Fractional discrete vortex solitons
FFractional discrete vortex solitons
Cristian Mej´ıa-Cort´es ∗ Programa de F´ısica, Facultad de Ciencias B´asicas,Universidad del Atl´antico, Puerto Colombia 081007, Colombia
Mario I. Molina
Departamento de F´ısica, Facultad de Ciencias,Universidad de Chile, Casilla 653, Santiago, Chile a r X i v : . [ n li n . PS ] F e b bstract We examine the existence and stability of nonlinear discrete vortex solitons in a square latticewhen the standard discrete Laplacian is replaced by a fractional version. This creates a new,effective site-energy term, and a coupling among sites, whose range depends on the value of thefractional exponent α , becoming effectively long-range at small α values. At long-distance, it canbe shown that this coupling decreases faster than exponential: ∼ exp( −| n | ) / (cid:112) | n | . In general, weobserve that the stability domain of the discrete vortex solitons is extended to lower power levels, asthe α coefficient diminishes, independently of their topological charge and/or pattern distribution. I. INTRODUCTION
Vortices are objects characterized by a spatially-localized distribution of field intensities,together with a nontrivial phase distribution. This phase circulates around a singular point,or central core, changing by 2 πS times in each closed loop around it (where S is an integernumber). Integer S is known as the topological charge of the vortex. The sign of S deter-mines the direction of power flow. In optics, this type of solution is also known as a vortexbeam and has arisen considerable interest given their potential technological applications.Optical vortices have been envisioned as a mean to codify information using their topologicalcharge value in classical [1] and quantum [2] regimes. Also, a stable vortex is capable ofdelivering its orbital angular momentum (OAM) to a nearby object, given way to one ofits most remarkable applications: optical tweezers in biophotonics, where they are usefuldue to their ability to influence the motion of living cells, virus, and molecules [3–5]. Otherapplications can be found in optical systems communications [6] and spintronics [7].A particular domain where discrete vortex solitons can be found, is in the discrete non-linear Schr¨odinger (DNLS) equation [8–10], whose dimensionless form can be written as: i dC n dt + (cid:88) m C m + χ | C n | C n = 0 , (1)where C n is, for instance, the amplitude of an optical or electronic field, χ is the nonlinearcoefficient, and the sum is usually restricted to nearest-neighbor lattice sites. The DNLSequation has proven useful in describing a variety of phenomena in nonlinear physics, such ∗ [email protected]
2s the transversal propagation of light in waveguide arrays [11–13], propagation of exci-tations in a deformable medium [14, 15], self-focusing and collapse of Langmuir waves inplasma physics [16, 17], dynamics of Bose-Einstein condensates inside coupled magneto-optical traps [18, 19], and description of rogue waves in the ocean [19] among others. Itsmain features include the existence of localized nonlinear solutions in 1D and 2D, usuallyreferred to as discrete solitons, with families of stable and unstable states, the existence of aselftrapping transition [20, 21] of an initially localized excitation, and a degree of excitationmobility in 1D [13]. For the DNLS equation, the existence and observation of discrete vor-tex solitons in Eq. (1) for several lattices have been reported in several works. For a squaregeometry and Kerr nonlinearity [Eq. (1)] it was found that the discrete vortex is stable when χ is larger than a critical value [22–24]. For saturable nonlinearity, discrete vortices havebeen experimentally observed in a square lattice [25]. They have also been studied in anonlinear anisotropic Lieb lattice, which possesses a flat band [26]. For a hexagonal latticein a self-focusing photorefractive crystal, vortices with S = 1 have been found but provenunstable, while for S = 2 a range of stability can be found [27–29]. Discrete vortices livingat the boundary between a square and hexagonal lattice with photorefractive nonlinearity,have also been found [30].Another field with substantial recent interest is that of fractional calculus. Its origin datesback to the firsts observations that the usual integer-order derivative could be exended toa fractional-order derivative, that is, ( d n /dx n ) → ( d α /dx α ), for real α , which is knownas the fractional exponent. The field has a long history dating back to letters exchangedbetween L’Hopital and Leibnitz, followed by later contributions by Euler, Laplace, Riemann,Liouville, and Caputo, to name some. Several formalisms have been derived to treat thesefractional derivatives, each one having its advantages and shortcomings. In the popularRiemann-Liouville formalism [31–34], the α -th derivative of a function f ( x ) can be formallyexpressed as (cid:18) d α dx α (cid:19) f ( x ) = 1Γ(1 − α ) ddx (cid:90) x f ( x (cid:48) )( x − x (cid:48) ) α dx (cid:48) , (2)for 0 < α <
