattice breathers have their counterparts in nonlinear homogeneous media. In homogeneous media, however, another type of self-localized particle-like waves was recently proposed: self-trapped leaky waves: soleakons [30

30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

]. A soleakon induces a waveguide through the nonlinearity and populates its leaky mode self-consistently. A numerical example of one-dimensional soleakons in a homogeneous medium was demonstrated in Ref. [30

30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

]. These soleakons display stable propagation, largely maintaining their intensity profiles for very long propagation distances (orders of magnitude larger than their diffraction lengths). They eventually disintegrate when their localized power decreases to a critical level. It was also shown that soleakons can exhibit coherent and resonance interaction with another faraway soleakon or with the continuum radiation – properties that do not exist in solitons and breathers. In that paper, the soleakon beam induced a double-barrier W structure waveguide which is known to support slowly attenuating leaky modes [31

31. N. Moiseyev, P. R. Certain, and F. Weinhold, “Resonance properties of complex-rotated hamiltonians,” Mol. Phys. 36(6), 1613–1630 (1978). [CrossRef]

]. The W-shape waveguide was self-induced by combination of nonlocal self-defocusing and local self-focusing nonlinearities. This combination can be realized in optics for example by using nonlocal thermal or molecular reorientational and simultaneously Kerr nonlinearities [32

32. H. C. Gurgov and O. Cohen, “Spatiotemporal pulse-train solitons,” Opt. Express 17(9), 7052–7058 (2009). [CrossRef] [PubMed]

34

34. O. Lahav, H. C. Gurgov, P. Sidorenko, O. Peleg, L. Levi, A. Fleischer, and O. Cohen, “Self-phase modulation spectral broadening in two-dimensional spatial solitons: toward three-dimensional spatiotemporal pulse-train solitons,” Opt. Lett. 37(24), 5196–5198 (2012). [CrossRef] [PubMed]

]. Beyond optics, Bose Einstein condensate can display simultaneous nonlocal nonlinearity through dipole-dipole interaction and local self-focusing by boson-boson scattering [35

35. S. Giovanazzi, A. Gorlitz, and T. Pfau, “Ballistic expansion of a dipolar condensate,” J. Opt. B 5(2), S208–S211 (2003). [CrossRef]

, 36

36. A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, “Comparing contact and dipolar interactions in a Bose-Einstein condensate,” Phys. Rev. Lett. 97(25), 250402 (2006). [CrossRef] [PubMed]

]. Still, the requirement for a proper superposition of wide negative and narrow positive nonlinearities is a major restricting factor in the experimental obtainability and impact of soleakons.

Here, we propose and demonstrate numerically two-dimensional soleakons that propagate in arrays of slab wave-guides. Two types of lattice soleakons are predicted: discrete soleakons and Bragg soleakons. Discrete soleakons are supported by combination of nonlocal defocusing and local focusing nonlinearities that jointly induce a ring-barrier wave-guide structure. This waveguide gives rise to slowly-decaying leaky modes that reside within the first band of the lattice transmission spectra. The leakage rate of discrete soleakons increases during propagation. Consequently, they eventually disintegrate abruptly, emitting all their power to delocalized radiation. The predicted Bragg soleakons are supported by local self-focusing nonlinearity only. This means that Bragg soleakons are much more universal than discrete soleakons and soleakons in homogeneous media that require combination of negative nonlocal and positive local nonlinearities. The leakage rate of Bragg soleakons decreases during propagation, hence, Bragg soleakons continue to propagate without disintegration.

The paper is organized as follows. The concept of leaky modes and soleakons is discussed in section 2. The linear transmission spectrum of array of slab wave-guides is analyzed in section 3. The model and methods that we used to find lattice soleakons are presented in section 4. The properties of the discrete and Bragg soleakons are described in section 5. Finally, in section 6 we conclude and suggest directions for future studies.

