arXiv subjects

Mario I. Molina

Publications and source records attributed to Mario I. Molina.

At least 37 records · Page 2Linked to original sources

arXiv2020

Magnetic metamaterials with correlated disorder

We examine the transport of magnetic energy in a simplified model of a magnetic metamaterial, consisting of a one-dimensional array of split-ring resonators, in the presence of correlated disorder in the resonant frequencies. The computation of the average participation ratio (PR) reveals that on average, the modes for the correlated disorder system are less localized than in the uncorrelated case. The numerical computation of the mean square displacement of an initially localized magnetic excitation for the correlated case shows a substantial departure from the uncorrelated (Anderson-like) case. A long-time asymptotic fit $\langle n^2\rangle \sim t^α$ reveals that, for the uncorrelated system $α\sim 0$, while for the correlated case $α>0$, spanning a whole range of behavior ranging from localization to super-diffusive behavior. The transmission coefficient of a plane wave across a single magnetic dimer reveals the existence of well-defined regions in disorder strength-magnetic coupling space, where unit transmission for some wavevector(s) is possible. This implies, according to the random dimer model (RDM) of Dunlap et al., a degree of mobility. A comparison between the mobilities of the correlated SRR system and the RDM shows that the RDM model has better mobility at low disorder while our correlated SRR model displays better mobility at medium and large disorder.

cond-mat.dis-nn↗

arXiv2020

The Two-Dimensional Fractional Discrete Nonlinear Schrodinger Equation

We study a fractional version of the two-dimensional discrete nonlinear Schrödinger (DNLS) equation, where the usual discrete Laplacian is replaced by its fractional form that depends on a fractional exponent $s$ that interpolates between the case of an identity operator ($s=0$) and that of the usual discrete 2D Laplacian ($s=1$). This replacement leads to a long-range coupling among sites that, at low values of $s$, decreases the bandwidth and leads to quasi-degenerate states. The mean square displacement of an initially-localized excitation is shown to be ballistic at all times with a `speed' that increases monotonically with the fractional exponent $s$. We also compute the nonlinear modes and their stability for both, bulk and surface modes. The modulational stability is seen to increase with an increase in the fractional exponent. The trapping of an initially localized excitation shows a selftrapping transition as a function of nonlinearity strength, whose threshold increases with the value of $s$. In the linear limit, there persists a linear trapping at small $s$ values. This behavior is connected with the decrease of the bandwidth and its associated increase in quasi-degeneracy.

nlin.PS↗

arXiv2020

Localized vortex beams in anisotropic Lieb lattices

We address the issue of nonlinear modes in a two-dimensional waveguide array, spatially distributed in the Lieb lattice geometry, and modeled by a saturable nonlinear Schrödinger equation. In particular, we analyze the existence and stability of vortex-type solutions finding localized patterns with symmetric and asymmetric profiles, ranging from topological charge S=1 to S=3. By taking into account the presence of anisotropy, which is inherent to experimental realization of waveguide arrays, we identify different stability behaviors according to their topological charge. Our findings might give insight on experimental feasibility to observe these kind of vortex states.

physics.optics↗

arXiv2020

Fractional dynamics in nonlinear magnetic metamaterials

We examine the existence of nonlinear modes and their temporal dynamics, in arrays of split-ring resonators, using a fractional extension of the Laplacian in the evolution equation. We find a closed-form expression for the dispersion relation as a function of the fractional exponent as well as an exact expression for the critical coupling between rings, beyond which no fractional magnetoinductive wave can exist. We also find the low-lying families of bulk and surface nonlinear modes and their bifurcation diagrams. Here the phenomenology is similar for all exponents and resembles what has been observed in other discrete evolution equations, such as the DNLS. The propagation of an initially localized magnetic excitation is always ballistic, with a `speed' that is computed in exact form as a function of the fractional exponent. For a given exponent, it increases with an increase in coupling up to a critical coupling value, beyond which the ballistic speed could diverge inside the fractional interval $[0,1]$. Examination of the modulational instability shows that it tends to increase with an increase in the fractional exponent, where the decay proceeds via the formation of filamentary structures that merge eventually and form pure radiation. The dynamical selftrapping around an initially localized excitation increases with the fractional exponent, but it also shows a degree of trapping in the linear limit. This trapping increases with a decrease in the exponent and can be explained by near-degeneracy considerations.

