Search

In this paper, we investigate a two-dimensional photonic array featuring a circular shape and an alternating gain and loss pattern. Our analysis revolves around determining the presence and resilience of optical ring modes with varying vorticity values. This investigation is conducted with respect to both the array's length and the strength of the non-Hermitian parameter. For larger values of array's length, we observe a reduction in the stability domain as the non-Hermitian parameter increases. Interestingly, upon an increasing of vorticity of the optical modes full stability windows emerge for shorter lattice sizes regime.

We study the spectrum and transmission coefficient of plane waves propagating along square ribbons of varying widths, containing a square-shaped, PT-symmetric impurity region. We start with a zero-width ribbon (1D chain) and place a PT symmetric dimer. The spectrum is computed numerically and the instability gain is computed as a function of the gain/loss dimer strength. The transmission coefficient is obtained in closed form and examined as a function of wavevector and the gain/loss parameter. Next, we study a ribbon in a narrow ladder configuration containing a square PT impurity. As before, we compute the instability gain numerically and the transmission coefficient in closed form for the two possible input modes. Finally, we repeat the calculations for a wider ladder ribbon containing a Lieb-like impurity in a PT configuration. For all cases and transmission channels, we obtain transmission divergences in wavevector-gain/loss parameter space, whose number increases with the width of the ribbon, Comment: 7 pages, 4 figures

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 close 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 $\Delta$, Comment: 6 pages, 6 figures

We examine the stability of a 1D electrical transmission line in the simultaneous presence of PT-symmetry and fractionality. The array contains a binary gain/loss distribution $\gamma_{n}$ and a fractional Laplacian characterized by a fractional exponent $\alpha$. For an infinite periodic chain, the spectrum is computed in closed form, and its imaginary sector is examined to determine the stable/unstable regions as a function of the gain/loss strength and fractional exponent. In contrast to the non-fractional case where all eigenvalues are complex for any gain/loss, here we observe that a stable region can exist when gain/loss is small, and the fractional exponent is below a critical value, $0 < \alpha < \alpha_{c1}$ . As the fractional exponent is decreased further, the spectrum acquires a gap with two nearly-flat bands. We also examined numerically the case of a finite chain of size N. Contrary to what happens in the infinite chain, here the stable region always lies above a critical value $\alpha_{c2} < \alpha < 1$. An increase in gain/loss or $N$ always reduces the width of this stable region until it disappears completely., Comment: 6 pages, 6 figures

We study the spectral stability of a 2D discrete Schr\"{o}dinger equation on a square lattice, in the simultaneous presence of a fractional Laplacian and $\cal{PT}$ symmetry. For that purpose, we compute the plane-wave spectrum in closed form, as a function of the gain/loss parameter and the fractional exponent. Examination of the spectrum reveals that an increase of the gain/loss parameter favors the early appearance of complex eigenvalues, thus is, the onset of a broken ${\cal PT}$ symmetry. On the other hand, as the fractional exponent decreases from unity, at a critical value a gap opens up separating the upper and lower bands, and the spectrum becomes real. Further decrease of the exponent increases the width of the gap and the system remains in the $\cal{PT}$-symmetric phase down to a vanishing value of the fractional exponent. Examination of the density of states and the participation ratio reinforce these observations and lead one to conclude that, unlike the standard, non-fractional case where the binary lattice is always in the broken $\cal{PT}$ phase, for the fractional case it is possible to have a symmetric $\cal{P}{\cal T}$ phase in the presence of a finite gain/loss parameter and a small enough fractional exponent., Comment: 6 pages, 4 figures

We examine analytically and numerically the effect of fractionality on a saturable bulk and surface impurity embedded in a 1D lattice. We use a fractional Laplacian introduced previously by us, and by the use of lattice Green functions we are able to obtain the bound state energies and amplitude profiles, as a function of the fractional exponent $s$ and saturable impurity strength $\chi$ for both, surface and bulk impurity. The transmission is obtained in closed form as a function of $s$ and $\chi$, showing strong deviations from the standard case, at small fractional exponent values. The selftrapping of an initially-localized excitation is qualitatively similar for the bulk and surface mode, but in all cases complete confinement is obtained at $s\rightarrow 0$, as shown theoretically and observed numerically., Comment: 7 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2108.10974

We examine the stability domains of a 1D discrete Schr\"{o}dinger equation in the simultaneous presence of parity-time ($\cal{PT}$) symmetry and fractionality. Direct numerical examination of the eigenvalues of the system reveals that, as the fractional exponent is decreased away from unity (the standard case), the instability gain increases abruptly past a critical value. Also, as the length of the system increases, the stable fraction decreases as well. Also, for a fixed fractional exponent and lattice size, an increase in gain/loss also brings about an abrupt increase in the instability gain. Finally, the participation ratio of the modes is seen to decrease with an increase of the gain/loss parameter and with a decrease of the fractional exponent, evidencing a tendency towards localization., Comment: 5 pages, 4 figures

We examine the interplay between disorder and fractionality in a one-dimensional tight-binding Anderson model. In the absence of disorder, we observe that the two lowest energy eigenvalues detach themselves from the bottom of the band, as fractionality $s$ is decreased, becoming completely degenerate at $s=0$, with a common energy equal to a half bandwidth, $V$. The remaining $N-2$ states become completely degenerate forming a flat band with energy equal to a bandwidth, $2V$. Thus, a gap is formed between the ground state and the band. In the presence of disorder and for a fixed disorder width, a decrease in $s$ reduces the width of the point spectrum while for a fixed $s$, an increase in disorder increases the width of the spectrum. For all disorder widths, the average participation ratio decreases with $s$ showing a tendency towards localization. However, the average mean square displacement (MSD) shows a hump at low $s$ values, signaling the presence of a population of extended states, in agreement with what is found in long-range hopping models., Comment: 6 pages, 5 figures

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 $0<\alpha<1$ where $\alpha=1$ 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., Comment: 7 pages, 5 figures

We study the scattering properties of a bi-inductive electrical lattice consisting of a one-dimensional array of coupled LC units. For an initially localized electrical excitation, and in the absence of any impurity, we compute in closed form the mean square displacement of an initially localized electrical excitation for the cases of an infinite and semi-infinite lattice, obtaining a ballistic propagation under very general conditions. For the transport of extended excitations, we compute in closed form the transmission coefficient of electro-inductive plane waves across an impurity region, containing a number of side-coupled units, or a single internal impurity with coupling to first-and second nearest neighbors, looking for the presence of Fano resonances (FRs). For all cases examined, we obtain a closed-form expression for the position of the FR in terms of the relative strengths of the inductive couplings involved. For the case of two, identical side-coupled impurities, the position of the single FR turns out to be independent of the relative distance between the two impurities., Comment: 6 pages, 6 figures

We use a lattice Green function approach to study the stationary modes of a linear/nonlinear (Kerr) impurity embedded in a periodic one-dimensional lattice where we replace the standard discrete Laplacian by a fractional one. The energies and the mode profiles are computed in closed form, for different fractional exponents and different impurity strengths. The energies of the impurity mode lie outside the linear band whose bandwidth decreases steadily as the fractional exponent decreases. For any fractional exponent values, there is always a single bound state for the linear impurity while for the nonlinear (Kerr) case, up to two bound states are possible, for impurity strengths above certain threshold. The energy of the linear mode (or that of the upper energy nonlinear one), becomes directly proportional to the impurity strength at large impurity strengths. The transmission of plane waves is also computed in closed form for several fractional exponents, and various impurity strengths. We observe that fractionality tends to increase the overall transmission. The selftrapping transition for the nonlinear impurity shifts to lower nonlinearity values as the fractional exponent is decreased. In both cases, linear and nonlinear, we observe a form of trapping at zero impurity strength, which can be explained by the near-degeneracy of the spectrum in the limit of a small fractional exponent., Comment: 7 pages, 5 figures

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., Comment: 9 pages, 16 figures