2. Leaky modes and soleakons

Before moving to lattice soleakons, it is instructive to explain the concept of leaky modes [37

37. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

] and soleakons in homogeneous media [30

30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

]. A leaky mode is localized in the vicinity of the waveguide, exhibiting monotonic decay for a finite distance in a transverse direction, and is oscillatory everywhere beyond that finite distance. During propagation, the localized power in a leaky mode gradually leaks out to the continuum at a constant rate that is given by twice the imaginary part of the mode complex propagation constant. This attenuation rate can be very small, yielding slowly-attenuating leaky modes. Interestingly, the real part of the propagation constant resides within a band of radiation modes. As such, leaky mode is resonant with corresponding radiation modes which comprise its spatial spectrum. In order to excite a leaky mode, one has to excite properly its localized section, which resembles a bound state. Because a leaky mode is not a true eigen-mode, the radiation modes comprising it (‘in-resonance’ radiation modes) dephase, hence radiation is constantly emitted away.

A soleakon induces a waveguide through the nonlinearity and populates its slowly-attenuating leaky mode self-consistently and robustly. Soleakons and their self-induced waveguides are non-stationary because they continuously leak power to the continuum. Self-consistency of soleakon means that during propagation, the localized section of the soleakon beam self-adjusts in such a way that it always fully populates a slowly-attenuating leaky mode of its self-induced waveguide. Importantly, a soleakon should satisfy the above self-consistency condition for propagation distance that is much larger than the diffraction length of the beam under linear propagation condition, LC. Otherwise, the beam will not exhibit particle-like features such as collisions and therefore cannot be called a soleakon. Like solitons and breathers, soleakons should be stable to noise. Here we explore soleakons in periodic potential: lattice soleakons.

It is worth mentioning that soleakons share some properties with embedded solitons [38

38. A. R. Champneys, B. A. Malomed, and M. J. Friedman, “Thirring solitons in the presence of dispersion,” Phys. Rev. Lett. 80(19), 4169–4172 (1998). [CrossRef]

41

41. X. Wang, Z. Chen, J. Wang, and J. Yang, “Observation of in-band lattice solitons,” Phys. Rev. Lett. 99(24), 243901 (2007). [CrossRef] [PubMed]

]: the propagation constants of both entities reside in the continuum spectra of radiation modes. However embedded solitons remain orthogonal to the ‘in-resonance’ radiation modes during propagation and therefore can be viewed as self-induced bound states in the continuum. In contrast with the embedded solitons, soleakons are self-induced leaky modes that are not orthogonal to the resonant radiation modes and therefore radiate their power to infinity, slowly attenuating during the propagation.

3. Transmission spectrum

All lattice soleakons are nonlinear waves that are trapped by their self-induced defects in periodic potentials. They are universal entities that can be excited in many nonlinear lattices. For concreteness, we analyze optical lattice soleakons in waveguide arrays and use the corresponding terminology. Specifically, we assume a bulk media with linear refractive index change in the form of array of slab sinusoidal wavegudes: Δn(x,y,z)=n0+Δn0cos2(πx/D), where n0=2.2is the homogeneous index, Δn0=3×103 and D=3μm are the amplitude and periodicity of the index modulation, respectively [Fig. 1(a)
Fig. 1 (a) Refractive index change in the array of slab waveguides. (b) Band structure of the array of slab waveguides. Propagation constants of linear radiation modes of the first (solid blue curves) and second (dash brown curves) band labeled by corresponding values ofkyD/π. The brown region displays the gap for modes withky=0. Radiation modes with ky0 reside in this gap, forming a semi-infinite band.
]. The linear modes of this structure are given by a product between a one-dimensional Flouqet-Bloch wave in x-axis and a plane wave in y-axis.

