Power propagation in homogeneous isotropic frequency-dispersive left-handed media

From solitons to rogue waves in nonlinear left-handed metamaterials

In the present work, we explore soliton and roguelike wave solutions in the transmission line analog of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltage-dependent, symmetric capacitance motivated by recently developed ferroelectric barium strontium titanate thin-film capacitor designs. We develop both the corresponding nonlinear dynamical lattice and its reduction via a multiple scales expansion to a nonlinear Schrödinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wave number) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS model is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely, Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.

Time-driven superoscillations with negative refraction

The flat-lens concept based on negative refraction proposed by Veselago in 1968 has been mostly investigated in the monochromatic regime. It was recently recognized that time development of the superlensing effect discovered in 2000 by Pendry is yet to be assessed and may spring surprises: Time-dependent illumination could improve the spatial resolution of the focusing. We investigate dynamics of flexural wave focusing by a 45°-tilted square lattice of circular holes drilled in a duralumin plate. Time-resolved experiments reveal that the focused image shrinks with time below the diffraction limit, with a lateral resolution increasing from 0.8λ to 0.35λ, whereas focusing under harmonic excitation remains diffraction limited. Modal analysis reveals the role in pulse reconstruction of radiating lens resonances, which repeatedly self-synchronize at the focal spot to shape a superoscillating field.

Transient chaos in optical metamaterials

We investigate the dynamics of light rays in two classes of optical metamaterial systems: (1) time-dependent system with a volcano-shaped, inhomogeneous and isotropic refractive-index distribution, subject to external electromagnetic perturbations and (2) time-independent system consisting of three overlapping or non-overlapping refractive-index distributions. Utilizing a mechanical-optical analogy and coordinate transformation, the wave-propagation problem governed by the Maxwell's equations can be modeled by a set of ordinary differential equations for light rays. We find that transient chaotic dynamics, hyperbolic or nonhyperbolic, are common in optical metamaterial systems. Due to the analogy between light-ray dynamics in metamaterials and the motion of light in matter as described by general relativity, our results reinforce the recent idea that chaos in gravitational systems can be observed and studied in laboratory experiments.

Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials

Nonlinear propagation characteristics are investigated theoretically in a one-dimensional photonic band-gap structure containing indefinite metamaterials. It is found that optical bistability can be achieved at very low threshold values of normalized input intensity in this composite structure when the indefinite metamaterial is a cut-off, anticut-off, and never cut-off medium. It is also found that there exists corresponding bistable lateral shift which is strongly dependent on the parameters of indefinite metamaterials.

Plasmon polaritons in photonic metamaterial superlattices: absorption effects

We discuss the propagation of electromagnetic waves in layered structures made up of alternate layers of air and metamaterials. The role played by absorption on the existence of electric and magnetic plasmon polaritons is investigated. Results show that plasmon-polariton modes are robust even in the presence of rather large absorption.

Achieving subdiffraction imaging through bound surface states in negative refraction photonic crystals in the near-infrared range

We report the observation of imaging beyond the diffraction limit due to bound surface states in negative refraction photonic crystals. We achieve an effective negative index figure of merit [-Re(n)/Im(n)] of at least 380, approximately 125x improvement over recent efforts in the near-infrared range, with a 0.4 THz bandwidth. Supported by numerical and theoretical analyses, the observed near-field resolution is 0.47lambda, clearly smaller than the diffraction limit of 0.61lambda. Importantly, we show this subdiffraction imaging is due to the resonant excitation of surface slab modes.

Experimental observation of left-handed behavior in an array of standard dielectric resonators

We demonstrate that by utilizing displacement currents in simple dielectric resonators instead of conduction currents in metallic split-ring resonators and by additionally exciting the proper modes, left-handed properties can be observed in an array of high dielectric resonators. Theoretical analysis and experimental measurements show that the modes, as well as the subwavelength resonance, play an important role in the origin of the left-handed properties. The proposed implementation of a left-handed metamaterial, based on a purely dielectric configuration, opens the possibility of realizing media at terahertz frequencies since scaling issues and losses, two major drawbacks of metal-based structures, are avoided.

