Query: Wave transmission in nonlinear lattices

Candidate papers:

abstract: Abstract The interplay of nonlinearity with lattice discreteness leads to phenomena and propagation properties quite distinct from those appearing in continuous nonlinear systems. For a large variety of condensed matter and optics applications the continuous wave approximation is not appropriate. In the present review we discuss wave transmission properties in one dimensional nonlinear lattices. Our paradigmatic equations are discrete nonlinear Schrodinger equations and their study is done through a dynamical systems approach. We focus on stationary wave properties and utilize well known results from the theory of dynamical systems to investigate various aspects of wave transmission and wave localization. We analyze in detail the more general dynamical system corresponding to the equation that interpolates between the non integrable discrete nonlinear Schrodinger equation and the integrable Albowitz Ladik equation. We utilize this analysis in a nonlinear Kronig Penney model and investigate transmission and band modification properties. We discuss the modifications that are effected through an electric field and the nonlinear Wannier Stark localization effects that are induced. Several applications are described, such as polarons in one dimensional lattices, semiconductor superlattices and one dimensional nonlinear photonic band gap systems.
author_names: Dirk Hennig

G P Tsironis
title: Wave transmission in nonlinear lattices
year: 1999

abstract: Abstract We propose a 3D printed graded lattice made of hollow elliptical cylinders (HECs) as a new way to design impact mitigation systems. We observe asymmetric dynamics in the graded HEC chains with increasing and decreasing stiffness. Specifically, the increasing stiffness chain shows an acceleration of the propagating waves, while the decreasing stiffness chain shows the opposite. From the standpoint of impact mitigation, the decreasing stiffness chain combined with the strain softening behavior of HECs results in an order of magnitude improvement in force attenuation compared to the increasing stiffness chain. We extend this finding to the graded 2D arrays and demonstrate a similar trend of wave transmission efficiency contrast between the increasing and decreasing stiffness lattices. The 3D printed HEC lattices shown in this study can lead to the development of a new type of impact mitigating and shock absorbing structures.
author_names: Hyunryung Kim

Eunho Kim

Jinkyu Yang
title: Nonlinear wave propagation in 3D printed graded lattices of hollow elliptical cylinders
venue: Journal of the Mechanics and Physics of Solids
year: 2019

abstract: One dimensional lattice with an on site cubic quintic nonlinear response described by a cubic quintic discrete nonlinear Schrodinger equation is tested for asymmetric wave propagation. The lattice is connected to linear side chains. Asymmetry is introduced by breaking the mirror symmetry of the lattice with respect to the center of the nonlinear region. Three cases corresponding to dimer, trimer and quadrimer are discussed with focus on the corresponding diode like effect. Transmission coefficients are analytically calculated for left and right moving waves via backward transfer map. The different transmission coefficients for the left and right moving waves impinging the lattice give rise to a diode like effect which is tested for different variations in asymmetry and site dependent coefficients. We show that there is a higher transmission for incoming waves with lower wavenumbers as compared to the waves with comparatively larger wavenumbers and a diode like effect improves by increasing the nonlinear layers. We also show that in the context of transport through such lattices, the cooperation between cubic and quintic nonlinear response is not "additive" Finally, we numerically analyse Gaussian wave packet dynamics impinging on the CQDNLS lattice for all three cases.
author_names: Muhammad Abdul Wasay
title: Asymmetric wave transmission through one dimensional lattices with cubic quintic nonlinearity
venue: Scientific Reports
year: 2018

abstract: Recent progress in topological mechanics has revealed a family of Maxwell lattices that exhibit topologically protected floppy edge modes. These modes lead to a strongly asymmetric elastic wave response. In this paper, we show how topological Maxwell lattices can be used to realize nonreciprocal transmission of elastic waves. Our design leverages the asymmetry associated with the availability of topological floppy edge modes and the geometric nonlinearity built into the mechanical systems' response to achieve the desired nonreciprocal behavior, which can be further utilized to form a phonon diode via the addition of on site pinning potentials that blocks the linear transmission and only allows the signal to transmit in one way. Finally, the nonreciprocal wave transmission can be switched on and off via topological phase transitions, paving the way to the design of cellular metamaterials that can serve as tunable topologically protected phonon diodes.
author_names: Di Zhou

Jihong Ma

Kai Sun

Stefano Gonella

Xiaoming Mao
title: Switchable phonon diodes using nonlinear topological Maxwell lattices
year: 2020

abstract: We consider a discrete model that describes a linear chain of particles coupled to an isolated ring composed of N defects. This simple system can be regarded as a generalization of the familiar Fano Anderson model. It can be used to model discrete networks of coupled defect modes in photonic crystals and simple waveguide arrays in two dimensional lattices. The analytical result of the transmission coefficient is obtained, along with the conditions for perfect reflections and transmissions due to either destructive or constructive interferences. Using a simple example, we further investigate the relationship between the resonant frequencies and the number of defects N, and study how to affect the numbers of perfect reflections and transmissions. In addition, we demonstrate how these resonance transmissions and refections can be tuned by one nonlinear defect of the network that possesses a nonlinear Kerr like response.
author_names: Cun-Xi Zhang