These propagating modes are completely delocalized in both x and y directions. Within the paraxial approximation, the propagation constant of the mode, βBlochq(kx,ky), depends on the Bloch wave-number, kx, the band number, q and the plane wave wave-number,ky:
βBlochq(kx,ky)=βBlochq(kx,0)ky2/2k0,
(1)
wherek0=2πn0/λis wave-number,λ=0.5μmis the wave-length of light in vacuum and n0=2.2 is the homogeneous index. Figure 1(b) shows two families of curves representing the propagation constants of the first βBloch1 (solid blue curves) and second βBloch2 (dash brown curves) bands versus kx for the modes with several ky’s. For a constant ky, the transmission spectra of the waveguide array is divided into bands that are separated by gaps in which propagating modes do not exist. Such a gap for modes with ky=0 is shown by the brown region in Fig. 1(b). However, as shown in Fig. 1(b), these gaps are full with propagating modes with other ky’s. In other words, the transmission spectrum of our structure consists of a semi-infinite band that is full with propagation waves and a semi-infinite gap above it.

4. Model and methods

5. Discrete and Bragg soleakons

The discrete soleakons in the array of slab wave-guides presented above are similar to the soleakons in homogeneous media [30

30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

] in that they both require a combination of nonlocal defocusing with local focusing nonlinearities and decay at increasing rate during propagation. Next, we show Bragg soleakons that exhibit properties that are profoundly different from those of the homogeneous and discrete soleakons. Bragg soleakons do not require the combination of nonlocal defocusing with local focusing nonlinearities and can be realized in array of slab waveguides with only saturable self-focusing. Propagation constants of these soleakons are shifted from the upper edge of the second band upward into the semi-infinite continuum of the first band. They radiate power into specific angles and decay at a decreasing rate and therefore do not disintegrate.

To find Bragg soleakons we assumed the following nonlinearity δn=δn1|ψ|2/(1+ς|ψ|2), where δn1=107cm2/Wis strength of saturable self-focusing nonlinearity and ς=8×105cm2/W is saturation coefficient. We start with determining the initial peak intensity |A(0)|2=2.8×104W/cm2 and find the initial beam as localized eigen-function of Eq. (3) whose eigen-value is shifted from the upper edge of the second band upward into the first band. The initial intensity profile of the Bragg soleakon is presented in Fig. 3(a)
Fig. 3 (a) Intensity profile of Bragg soleakon at z = 0 (logarithmic scale); (b) Intensity profile of beam in a linear lattice (δn=0) at z = 0.05cm = 2 × LC; Intensity profiles of soleakon (logarithmic scale) at z = 8 (c) and at z = 50cm (d); (e) localized power versus propagation distance obtained by adiabatic (blue dashed curve) and exact (red solid curve) methods and for linear defect (black dash-dot curve). (f) Soleakon decay rate versus localized power; (g) Propagation constant of the Bragg soleakon (red cross) on the background of linear band structure. The propagation constant of the soleakon is shifted from the upper edge of the second band upward into the first band; (h) Fourier power spectrum of the Bragg soleakon wave-function (logarithmic scale). Narrow lines connecting hot-spots correspond to the radiation part of the soleakon. Normals (black dashed lines) point in the direction of radiation; (i) Soleakon widths x2=+dx+dyx2|ψ(x,y)|2/|ψ|max2 (blue solid curve) and y2=+dx+dyy2|ψ(x,y)|2/|ψ|max2 (red dashed curve) versus localized power; (j) Fourier power spectrum of the soleakon wave-function vs. ky at kx=π/d and z = 0 (red solid curve) and z = 50cm (blue dashed curve). Arrows point to the minimal values of resonant plane wave-numbers kyR.
. The power of the localized component (defined above) is Plocalized=10mW. The diffraction length was estimated by linear propagation of initial beam which was modeled by solving Eq. (2) with δn=0 [Fig. 3(b) shows the intensity at z = 0.05cm = 2 × LC]. The nonlinear propagation of the beam was than evaluated using the ‘exact’ method. The intensity patterns at z = 8cm (320×LC) and z = 50cm (2000×LC) [Figs. 3(c) and 3(d), respectively] show that, similarly to the Discrete soleakon, Bragg soleakon broadens but stays localized for propagation distance that is many times larger than the diffraction length. We then calculated the evolution of the initial beam using the adiabatic method. Figure 3(e) shows the localized power vs. the propagation distance obtained by exact (blue dashed curve) and adiabatic (red solid curve) methods. The fine matching between two methods confirms that the beam is indeed a soleakon. Black dash-dot curve in Fig. 3(e) represents the localized power when the initial beam propagates linearly in a fixed defect that corresponds to the self-induced defect at z = 0. As shown the Bragg soleakon attenuates slower than the linear case and continues to propagate without disintegration. This results from the fact that the leakage rate of the Bragg soleakon decreases with localized power [Fig. 3(f)]. Figures 3(g)-3(h) display properties of the self-induced leaky modes corresponding to Bragg soleakon at z = 0. Real part of its propagation constant Re(βSoleakon)[red cross in Fig. 3(g)] resides in the region filled by radiation modes from the first band with nonzero ky. Its power spectrum [Fig. 3(h)] consists of intense humps that correspond to the localized section and thin lines between them, which correspond to the radiation part of the soleakon. The two most intense humps are centered around kx=±π/D, because Bragg soleakon bifurcates from the upper edge of the second band and hence is Bragg-matched with the lattice.