Construction of a polarization insensitive lens from a quasi-isotropic metamaterial slab

We propose to employ the quasi-isotropic metamaterial (QIMM) slab to construct a polarization insensitive lens, in which both E- and H-polarized waves exhibit the same refocusing effect. For shallow incident angles, the QIMM slab will provide some degree of refocusing in the same manner as an isotropic negative index material. The refocusing effect allows us to introduce the ideas of paraxial beam focusing and phase compensation by the QIMM slab. On the basis of angular spectrum representation, a formalism describing paraxial beams propagating through a QIMM slab is presented. Because of the negative phase velocity in the QIMM slab, the inverse Gouy phase shift and the negative Rayleigh length of paraxial Gaussian beam are proposed. We find that the phase difference caused by the Gouy phase shift in vacuum can be compensated by that caused by the inverse Gouy phase shift in the QIMM slab. If certain matching conditions are satisfied, the intensity and phase distributions at object plane can be completely reconstructed at image plane. Our simulation results show that the superlensing effect with subwavelength image resolution could be achieved in the form of a QIMM slab.

Group front evolution of Gaussian beam refracted from a right- to left-handed medium

Amplitude enhancement in a group front of continuous wave (CW) Gaussian beam refracted at the boundary of right- and left-handed media is observed. Behind the interface plane in a high dispersion double negative medium the individual Fourier components of the beam diffract at different angles and have diversified phase speeds. This results in the group front build-up that propagates on with the beam and moves sideways with respect to the group velocity direction, where energy is transported. The enhancement is illustrated with 2-D simulations using finite difference time domain (FDTD) method.

Delay times and detector times for optical pulses traversing plasmas and negative refractive media

We show that arrival times for electromagnetic pulses measured through the rate of absorption in an ideal impedance matched detector are equivalent to the arrival times using the average flow of optical energy as proposed by Peatross [Phys. Rev. Lett. 84, 2370 (2000)]. We then investigate the transport of optical pulses through dispersive media with negative dielectric permittivity and negative refractive index choosing the geometry such that no resonant effects come into play. For evanescent waves, the definitions of the group delay, and the reshaping delay get interchanged in comparison to propagating waves. The total delay times for the evanescent waves can be negative in an infinite plasma medium even for broadband pulses. The total delay time is, however, positive for broadband pulses in the presence of an interface when the radiation is detected outside the plasma. We find evidence of the Hartman effect for pulses when the distance traversed in the plasma is much smaller than the free space pulse length. We also show that for a negative refractive index medium (NRM) with epsilon(omega)=mu(omega) the reshaping delay for propagating waves is identically zero. The total delay time in NRM is otherwise dominated by the reshaping delay time, and for broadband pulses in NRM the total delay time is subluminal.

Conservation of electromagnetic momentum and radiation forces exerted on left-handed material interfaces

It is shown that the boundary conditions at left-handed material interfaces cause reversal in the propagation direction of the parallel-to-the-surface electromagnetic energy and momentum fluxes. First-principle examination excludes the possibility of induced surface wave excitation as a way of providing radiation momentum conservation. Thus the imparted net change in electromagnetic momentum should cause a recoil force parallel to the surface, which is unique to left-handed interfaces. The shear force exerted on a left-handed material interface is computed. The magnitude of this force is found detectable by experiments.

Exact solution to plane-wave scattering by an ideal "left-handed" wedge

An exact analytical solution to the problem of plane-wave diffraction by a penetrable left-handed medium (LHM) epsilon = micro = -1 wedge of arbitrary angle (subject to valid physical constraints) is presented. Standard analysis involving discontinuous angular eigenfunctions and even/odd symmetry decomposition resulted in a discrete spectrum leading to a series solution resembling the traditional perfect electric conductor wedge solution but exhibiting the expected negative refraction phenomenology. Numerical results are presented, some of which seemed paradoxical but are explainable by classical means. A new type of illusory edge radiation is observed and explained. Also, a novel edge-launched interface standing wave is observed on the directly illuminated side. The exact analytical solution is verified by comparison with finite-difference time-domain simulation on causal LHM materials.

Resolving the wave vector in negative refractive index media

We address the general issue of resolving the wave vector in complex electromagnetic media including negative refractive media. This requires us to make a physical choice of the sign of a square root imposed merely by conditions of causality. By considering the analytic behavior of the wave vector in the complex plane, it is shown that there are a total of eight physically distinct cases in the four quadrants of two Riemann sheets.