nlin.PS↗

arXiv2020

Discrete embedded modes in the continuum in 2D lattices

We study the problem of constructing bulk and surface embedded modes (EMs) inside the quasi-continuum band of a square lattice, using a potential engineering approach à la Wigner and von Neumann. Building on previous results for the one-dimensional (1D) lattice, and making use of separability, we produce examples of two-dimensional envelope functions and the two-dimensional (2D) potentials that produce them. The 2D embedded mode decays like a stretched exponential, with a supporting potential that decays as a power law. The separability process can cause that a 1D impurity state (outside the 1D band) can give rise to a 2D embedded mode (inside the band). The embedded mode survives the addition of random perturbations of the potential; however, this process introduces other localized modes inside the band, and causes a general tendency towards localization of the perturbed modes.

cond-mat.mes-hall↗

arXiv2018

Seltrapping in flat band lattices with nonlinear disorder

We study the selftrapping properties of an initially localized excitation in several flat band lattices, in the presence of nonlinear (Kerr) disorder. In the weak nonlinearity regime, the dynamics is controlled by the degeneracy of the bands leading to a linear form of selftrapping. In the strong nonlinearity regime, the dynamics of the excitations depends strongly on the local environment around the initial excitation site that leads to a highly fluctuating selfrapping profile. It is shown that the spreading of the wavefunction in the presence of random nonlinearity, is always completely inhibited about fifty percent of all cases. The role of degeneracy associated with the flatbands is explained with the help of the $n$-simplex system, which can be solved in closed form.

nlin.PS↗

arXiv2017

Solitons in a modified discrete nonlinear Schroedinger equation

We study the bulk and surface nonlinear modes of the modified one-dimensional discrete nonlinear Schroedinger (mDNLS) equation. A linear and a modulational stability analysis of the lowest-order modes is carried out. While for the fundamental bulk mode there is no power threshold, the fundamental surface mode needs a minimum power level to exist. Examination of the time evolution of discrete solitons in the limit of strongly localized modes, suggests ways to manage the Peierls- Nabarro barrier, facilitating in this way a degree of steering. The long-time propagation of an initially localized excitation shows that, at long evolution times, nonlinear effects become negligible and as a result, the propagation becomes ballistic. The similarity of all these results to the ones obtained for the DNLS equation, points out to the robustness of the discrete soliton phenomenology.

nlin.PS↗

arXiv2017

Compact modes in quasi one dimensional coupled magnetic oscillators

In this work we study analytically and numerically the spectrum and localization properties of three quasi-one-dimensional (ribbons) split-ring resonator arrays which possess magnetic flatbands, namely, the stub, Lieb and kagome lattices, and how their spectra is affected by the presence of perturbations that break the delicate geometrical interference needed for a magnetic flatband to exist. We find that the Stub and Lieb ribbons are stable against the three types of perturbations considered here, while the kagome ribbon is, in general, unstable. When losses are incorporated, all flatbands remain dispersionless but become complex, with the kagome ribbon exhibiting the highest loss rate.

cond-mat.dis-nn↗

arXiv2015

Flat bands and PT-symmetry in quasi-one-dimensional lattices

We examine the effect of adding PT-symmetric gain and loss terms to quasi 1D lattices (ribbons) that possess flat bands. We focus on three representative cases: (a) The Lieb ribbon, (b) The kagome ribbon, and (c) The stub Ribbon. In general we find that the effect on the flat band depends strongly on the geometrical details of the lattice being examined. One interesting and novel result that emerge from an analytical calculation of the band structure of the Lieb ribbon including gain and loss, is that its flat band survives the addition of PT-symmetry for any amount of gain and loss, while for the other two lattices, any presence of gain and loss destroys the flat bands. For all three ribbons, there are finite stability windows whose size decreases with the strength of the gain and loss parameter. For the Lieb and kagome cases, the size of this window converges to a finite value. The existence of finite stability windows, plus the constancy of the Lieb flat band are in marked contrast to the behavior of a pure one-dimensional lattice.

quant-ph↗

arXiv2014

Observation of bound states in Lieb photonic lattices

We present the first experimental demonstration of a new type of bound states in the continuum, namely, compacton-like linear states in flat bands lattices. To this end, photonic Lieb lattices are employed, which exhibit three tight-binding bands, with one being perfectly flat. Our results could be of great importance for fundamental physics as well as for various applications concerning imaging and data transmission.