6. Conclusion

In conclusions, we predicted and demonstrated numerically lattice soleakons (discrete and Bragg): robust self-trapped leaky waves that induce defects in the lattice and populate their leaky modes (resonance states) self-consistently. Lattice soleakons exhibit stable propagation, largely maintaining their intensity profiles, for very long propagation distances (orders of magnitude larger than their diffraction lengths). We anticipate that lattice soleakons will be experimentally demonstrated in several physical systems, including optics and Bose Einstein condensates. Importantly, the fact that Bragg soleakons are supported by only local self-focusing nonlinearity significantly extends the range of physical systems in which soleakons exist and can be explored experimentally. In this respect, Bragg soleakons are probably more universal than discrete soleakons or the homogeneous soleakons of Ref. 30

30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

. We also expect that lattice soleakons can exhibit wealth of intrinsic dynamics (e.g. multi-mode vector soleakons and incoherent soleakons) and of extrinsic dynamics (e.g. moving and accelerating soleakons). The fact that soleakons interact strongly and selectively with radiation modes and with other soleakons, that are possibly far away, may give rise to new phenomena and applications that do not exist with lattice solitons.

Acknowledgments

We acknowledge the support of the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation.

References and links

1.

A. A. Ovchinnikov, “Localized long-lived vibrational states in molecular crystals,” Zh. Exp. Theor. Phys. 57, 263–270 (1969).

2.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]

3.

A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61(8), 970–973 (1988). [CrossRef] [PubMed]

4.

A. S. Davydov, “The theory of contraction of proteins under their excitation,” J. Theor. Biol. 38(3), 559–569 (1973). [CrossRef] [PubMed]

5.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58(2), 160–163 (1987). [CrossRef] [PubMed]

6.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]

7.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62(15), 1746–1749 (1989). [CrossRef] [PubMed]

8.

J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18(16), 1302–1304 (1993). [CrossRef] [PubMed]

9.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90(25), 253902 (2003). [CrossRef] [PubMed]

10.

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]

11.

M. Sato, B. E. Hubbard, and A. J. Sievers, “Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays,” Rev. Mod. Phys. 78(1), 137–157 (2006). [CrossRef]

12.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, “Intrinsic localized modes in parametrically driven arrays of nonlinear resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4), 046202 (2009). [CrossRef] [PubMed]

13.

E. Trías, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in a josephson array,” Phys. Rev. Lett. 84(4), 741–744 (2000). [CrossRef] [PubMed]

14.

N. K. Efremidis and D. N. Christodoulidis, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67(6), 063608 (2003). [CrossRef]

15.

B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. 92(23), 230401 (2004). [CrossRef] [PubMed]

16.

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).

17.

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(16), 3383–3386 (1998). [CrossRef]

18.

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(2), 023902 (2003). [CrossRef] [PubMed]

19.

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]

20.

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28(9), 710–712 (2003). [CrossRef] [PubMed]

21.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92(12), 123902 (2004). [CrossRef] [PubMed]

22.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29(13), 1530–1532 (2004). [CrossRef] [PubMed]

23.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92(9), 093904 (2004). [CrossRef] [PubMed]

24.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]

25.