Brewster angle with a negative-index material

The demonstration and confirmation of metamaterials with simultaneous negative permittivity and permeability, and thus a negative refractive index, has resulted in a surge of interest in the reflection and refraction phenomena at the interfaces of these so-called negative-index materials (NIMs). We present a systematic study of the Brewster angle, i.e., the angle of incidence at which no reflection occurs, for both TE and TM waves scattering at the interface between two semi-infinite planar media, one of which may be a NIM. Detailed physical explanations that account for the Brewster angle for a plane wave incident upon a NIM are provided under the framework of the Ewald-Oseen extinction theorem, considering the reemission of induced electric and magnetic dipoles. The conditions under which the Brewster angle exists are concisely summarized in a map of different material parameter regimes.

Scattering properties of an impedance-matched, ideal, homogeneous, causal "left-handed" sphere

The plane-wave scattering properties of a sphere of material having an ideal, homogeneous, and causal permittivity epsilon(f), and permeability mu(f) were investigated through detailed three-dimensional finite-difference time-domain, method-of-moments, and series-solution simulations. A Lorentzian functional form was chosen for epsilon(f) and mu( f), as it yields causal responses and allows us to study the physics of the left-handed-medium (LHM) regime. Our interest lies mainly in the frequency range where negative refraction [Re(n) < 0] is observed. We found that when operating in the LHM regime, an impedance-matched sphere responds with scattering features strikingly different from those found in ordinary materials. In particular, we found zero back-scattering and forward scattering that exceeds that of a metal sphere of similar size. The equality of E- and H-plane patterns was proved analytically and numerically, and the possibility of internal subwavelength focusing with a zero index sphere is also reported.

Left-handed materials composed of only S-shaped resonators

We analyze an S-shaped inclusion for the realization of metamaterials exhibiting left-handed properties. Unlike most of the conventional inclusions used so far that are composed of two separate geometries-typically a split ring and a rod-the inclusion proposed in this paper is made of only one S-shaped element which yields an overlapping negative permittivity and negative permeability response over a frequency band of about 2.6 GHz. By adopting this geometry, we manage to lower the negative permittivity frequency band down to the level of the negative permeability frequency band, thus allowing the overlapping to occur. Therefore, the structure works as a stand alone and does not require the use of an additional rod. A theoretical analysis is carried out to study this inclusion and numerical simulations, as well as a Snell refraction experiment, clearly show that the material indeed exhibits a negative index of refraction at some frequencies. The simple pattern of the inclusion, the wide left-handed frequency band exhibited, and the low losses measured indicate the superiority of this inclusion in the realization of left-handed metamaterials.

Photon tunneling in composite layers of negative- and positive-index media

The photon tunneling phenomena in the composite barrier of both positive index material and negative index medium (NIM) are analyzed. A negative barrier-length concept is introduced for the tunneling photon in the NIM, which facilitates the analysis of a composite barrier. In addition to transmission and reflection coefficients, some comments on the post-tunneling position and stationary-phase tunneling time are given.

Effective medium theory of left-handed materials

We analyze the transmission and reflection data obtained through transfer matrix calculations on metamaterials of finite lengths, to determine their effective permittivity epsilon and permeability micro. Our study concerns metamaterial structures composed of periodic arrangements of wires, cut wires, split ring resonators (SRRs), closed SRRs, and both wires and SRRs. We find that the SRRs have a strong electric response, equivalent to that of cut wires, which dominates the behavior of left-handed materials (LHM). Analytical expressions for the effective parameters of the different structures are given, which can be used to explain the transmission characteristics of LHMs. Of particular relevance is the criterion introduced by our studies to identify if an experimental transmission peak is left or right handed.

Metamaterials and negative refractive index

Recently, artificially constructed metamaterials have become of considerable interest, because these materials can exhibit electromagnetic characteristics unlike those of any conventional materials. Artificial magnetism and negative refractive index are two specific types of behavior that have been demonstrated over the past few years, illustrating the new physics and new applications possible when we expand our view as to what constitutes a material. In this review, we describe recent advances in metamaterials research and discuss the potential that these materials may hold for realizing new and seemingly exotic electromagnetic phenomena.