1. For the case of the laplacian operator ∆ = ∂ /∂x + ∂ /∂y , its fractionalform ( − ∆) α in two dimensions can be expressed as [34]( − ∆) α f ( x ) = L ,α (cid:90) f ( x ) − f ( y ) | x − y | α dy, (3)3ith, L ,α = 16 Γ(1 + α ) π | Γ( − α ) | , (4)where Γ( x ) is the Gamma function and 0 < α < α = 1, there is a stabilizing effect on this kind of helical modes, i. e., the power thresholdbecomes reduced. II. MODEL
Let us consider a square lattice, where the kinetic energy term in Eq.(1), (cid:80) m C m , can bewritten as 4 C n +∆ n C n , where ∆ n corresponds a the well-known expression for the discretizedLaplacian ∆ n C n = C p +1 ,q + C p − ,q − C p,q + C p,q +1 + C p,q − , (5)where n = ( p, q ). Equation (1) can then rewritten as i dC n dt + 4 C n + ∆ n C n + χ | C n | C n = 0 . (6)Let us now replace the Laplacian ∆ n by its fractional form (∆ n ) α , and given by [48, 49](∆ n ) α C n = (cid:88) m (cid:54) = n ( C m − C n ) K α ( n − m ) (7)where, K α ( m ) = 1 | Γ( − α ) | (cid:90) ∞ e − t I m (2 t ) I m (2 t ) t − − α dt (8)4ith m = ( m , m ) and I m ( x ) is the modified special Bessel function. An equivalent expres-sion for (∆ n ) α is (∆ n ) α C j = L ,α (cid:88) m (cid:54) = j ( C m − C j ) G , , (cid:16) / , − ( j − m α,j − m α )1 / α,j − m , − ( j − m (cid:12)(cid:12)(cid:12) (cid:17) , (9)where j = ( j , j ) and m = ( m , m ), and G ( ... ) is the Meijer G-function. As we can see,the symmetric kernel K α ( m ) = K α ( − m ) plays the role of a long-ranged coupling. Near α = 1, K ( m ) → δ m , u where u = (1 ,
0) or u = (0 , i dC n dt + 4 C n + (cid:88) m (cid:54) = n ( C m − C n ) K α ( m − n ) + χ | C n | C n = 0 , (10)With a bit of algebraic manipulations, it is possible to prove that Eq. (10) has two conservedquantities namely, the power P = (cid:88) n | C n ( t ) | (11)and the Hamiltonian, H = (cid:80) n (4 − (cid:80) m (cid:54) = n K α ( n − m )) | C n | (cid:80) n (cid:80) m (cid:54) = n K α ( n − m ) C ∗ n C m + ( χ/ (cid:80) n | C n | . (12)These relations prove useful when monitoring the accuracy of numerical computations.Now let us consider stationary modes defined by C n ( t ) = e iλt φ n , which obey( − λ + 4) φ n + (cid:88) m (cid:54) = n ( φ m − φ n ) K α ( m − n ) + χ | φ n | φ n = 0 , (13)where φ n is the field amplitude that defines a complex spatial profile of the solution, and λ is the (eigenvalue) propagation constant. It should be mentioned that, when dealing with afinite square lattice, in expressions (6) and (13) the term 4 is to be replaced by 3 (2) when n falls at the edge (corner). Figure 1 shows the effective site energy (cid:15) ( n ) = 4 − (cid:80) m (cid:54) = n K α ( m − n )and effective coupling K α ( m − n ). We can see that, as α decreases, the range of the couplingbetween two distant sites increases. In particular, for n = 0 and along the main diagonal m =( m, m ), its value can be shown to approach K α ( m ) ∼ | Γ( − α ) | − m √ m ( n → ∞ ) , (14)i.e., faster than exponential. 5 K s ( n ) K s ( n ) K s ( n ) ↵ ↵ ↵ (cid:1) ( n ) (cid:1) ( n ) (cid:1) ( n ) FIG. 1. Top row: Effective coupling K α ( n − m ) between m = (0 ,
0) and sites n = ( n,
0) (leftcolumn), n = ( n, n ) (middle column), and n = ( n, n ) (right column). Bottom row: Effective siteenergy (cid:15) ( n ) for several fractional exponents α and n = ( n,
0) (left column), n = ( n, n ) (middlecolumn), and n = ( n, n ) (right column). Number of sites = 10 ×
10. The numbers on each curvedenote the value of the fractional exponent.
III. DISCRETE VORTEX SOLITONS
Let us examine the nonlinear stationary modes given as complex solutions of Eq. (13) andcharacterized by a nontrivial distribution of the phases. They form a set of N × N nonlinearalgebraic equations for the amplitudes { φ n } . The form of the nonlinear term chosen here isof the Kerr type (cubic), although other forms can be used, such as the saturable nonlinear-ity [26]. Numerical solutions are obtained by the use of a multidimensional Newton-Raphsonscheme, using as a seed a solution in the form φ n = A n exp( iSθ n ), where S is the topological6 FIG. 2. 4-sites discrete vortex with S = 1 and exponent α = 0 .
2. Top left: Real part. Top right:Imaginary part. Bottom left: Amplitude profile. Bottom right: Phase profile. ( λ = 6) charge and θ n is the azimuthal angle of the n th site, with a highly localized distributionfor A n . This ansatz is obtained from the decoupled limit, also known as the anticontinuouslimit, where each site becomes decoupled from each other. We use a finite N × N latticewith open boundary conditions. Figures 2 and 3 show examples of two different discretevortex solitons with fractional exponent α = 0 .