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5777–5782 (2000). [CrossRef] [PubMed]

26.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003). [CrossRef] [PubMed]

27.

A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91(11), 113902 (2003). [CrossRef] [PubMed]

28.

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(22), 223901 (2004). [CrossRef] [PubMed]

29.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature 433(7025), 500–503 (2005). [CrossRef] [PubMed]

30.

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80(4), 041801 (2009). [CrossRef]

31.

N. Moiseyev, P. R. Certain, and F. Weinhold, “Resonance properties of complex-rotated hamiltonians,” Mol. Phys. 36(6), 1613–1630 (1978). [CrossRef]

32.

H. C. Gurgov and O. Cohen, “Spatiotemporal pulse-train solitons,” Opt. Express 17(9), 7052–7058 (2009). [CrossRef] [PubMed]

33.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, “Accessible light bullets via synergetic nonlinearities,” Phys. Rev. Lett. 102(20), 203903 (2009). [CrossRef] [PubMed]

34.

O. Lahav, H. C. Gurgov, P. Sidorenko, O. Peleg, L. Levi, A. Fleischer, and O. Cohen, “Self-phase modulation spectral broadening in two-dimensional spatial solitons: toward three-dimensional spatiotemporal pulse-train solitons,” Opt. Lett. 37(24), 5196–5198 (2012). [CrossRef] [PubMed]

35.

S. Giovanazzi, A. Gorlitz, and T. Pfau, “Ballistic expansion of a dipolar condensate,” J. Opt. B 5(2), S208–S211 (2003). [CrossRef]

36.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, “Comparing contact and dipolar interactions in a Bose-Einstein condensate,” Phys. Rev. Lett. 97(25), 250402 (2006). [CrossRef] [PubMed]

37.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

38.

A. R. Champneys, B. A. Malomed, and M. J. Friedman, “Thirring solitons in the presence of dispersion,” Phys. Rev. Lett. 80(19), 4169–4172 (1998). [CrossRef]

39.

J. Yang, B. A. Malomed, and D. J. Kaup, “Embedded solitons in second-harmonic-generating systems,” Phys. Rev. Lett. 83(10), 1958–1961 (1999). [CrossRef]

40.

J. Yang, “Fully localized two-dimensional embedded solitons,” Phys. Rev. A 82(5), 053828 (2010). [CrossRef]

41.

X. Wang, Z. Chen, J. Wang, and J. Yang, “Observation of in-band lattice solitons,” Phys. Rev. Lett. 99(24), 243901 (2007). [CrossRef] [PubMed]

42.

G. Agraval, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

OCIS Codes
(190.5940) Nonlinear optics : Self-action effects
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 30, 2013
Revised Manuscript: July 15, 2013
Manuscript Accepted: July 15, 2013
Published: August 14, 2013

Citation
Maxim Kozlov, Ofer Kfir, and Oren Cohen, "Self-trapped leaky waves in lattices: discrete and Bragg soleakons," Opt. Express 21, 19690-19700 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19690