Finite-size effects of a left-handed material slab on the image quality

The characteristics of an imaging system formed by a left-handed material (LHM) slab of finite length are studied, and the influence of the finite length of the slab on the image quality is analyzed. Unusual phenomena such as surface bright spots and negative energy stream at the image side are observed and explained as the cavity effects of surface plasmons excited by the evanescent components of the incident field. For a thin LHM slab, the cavity effects are found rather sensitive to the length of the slab; the bright spots on the bottom surface of the slab may stretch to the image plane and degrade the image quality.

Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative

We analyze refraction of electromagnetic wave packets on passing from an isotropic positive to an isotropic negative-refractive-index medium. We definitively show that in all cases the energy is always refracted negatively. For localized wave packets, the group refraction is also always negative.

Resonant and antiresonant frequency dependence of the effective parameters of metamaterials

We present a numerical study of the electromagnetic response of the metamaterial elements that are used to construct materials with negative refractive index. For an array of split ring resonators (SRR) we find that the resonant behavior of the effective magnetic permeability is accompanied by an antiresonant behavior of the effective permittivity. In addition, the imaginary parts of the effective permittivity and permeability are opposite in sign. We also observe an identical resonant versus antiresonant frequency dependence of the effective materials parameters for a periodic array of thin metallic wires with cuts placed periodically along the length of the wire, with roles of the permittivity and permeability reversed from the SRR case. We show in a simple manner that the finite unit cell size is responsible for the antiresonant behavior.

Total negative refraction in real crystals for ballistic electrons and light

It is found that there exists a category of material interfaces, readily available, that not only can provide total refraction (i.e., zero reflection) but can also give rise to amphoteric refraction (i.e., both positive and negative refraction) for electromagnetic waves in any frequency domain as well as for ballistic electron waves. These two unusual phenomena are demonstrated experimentally for the propagation of light through such an interface.

Causality and double-negative metamaterials

The causality of waves propagating in a double-negative (DNG) metamaterial (epsilon (r)<0 and mu(r)<0) has been investigated both analytically and numerically. By considering the one-dimensional electromagnetic problem of a pulsed current sheet radiating into a DNG medium, it is shown that causality is maintained in the presence of a negative index of refraction only if the DNG medium is dispersive. A Drude model DNG medium is used in this study. Spectrograms of the wave phenomena in the dispersive DNG medium show that the higher frequency components, which create the leading edge of the electromagnetic signals and see a double positive (DPS) medium (epsilon (r)>0 and mu(r)>0), arrive causally before the negative index effects germinate completely. Comparisons with approximate analytical results demonstrate the presence of the negative index of refraction properties in the continuous wave portion of the signals. This dynamic pulse reshaping between the positive and negative index of refraction wave components causes an apparent delay in the realization of the negative index of refraction properties. Pulse broadening of the signal tails is associated with both dispersion and a larger negative index of refraction seen by the associated wave components.

Influence of the dispersive properties of metals on the transmission characteristics of left-handed materials

We study numerically the influence of the frequency dispersion of the dielectric function of metals on the physical properties of negative-refractive-index metamaterials. A numerical analysis is performed using the transfer matrix formalism in conjunction with the finite-difference time-domain method. We analyze the dependence of the transmission and absorption properties of a slab of split-ring-type resonators on the parameters characterizing the frequency dispersion of the metallic dielectric function: plasma frequency and damping frequency. Then, using these transmission and reflection coefficients, we show that the refractive index remains negative near the resonant frequency of the rings, despite the presence of frequency dispersion. We also determine the dependence of the position and width of the band gaps of a slab of such a metamaterial on the material dispersion. Finally, we also discuss the influence of the shape of the split-ring resonators on the transmission and reflection coefficients. The calculations are performed for both two- and three-dimensional structures.

Experimental verification and simulation of negative index of refraction using Snell's law

We report the results of a Snell's law experiment on a negative index of refraction material in free space from 12.6 to 13.2 GHz. Numerical simulations using Maxwell's equations solvers show good agreement with the experimental results, confirming the existence of negative index of refraction materials. The index of refraction is a function of frequency. At 12.6 GHz we measure and compute the real part of the index of refraction to be -1.05. The measurements and simulations of the electromagnetic field profiles were performed at distances of 14lambda and 28lambda from the sample; the fields were also computed at 100lambda.