Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

Vicencio, Rodrigo A.; Johansson, Magnus

doi:10.1103/physreve.73.046602

Cited by 77 publications

(133 citation statements)

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“…For the same reason, fully discrete 2D solitons in lattices with the cubic onsite nonlinearity demonstrate no mobility either. 2D discrete solitons are effectively mobile in settings with the saturable [45] or quadratic (second-harmonic-generating) [46] nonlinearity, where the collapse does not occur (in the continuum limit), allowing for the existence of stable broad solitons.…”

Section: Introduction and The Modelmentioning

confidence: 99%

Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides

Blit

Malomed

2012

Phys. Rev. A

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We consider shapes and dynamics of semi-discrete solitons (SDSs) in the known model of the set of linearly-coupled waveguides with the intrinsic cubic nonlinearity. The model applies to the description of a planar array of optical fibers, or of a stack of parallel planar waveguides, in the temporal and spatial domains, respectively, as well as to the selfattractive Bose-Einstein condensate (BEC) loaded into an array of parallel tunnel-coupled cigar-shaped traps. It was found previously that the interplay of the group-velocity dispersion, discrete diffraction (in the longitudinal and transverse directions, respectively) and intrinsic self-focusing gives rise to SDSs in the array. We here develop the variational approximation (VA) and additional analytical methods for the description of the SDSs, and study their mobility and collisions by means of systematic simulations. The VA and an exact solution of the linearized equation in the cores adjacent to the central one produce an accurate description for the family of stable fundamental onsite-centered SDS solutions, as well of surface SDSs in the semi-infinite array. The VA is also presented for transversely unstable intersite-centered solitons. In simulations, the solitons are not mobile in the discrete direction (non-soliton semi-discrete modes may be mobile across the array). Collisions between SDSs traveling in the longitudinal direction feature a threshold separating the passage and merger or destruction. The exact shape of the threshold, considered as a function of the solitons' energy, features irregularities, while its average form is explained analytically. "Shifted" collisions between SDSs centered at adjacent cores are studied too.

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Section: Introduction and The Modelmentioning

confidence: 99%

Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides

Blit

Malomed

2012

Phys. Rev. A

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“…That can be understood as the localization of a discrete optical soliton near the surface [4] for powers exceeding a certain threshold value, for which the repulsive effect of the surface is balanced. A similar effect of light localization near the edge of the waveguide array and the formation of surface gap solitons have been predicted and observed for defocusing nonlinear media [5,6].It is important to analyze how the properties of nonlinear surface waves are modified by the lattice dimensionality, and the first studies of different types of discrete surface solitons in two-dimensional lattices [7,8,9,10] revealed, in particular, that the presence of a surface increases the stability region for two-dimensional (2D) discrete solitons [10] and the threshold power for the edge surface state is slightly higher than that for the corner soliton [9].In this Letter we consider anisotropic semi-infinite twodimensional photonic lattices and study the crossover between one-and two-dimensional surface solitons emphasizing the crucial effect of the lattice dimensionality on the formation of surface solitons.We consider a semi-infinite 2D lattice [shown schematically in Fig.2(a) below], described by the system of coupled-mode equations for the normalized amplitudes u n,m [11,12],where ξ is the normalized propagation distance. We de- fine the lattice coupling as follows:…”

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confidence: 99%

“…We consider a semi-infinite 2D lattice [shown schematically in Fig.2(a) below], described by the system of coupled-mode equations for the normalized amplitudes u n,m [11,12],…”

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Discrete surface solitons in two-dimensional anisotropic photonic lattices