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. A. Ovchinnikov, “Localized long-lived vibrational states in molecular crystals,” Zh. Exp. Theor. Phys.57, 263–270 (1969).
  2. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett.42(25), 1698–1701 (1979). [CrossRef]
  3. A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett.61(8), 970–973 (1988). [CrossRef] [PubMed]
  4. A. S. Davydov, “The theory of contraction of proteins under their excitation,” J. Theor. Biol.38(3), 559–569 (1973). [CrossRef] [PubMed]
  5. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58(2), 160–163 (1987). [CrossRef] [PubMed]
  6. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13(9), 794–796 (1988). [CrossRef] [PubMed]
  7. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett.62(15), 1746–1749 (1989). [CrossRef] [PubMed]
  8. J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett.18(16), 1302–1304 (1993). [CrossRef] [PubMed]
  9. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett.90(25), 253902 (2003). [CrossRef] [PubMed]
  10. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express13(6), 1780–1796 (2005). [CrossRef] [PubMed]
  11. M. Sato, B. E. Hubbard, and A. J. Sievers, “Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays,” Rev. Mod. Phys.78(1), 137–157 (2006). [CrossRef]
  12. E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, “Intrinsic localized modes in parametrically driven arrays of nonlinear resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.80(4), 046202 (2009). [CrossRef] [PubMed]
  13. E. Trías, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in a josephson array,” Phys. Rev. Lett.84(4), 741–744 (2000). [CrossRef] [PubMed]
  14. N. K. Efremidis and D. N. Christodoulidis, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A67(6), 063608 (2003). [CrossRef]
  15. B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett.92(23), 230401 (2004). [CrossRef] [PubMed]
  16. F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys.52, 555–600 (1928).
  17. 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(16), 3383–3386 (1998). [CrossRef]
  18. 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(2), 023902 (2003). [CrossRef] [PubMed]
  19. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature422(6928), 147–150 (2003). [CrossRef] [PubMed]
  20. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett.28(9), 710–712 (2003). [CrossRef] [PubMed]
  21. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004). [CrossRef] [PubMed]
  22. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett.29(13), 1530–1532 (2004). [CrossRef] [PubMed]
  23. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett.92(9), 093904 (2004). [CrossRef] [PubMed]
  24. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett.91(21), 213906 (2003). [CrossRef] [PubMed]
  25. S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(44 Pt B), 5777–5782 (2000). [CrossRef] [PubMed]
  26. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett.91(11), 113901 (2003). [CrossRef] [PubMed]
  27. A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett.91(11), 113902 (2003). [CrossRef] [PubMed]
  28. 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(22), 223901 (2004). [CrossRef] [PubMed]
  29. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature433(7025), 500–503 (2005). [CrossRef] [PubMed]
  30. O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A80(4), 041801 (2009). [CrossRef]
  31. N. Moiseyev, P. R. Certain, and F. Weinhold, “Resonance properties of complex-rotated hamiltonians,” Mol. Phys.36(6), 1613–1630 (1978). [CrossRef]
  32. H. C. Gurgov and O. Cohen, “Spatiotemporal pulse-train solitons,” Opt. Express17(9), 7052–7058 (2009). [CrossRef] [PubMed]
  33. I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, “Accessible light bullets via synergetic nonlinearities,” Phys. Rev. Lett.102(20), 203903 (2009). [CrossRef] [PubMed]
  34. O. Lahav, H. C. Gurgov, P. Sidorenko, O. Peleg, L. Levi, A. Fleischer, and O. Cohen, “Self-phase modulation spectral broadening in two-dimensional spatial solitons: toward three-dimensional spatiotemporal pulse-train solitons,” Opt. Lett.37(24), 5196–5198 (2012). [CrossRef] [PubMed]
  35. S. Giovanazzi, A. Gorlitz, and T. Pfau, “Ballistic expansion of a dipolar condensate,” J. Opt. B5(2), S208–S211 (2003). [CrossRef]
  36. A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, “Comparing contact and dipolar interactions in a Bose-Einstein condensate,” Phys. Rev. Lett.97(25), 250402 (2006). [CrossRef] [PubMed]
  37. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  38. A. R. Champneys, B. A. Malomed, and M. J. Friedman, “Thirring solitons in the presence of dispersion,” Phys. Rev. Lett.80(19), 4169–4172 (1998). [CrossRef]
  39. J. Yang, B. A. Malomed, and D. J. Kaup, “Embedded solitons in second-harmonic-generating systems,” Phys. Rev. Lett.83(10), 1958–1961 (1999). [CrossRef]
  40. J. Yang, “Fully localized two-dimensional embedded solitons,” Phys. Rev. A82(5), 053828 (2010). [CrossRef]
  41. X. Wang, Z. Chen, J. Wang, and J. Yang, “Observation of in-band lattice solitons,” Phys. Rev. Lett.99(24), 243901 (2007). [CrossRef] [PubMed]
  42. G. Agraval, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited