Linear impurity modes in an electrical lattice: Theory and experiment

Molina, Mario I.; English, Lars Q.; Chang, M. K.; Kevrekidis, P. G.

doi:10.1103/physreve.100.062114

Cited by 11 publications

(6 citation statements)

References 32 publications

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“…Other related applications abound in a diverse array of themes, ranging from defect modes in photonic crystals [9], optical waveguide arrays [10][11][12][13], superconductors [14], dielectric superlattices with embedded defect layers [15,16], electron-phonon interactions [16], granular crystals in materials science [17], and electrical transmission lines. In this last case, localized modes at capacitive impurity places have been experimentally observed in 1D [18].…”

Section: Introductionmentioning

confidence: 77%

“…Finally, when both spacings are equal, C 0 /C = 1 and ∆ = 0. The system described by equation ( 3) has been realized in the lab using an array of electrical units [18].…”

Section: The Modelmentioning

confidence: 99%

“…For instance, for problems involving boundaries, like impurities close to a surface, a more serious treatment is needed. An Reprinted with permission from [18], Copyright (2019) by the American Physical Society. elegant method for dealing with impurity problems is the technique of lattice Green functions [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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Fractional electrical impurity

Molina

2024

New J. Phys.

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We examine the localized mode and the transmission of plane waves across a capacitive impurity of strength $\Delta$, in a 1D bi-inductive electrical transmission line where the usual discrete Laplacian is replaced by a fractional one characterized by a fractional exponent $s$. In the absence of the impurity, the plane wave dispersion is computed in closed form in terms of hypergeometric functions. It is observed that the bandwidth decreases steadily, as $s$ decreases towards zero, reaching a minimum width at $s=0$. The localized mode energy and spatial profiles are computed in closed form v\`{i}a lattice Green functions. The profiles show a remnant of the staggered-unstaggered symmetry that is common in non-fractional chains. The width of the localized mode decreases with decreasing $s$, becoming completely localized at the impurity site at $s=0$. The transmission coefficient of plane waves across the impurity is qualitatively similar to its non-fractional counterpart ($s=1$), except at low $s$ values ($s\ll 1$). For a fixed exponent $s$, the transmission decreases with increasing $\Delta$.

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Section: Introductionmentioning

confidence: 77%

“…Finally, when both spacings are equal, C 0 /C = 1 and ∆ = 0. The system described by equation ( 3) has been realized in the lab using an array of electrical units [18].…”

Section: The Modelmentioning

confidence: 99%

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Fractional electrical impurity

Molina

2024

New J. Phys.

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“…and considerations of impurities [45], among many others. Here, we propose an electric line composed by resonant elements (LC oscillators) coupled by inductors as shown in Fig.…”

Section: A Concrete Realization Proposal: a Tailored Electric Circuit...mentioning

confidence: 99%

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

Palmero,

Molina,

Cuevas-Maraver

et al. 2021

Preprint

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For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly-decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrodinger equation, where the linear operator of the Schrodinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.

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“…In the case of a nonlinear impurity, this could simply consist of a unit with a nonlinear response immersed in an otherwise linear array of units. For instance, in an inductively-coupled array of electrical units, each consisting of an LC resonator, one can render one of the capacitors as nonlinear by inserting an appropriate nonlinear dielectric between the plates of the capacitor [32]. In a similar vein, in an optics context, the system of interest is a dielectric waveguide array, where one of the guides is judiciously doped with an element with strong polarizability.…”

Section: Introductionmentioning

confidence: 99%

Fractional nonlinear surface impurity in a 2D lattice

Molina

2021

Preprint

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We study the formation of localized modes around a generalized nonlinear impurity which is located at the boundary of a semi-infinite square lattice, and where we replace the standard discrete Laplacian by a fractional one, characterized by a fractional exponent 0 < α < 1 where α = 1 marks the standard, non-fractional case. We specialize to two impurity cases: impurity at an "edge" and impurity at a "corner" and use the formalism of lattice Green functions to obtain in closed form the bound state energy and its mode amplitude. It is found that, for any fractional exponent and for impurity strengths above a certain threshold, there is always a single bound state for the linear impurity, while for the nonlinear (cubic) case, up to two bound states are possible. At small fractional exponents, the energy of the impurity mode becomes directly proportional to the impurity strength.

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Linear impurity modes in an electrical lattice: Theory and experiment

Cited by 11 publications

References 32 publications

Fractional electrical impurity

Fractional electrical impurity

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

Fractional nonlinear surface impurity in a 2D lattice

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