et al. 2007

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We study nonlinear surface modes in two-dimensional anisotropic periodic photonic lattices and demonstrate that, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold power is lower at the surface, and two-dimensional discrete solitons can be generated easier near the lattice corners and edges. We analyze the crossover between effectively one-and two-dimensional regimes of the surface-mediated beam localization in the lattice. PACS numbers:Surface modes have been studied in different branches of physics; in guided wave optics surface states were predicted to exist at interfaces separating periodic and homogeneous dielectric media [1]. The interest in studying surface waves has been renewed recently because the interplay of discreteness and nonlinearity can facilitate the formation of discrete surface solitons [2,3] at the edge of the waveguide array. That can be understood as the localization of a discrete optical soliton near the surface [4] for powers exceeding a certain threshold value, for which the repulsive effect of the surface is balanced. A similar effect of light localization near the edge of the waveguide array and the formation of surface gap solitons have been predicted and observed for defocusing nonlinear media [5,6].It is important to analyze how the properties of nonlinear surface waves are modified by the lattice dimensionality, and the first studies of different types of discrete surface solitons in two-dimensional lattices [7,8,9,10] revealed, in particular, that the presence of a surface increases the stability region for two-dimensional (2D) discrete solitons [10] and the threshold power for the edge surface state is slightly higher than that for the corner soliton [9].In this Letter we consider anisotropic semi-infinite twodimensional photonic lattices and study the crossover between one-and two-dimensional surface solitons emphasizing the crucial effect of the lattice dimensionality on the formation of surface solitons.We consider a semi-infinite 2D lattice [shown schematically in Fig.2(a) below], described by the system of coupled-mode equations for the normalized amplitudes u n,m [11,12],where ξ is the normalized propagation distance. We de- fine the lattice coupling as follows:where α characterizes the lattice anisotropy.Linear lattice waves of the form u n,m (ξ) = u 0 sin(kn) sin(qm) exp(iβξ) satisfy the dispersion relation β kq = 2(cos k + α cos q). In the nonlinear case, we look for localized stationary solutions of the form u n,m (ξ) = u n,m exp(iλξ), where the amplitudes u n,m are real, and λ is the nonlinear propagation constant. For a given λ, localized solutions are found in a 15 × 15 lattice by using the Newton-Raphson method.We calculate the power threshold P th that characterizes the discrete solitons in 2D lattices [13]. We study three different modes: corner [ Fig. 1(a)], edge surface [ Fig. 1(b,c)] and central [ Fig. 1(d)] localized modes. The corner and edge modes represent 2D surface localized

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“…There, the robust, radiationless nature of the new moving pulses may be particularly appealing. Another relevant question is the extension of the present ideas to two-dimensional settings [29], where deciding which directions of wave propagation within the lattice can lead to radiationless pulses will also be of interest. …”

Section: Prl 97 124101 (2006) P H Y S I C a L R E V I E W L E T T E mentioning

confidence: 99%

Radiationless Traveling Waves in Saturable Nonlinear Schrödinger Lattices

et al. 2006

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The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities. DOI: 10.1103/PhysRevLett.97.124101 PACS numbers: 42.65.Tg, 05.45.Yv, 63.20.Pw Recently, the topic of discrete solitons (intrinsic localized modes) in photorefractive materials has received significant attention; see [1] for a review. This interest was initialized by the experimental realization of twodimensional periodic lattices in photorefractive crystals [2] in which solitons were observed [3,4]. Further work has revealed a wealth of additional coherent structures such as dipoles, quadrupoles, soliton trains, vector, necklace, and ring solitons; see, e.g., [5][6][7][8]. Since photorefractive materials feature the so-called saturable nonlinearity, these results for periodic lattices have spawned a parallel interest in genuinely discrete saturable nonlinear lattices [9][10][11]. One particularly intriguing result of these studies is that the stability properties of the localized modes are substantially different from their regular discrete nonlinear Schrödinger (DNLS) analogs. In DNLS, it is well known [12 -14] that site-centered localized modes are always stable, while intersite-centered modes are unstable (and are stabilized only in the continuum limit where these two branches degenerate into the well-known continuum sech-soliton of the integrable cubic NLS). For the photorefractive nonlinearity (i.e., the so-called Vinetskii-Kukhtarev model originally proposed in Ref. [15] and revisited in Refs. [9][10][11]), depending on the coupling strength, the intersite-centered modes may have lower energy than their on site counterparts. Hence, one should expect that the ensuing sign reversal of the so-called Peierls-Nabarro (PN) energy barrier E E IS ÿ E OS (where the subscripts denote intersite and on site, respectively) should cause an exchange between the linear stability properties of the two modes.A related, even more fundamental, question in nonlinear lattice models of DNLS type is whether exponentially localized self-supporting excitations that move with a constant wave speed can exist, the so-called moving discrete solitons. For continuum models, this question is in some sense trivial since the equations posed in a moving frame remain of the same fundamental type. Yet for lattices, passing to a moving frame leads to so-called advance-delay equations that are notoriously hard to analyze. One recent (negative) result in this direction [16] for the so-called Salerno model shows that, starting from the integrable Ablowitz-Ladik equation, mobile discrete solitary waves acquire nonvanishing tails as soon as parameters deviate from the integrable limit. Hence, exponentially localized fu...

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Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

Cited by 77 publications

References 35 publications

Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides

Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides

Discrete surface solitons in two-dimensional anisotropic photonic lattices

Radiationless Traveling Waves in Saturable Nonlinear Schrödinger Lattices

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