2, and two values of the topological charge, S = 1 and S = 2. The stability of the computed vortex solitons is carried out by a simplelinear stability analysis [50]Results from the above procedure are displayed in Fig. 4. They are summarized by mean7f power vs eigenvalue diagrams, for several values of the fractional exponent α . Amplitude(top left) and phase (bottom right) profiles for these kinds of stationary vortex solutions aredisplayed at the inset of each diagram. We see that for vortex beams with S = 1 and fourmain peaks, the off-site square (a) and diamond shape (b), increase their stability domainas the α coefficient diminish. Similar behavior can be observed for those stationary modesendowed with S = 2 and displaying six main peaks and hexagonal shape (c). However,for modes with eight peaks and on-site square shape (d), the stability domain displays apiecewise domain for high values of α . Here, we have employed a N × N square lattice FIG. 3. 6-sites discrete vortex with S = 2 and exponent α = 0 .
2. Top left: Real part. Top right:Imaginary part. Bottom left: Amplitude profile. Bottom right: Phase profile. ( λ = 6) IG. 4. P vs λ diagram of some vortex solitons, for several fractional exponents and topologicalcharges S = 1 (upper row) and S = 2 (lower row). Solid (dashed) lines represent stable (unstable)solutions. Blue, orange, green and violet lines correspond to α = 0 . , . , . λ = 12. with N = 17. As normally happens in the non-fractional case, families of modes, indistinctof α coefficient, exhibit a saddle-node bifurcation near to the linear band border. Here weonly calculate solutions belonging to the lower branch of the bifurcation point. Discretesolitons display here highly localized patterns, as expected for a cubic nonlinearity. We canobserve a smooth spiral phase for any loop enclosing the central core, in those solutions withsymmetric amplitude profiles matching their nominal topological charge. On the contrary,for the hexagonal asymmetric pattern, the topological charge only can be observable in the9egion where the field amplitude is significant.In all cases, without exception, the power curves shift down as α is decreased. Moreover,the main effect of small α values of fractionality is to diminish the power threshold to obtainstable solutions, which leads to increase the domain of stability of these helical modes.ofthese helical modes. Another observation concerns the limit α →
0. In that limit, the rangeof the coupling diverges and, as a result, all sites are coupled with each other. Assumingthat the amplitude at each site is nearly identical, the stationary equations (13) reduce to( − λ + 4) φ + χφ ≈
0. For φ (cid:54) = 0 we have (4 − λ ) + χφ ≈
0. Using P ∼ Zφ where Z isthe number of sites initially excited, we have P ≈ ( Z/χ )( λ − λ <
4, we must take φ = 0, which implies P = 0. This linear dependence can be clearly seen in all plots of Fig. 4at small α values. IV. CONCLUSIONS
In this work we considered the existence and stability of discrete vortex solitons of thediscrete nonlinear Schr¨odinger (DNLS) equation, when the usual Laplacian ∆ n is replacedby a fractional version (∆ n ) α with 0 < α <
1. We employed a square lattice and a Kerrnonlinearity and computed discrete vortex modes and their stability for different values ofthe fractional exponent α . Discrete vortex solitons reported here, namely, the diamond, offand on-site square and hexagonal shape, exist for any value of fractional exponent and S = 1and S = 2 topological charges. However, those with diamond and off-site square shape, areonly stable for S = 1. On the contrary, the on-site square and hexagonal shape cases arestable for S = 2. The existence and stability of these modes are strongly related to theirspatial distribution, as well as to the lattice geometry. In all cases examined, a decrease ofthe fractional exponent α causes the power stability curves to shift to lower values, whichcould be an intriguing feature since a lower power threshold can ease their experimentalobservation, hence, their potential usefulness in photonic applications.The fractional model that we delve here could be seen as an alternative approach todescribe for example photonic lattices with a long range coupling. The peculiar effectivelong-range coupling could be realized experimentally using a couple waveguide array, bymeans of a judicious coupling-engineering [51]. This kind of optical devices can be builtby femtosecond laser inscription, in amorphous [52] as well as crystalline dielectric materi-10ls [53], or in photorefractive crystals, where the linear refractive index can be modulatedexternally by light [54]. In both systems evanescent waves couple to neighbor waveguidesdetermining the transversal dynamics of light propagation.In general, the basic properties of discrete vortices observed before for the standardLaplacian exponent ( α = 1) are more or less maintained in the case a fractional Laplacian.This is itself interesting, since it suggests that the discrete vortex soliton properties arerobust against mathematical “perturbations”. Funding : National Laboratory for High Performance Computing (ECM-02); Fondo Na-cional de Desarrollo Cient´ıfico y Tecnol´ogico (1200120).
Acknowledgments : The authors acknowledges helpful discussions with L. Roncal.
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