physics.optics↗

arXiv2014

Self trapping transition for a nonlinear impurity within a linear chain

In the present work we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schrödinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied in the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations.

nlin.PS↗

arXiv2014

Embedded states in the continuum for ${\mathcal{P T}}$-symmetric systems

We introduce the novel concept of a bound state in the continuum (BIC) for a binary lattice satisfying the ${\mathcal{P T}}$ symmetry condition. We show how to build such state and the local potential necessary to sustain it. We find that an appropriate choice of the envelope function can bring the system from a ${\mathcal{P T}}$-symmetric phase into a Hamiltonian one. For more general envelope functions, the BIC can still be created but the bounded state will force the system to undergo the ${\mathcal{P T}}$ symmetry breaking transition.

nlin.PS↗

arXiv2014

Nonlinear multi-core waveguiding structures with balanced gain and loss

We study existence, stability, and dynamics of linear and nonlinear stationary modes propagating in radially symmetric multi-core waveguides with balanced gain and loss. We demonstrate that, in general, the system can be reduced to an effective ${\cal PT}$-symmetric dimer with asymmetric coupling. In the linear case, we find that there exist two modes with real propagation constants before an onset of the ${\cal PT}$-symmetry breaking while other modes have always the propagation constants with nonzero imaginary parts. This leads to a stable (unstable) propagation of the modes when gain is localized in the core (ring) of the waveguiding structure. In the case of nonlinear response, we show that an interplay between nonlinearity, gain, and loss induces a high degree of instability, with only small windows in the parameter space where quasi-stable propagation is observed. We propose a novel stabilization mechanism based on a periodic modulation of both gain and loss along the propagation direction that allows bounded light propagation in the multi-core waveguiding structures.

physics.optics↗

arXiv2013

Self-trapping transition in nonlinear cubic lattices

We explore the fundamental question of the critical nonlinearity value needed to dynamically localize energy in discrete nonlinear cubic (Kerr) lattices. We focus on the effective frequency and participation ratio of the profile to determine the transition into localization in one-, two-, and three-dimensional lattices. A simple and general criterion is developed, for the case of an initially localized excitation, to define the transition region in parameter space ("dynamical tongue") from a delocalized to a localized profile. We introduce a method for computing the dynamically excited frequencies, which helps us validate our stationary ansatz approach and the effective frequency concept. A general analytical estimate of the critical nonlinearity is obtained, with an extra parameter to be determined.We find this parameter to be almost constant for two-dimensional systems and prove its validity by applying it successfully to two-dimensional binary lattices.

nlin.PS↗

arXiv2013

The nonlinear magnetoinductive dimer

We examine a nonlinear magnetoinductive dimer and compute its linear and nonlinear symmetric, antisymmetric and asymmetric modes in closed-form, in the rotating-wave approximation. A linear stability analysis of these modes reveals that the asymmetric mode is always stable, for any allowed value of the coupling parameter and for both, hard and soft nonlinearity. A numerical computation of the dimer dynamics reveals a magnetic energy selftrapping whose threshold increases for increasing dimer coupling.

nlin.SI↗

arXiv2012

Localized modes in nonlinear photonic kagome nanoribbons

We examine localization of light in nonlinear (Kerr) kagome lattices in the shape of narrow strips of varying width. For the narrowest ribbon, the band structure features a flat band leading to linear dynamical trapping of an initially localized excitation. We also find a geometry-induced bistability of the nonlinear modes as the width of the strip is changed. A crossover from one to two dimensions localization behavior is observed as the width is increased, attaining two-dimensional behavior for relatively narrow ribbons.

nlin.PS↗

arXiv2012

Expansion of a wave-packet in lattices with disorder and nonlinearity

We show, theoretically and experimentally, the counterintuitive result that an increase of disorder can result in an enhanced spreading of an initially localized excitation. Moreover, we find that adding a focusing nonlinearity facilitates the expansion of the wave-packet even further by increasing its effective size. We find a clear transition between between the regions of enhanced spreading (weak disorder) and localization (strong localization) described by a "diffusion peak".

physics.optics↗

arXiv2012

Anderson localization in a periodic photonic lattice with a disordered boundary

We investigate experimentally the light evolution inside a two-dimensional finite periodic array of weakly- coupled optical waveguides with a disordered boundary. For a completely localized initial condition away from the surface, we find that the disordered boundary induces an asymptotic localization in the bulk, centered around the initial position of the input beam.

physics.optics↗