Xiu-Huan Ding

Rui Wang

Yun-Qing Zhou

Ling Min Kong
title: Fano resonance and wave transmission through a chain structure with an isolated ring composed of defects
year: 2012

abstract: We study numerically light beam propagation across uniform, linear, one dimensional photonic lattice possessing one nonlinear defect. Depending on the strength of nonlinear defect, input beam position and phase shift, different dynamical regimes have been identified. We distinguish input parameters set for which a regime of light propagation blockade by the nonlinear defect appears. Obtained results may be useful for all optical control of transmission of waves in interferometry.
author_names: S Kuzmanovic

Marija Stojanovic Krasic

Daniela M Milovic

Ana Radosavljevic

G Gligoric

Aleksandra Maluckov

Milutin Stepic
title: Defect induced wave packet dynamics in linear one dimensional photonic lattices
year: 2015

abstract: The interaction of a solitary wave with an interface formed by two strongly nonlinear noncohesive granular lattices displays rich behavior, characterized by the breakdown of continuum equations of motion in the vicinity of the interface. By treating the solitary wave as a quasiparticle with an effective mass, we construct an intuitive (energy and linear momentum conserving) discrete model to predict the amplitudes of the transmitted solitary waves generated when an incident solitary wave front, parallel to the interface, moves from a denser to a lighter granular hexagonal lattice. Our findings are corroborated with simulations. We then successfully extend this model to oblique interfaces, where we find that the angle of refraction and reflection of a solitary wave follows, below a critical value, an analogue of Snell's law in which the solitary wave speed replaces the speed of sound, which is zero in the sonic vacuum.
author_names: Abigail Tichler

Leopoldo R Gomez

Nitin Upadhyaya

Xander Campman

Vitali F Nesterenko

Vincenzo Vitelli
title: Transmission and reflection of strongly nonlinear solitary waves at granular interfaces.
venue: Physical review letters
year: 2013

abstract: We study the dynamics of waves in honeycomb lattices. Wave dynamics in honeycomb lattices exhibits unique phenomena that cannot be observed in any other periodic system, such as conical diffraction [1] and Klein tunneling. The origin of these intriguing phenomena is the conical intersection between the two bands and the presence of two atoms in each unit cell. These two properties make it possible to describe the dynamics of the Bloch modes by the massless Dirac equation the same equation describing neutrinos that are massless spin half particles. It it very well known that when particles (or a wave packet) incident upon an infinite potential step of height V0 with energy lower then the step are always reflected. However, Dirac particles (relativistic fermions) have non zero transmission probability [2] This remarkable phenomenon is named Klein tunneling, first proposed in 1929. Even more surprising is the scattering of quasi particles in graphene/carbon nanotubes that behave as massless fermions. The later do not exhibit ordinary Klein tunneling but instead they exhibit unit transmission probability [3, 4] Here, we study scattering of wave packets from potential steps in deformed honeycomb photonic lattices, and find that below a critical deformation in which a gap in the spectrum is formed the wave packet is completely transmitted, i.e. an optical beam exhibits unit transmission [5] On the other hand, beyond the critical deformation we find total reflection for a wave packet incident upon a step in the deformation direction [5] Hence, we witness a very sharp transition from unit transmission to total reflection, giving an indication for a possible quantum phase transition. This remarkable phenomenon occurring only in honeycomb lattices is completely linear. However, it is most interesting to also study the nonlinear dynamics of waves in honeycomb lattices. We study the nonlinear propagation of a circular wave packet from the Dirac cone, that under linear conditions evolves into two rings with zero intensity in the middle, i.e. conical diffraction [1] Surprisingly, we find that the same circular wave packet evolves into "triangular rings" (triangles with slightly curves lines) By projecting the output beam onto the Bloch modes of the system, we find that the nonlinearity is responsible for energy transfer between different Bloch modes, and eventually the wave packet is comprised of Bloch mode that do not reside inside the Dirac cone anymore. Hence, the nonlinearity breaks the Dirac dynamics in honeycomb lattices, giving rise to completely new phenomena.
author_names: Omri Bahat-Treidel

Or Peleg

Mordechai Segev
title: Nonlinear Waves in Honeycomb graphene like Photonic Lattices
year: 2010

abstract: We introduce a family of non Hermitian optical potentials that are given in terms of double exponential periodic functions. The center of \mathcal{PT} symmetry is not around zero and the potential satisfies a shifted \mathcal{PT}\text{ symmetry} relation at two distinct locations. Motivated by wave transmission through thin phase screens and gratings, we examine these refractive index modulations from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics, abrupt phase transitions in the eigenvalue spectrum, and exceptional points in the band structure are examined in detail. In addition, the nonlinear properties of wave propagation in Kerr nonlinearity media are studied. In particular, coherent structures such as lattice solitons are numerically identified by applying the spectral renormalization method. The spatial symmetries of such lattice solitons follow the shifted \mathcal{PT}\text{ symmetric} relations. Furthermore, such lattice solitons have a power threshold and their linear and nonlinear stabilities are critically dependent on their spatial symmetry point.
author_names: Justin T Cole

Konstantinos G Makris

Ziad Musslimani

Demetrios N Christodoulides

Stefan Rotter
title: Twofold PT symmetry in doubly exponential optical lattices
year: 2016

abstract: The wave propagation problem in the nonlinear periodic mass spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.
author_names: Xiuhui Hou

Zichen Deng

Jiaxi Zhou
title: Symplectic analysis for wave propagation in one dimensional nonlinear periodic structures
year: 2010