Simulation of isotropic acoustic metamaterials

The model use to study electromagnetic metamaterials in transverse electric mode was modified to study the pressure distribution in an acoustic metamaterial in a two dimensional geometry. An electromagnetic wave of 30 GH z in transverse magnetic mode at normal incidence propagating through a two dimensional isotropic semi infinite double negative metamaterial slab of 640 830  cells embedded in free space with low loss damping frequency 8 1 10 s  was studied by finite difference time dependent method with Yee’s algorithm with an explicit leapfrog scheme. The multiple cycle m-n-m pulses generating beams were used as sources. The simulations for refractive indices 1 n  and 6 n  at different time steps are presented. For 1 n  , the sub-wavelength imaging is apparent, diverging the waves from source into two sources, one inside and the other outside the slab. The reverse propagation is much more significant for 6 n  showing that the group velocity is much larger inside the metamaterial by the closely packed wave front, making the continuation of the wave pattern much more significant through the metamaterial into the normal space.


INTRODUCTION
Metamaterials are artificially fabricated materials designed to control, direct and manipulate light as well as sound waves in fluids and solids. These assemblies fashioned from replacing the molecules of conventional materials with artificial atoms on a scale much less than the relevant wavelength, derive their properties from their exactingly-designed structures. The core concept of metamaterial is the negative index of refraction for particular wavelengths, electromagnetic or sound waves propagating in reverse. The first metamaterials were developed for electromagnetic waves. Acoustic wave, a longitudinal wave have common physical concepts and in the two dimensional case, when there is only one polarization mode, the electromagnetic wave has scalar wave formulation and the optical and acoustic metamaterials share many similar implementation approaches as well. The controlling of the various forms of sound waves is mostly accomplished through the bulk modulus β, mass density ρ, and chirality. In negative index materials, the density and bulk modulus are analogies of the electromagnetic parameters, permittivity and permeability. Also materials have mass and intrinsic degrees of stiffness. Together these form a resonant system, and may be excited by appropriate sonic frequencies for pulses at audio frequencies.
An experimental acoustic metamaterial based on the transmission line model, as well as unit cells for isotropic and anisotropic metamaterials were proposed by Zang [1]. To simulate two dimensional electromagnetic materials in transverse electric mode, finite difference time domain (FDTD) method is widely used [2]- [4]. A dispersive FDTD scheme or a leap frog scheme has also been used to determine the values of electric and magnetic fields as well as induced electric current and induced magnetic current values inside the metamaterials [5], [6]. The notations used in these dispersive schemes are adopted from Taflove and Hagness [7]. The existence of an analogy between the acoustic metamaterial parameters and electromagnetic metamaterial parameters in transverse magnetic mode, first proposed by Cummer and Schurig [8] was used by Torrent and Dehesa [9].
In this work, the model used by Ziolkowski [4] to study electromagnetic metamaterials using transverse electric mode was modified to transverse magnetic mode to study the pressure distribution of an acoustic wave in metamaterials. The simulation were performed for electromagnetic wave of 30GHz in transverse magnetic mode at normal incidence through a two dimensional isotropic semi infinite double negative metamaterial slab of 640 830  cells embedded in free space with low loss damping frequency 81 10 s  by finite difference time dependent method with Yee's algorithm which calculates the electric and magnetic field in an explicit leapfrog scheme. For the simulations the mesh size used was 100μm with time steps of 22.39 ps . The analogy between acoustic and electromagnetic metamaterials is used to illustrate the behaviour of magnitude of z component of electric field distribution in electromagnetic metamaterial or magnitude of pressure distribution of acoustic metametrial for refraction indices 1 n  and 6 n  .

DRUDE METAMATERIAL MODEL
The pressure and the particle velocity are the parameters used to describe an acoustic wave which is a longitudinal wave. In the two dimensional Cartesian coordinate with z invariance, the time harmonic acoustic equations for fluid with anisotropic or isotropic medium are: where x u and y u are particle velocities in x and y directions, P is acoustic scalar pressure, x  and y  are the dynamic density along x and y directions and  is the dynamic compressibility. These lead to the pressure equation: In electromagnetism, both electric and magnetic fields are transverse waves. However the two wave systems have the common physical concepts such as wave vector, wave impedance and power flow. The manner in which the permittivity and permeability behave in electromagnetism is closely analogous to the behaviour of compressibility and density in an acoustic system. Moreover in a two dimensional case, when there is only one polarization mode the acoustic waves in fluid and the Maxwell equations for electromagnetic wave have identical form under certain variable exchange. An analogy has been proposed between acoustic and electromagnetic metamaterials in Cartesian coordinates [1] The boundary conditions in electromagnetism are that the normal component of electric field E z and tangential component of magnetic field H are continuous while at a fluid interface, the normal component of particle velocity y u and pressure P are continuous. Therefore the boundary conditions are preserved under this variable exchange. For normal incidence of acoustic waves, solids obey the same equations developed for fluids with the modification that the speed of sound in the solid is the bulk sound depending on the bulk modulus and the sheer modulus. For normal incidence, the boundary between two fluids is considered to be at j t k t t ta P P e    The relevant particle velocities are respectively. The boundary conditions require that at 0 The reflection and transmission coefficients are given by: The reflection coefficient of intensity I r and transmission coefficient of intensity I t are: where 1 c and 2 c are the velocities of sound in the two mediums [1]. The Drude metamaterial model is the most popular and widely used model in electromagnetic metamaterial simulations. The Lossy Drude polarization and magnetization models are used in the frequency domain in Drude metamaterial model and describe the electric permittivity and magnetic permeability as: For simplicity, it is assumed that the boundary of the domain  is a perfect conductor, leading to ˆ0 n E on  where n is the unit outward normal to  . Moreover, it is assumed that the initial conditions are

ILCPA Volume 65
The finite-difference time domain (FDTD) method is very popular in modelling electromagnetic fields. Yee's algorithm is based on the time-dependent Maxwell's curl equations and it couples the equations in order to solve for multiple field components simultaneously. The spatially staggered grid simplifies the contours involved in the curl equations, maintaining the continuity of the tangential components of the electric and magnetic fields, as well as simplifying the implementation of the boundary conditions. Yee's algorithm uses a fully explicit leapfrog scheme in time that also involved second-order central differences, so the field components are staggered temporally. This means the electric field is calculated before or after, but not simultaneously with the magnetic field thereby defining the leapfrog scheme. Any three dimensional vector field component can be written as ,, The Yee algorithm is a conditionally stable algorithm, and there exists a maximum allowable time step max t  in order for the algorithm to remain stable.
where c is the speed of light in vacuum. In one dimension, the Courant number (S) is defined as, / S c t x    and 1 S  . For , x y z      in order for the algorithm to remain stable 1/ 2 S  in two dimensional case and 1 / 3 S  in three dimensional case. Simulating wave propagation from scattering or waveguides requires an unbounded domain or a domain large enough so that waves do not reflect of the domain boundaries back into the computational domain and interfere with the wave propagation being analyzed. As it is impossible to have an unbounded domain in scientific computing, the absorbing boundary conditions (ABCs) have emerged. This is an additional domain surrounding the computational domain. It is connected to the computational domain boundary, but the fields are computed separately in this external domain. Here all tangential properties are preserved, and the fields are continuous across the boundary. It is made so that it absorbs waves that come in contact with that region. The external region is designed as a lossy material, so that it does not really absorb the wave but severely dampen the wave as it enters that region by removing its power so quickly that there's nothing left to reflect back off the outer boundary [7]. The Yee lattice used in FDTD is shown in figure 2. Equation 12 in the form given in equation 13 read:   1  11  2  22  3  1  1  1  1  1  1  2  2  2  2  2  2  2 , , , where  and   are the electric conductivity and the equivalent magnetic loss respectively.
x y z      is the cell size or mesh size, t  is the time step and m standing for the media is 2 for the two dimensional simulation, so that ,, aa b C C D and D b are arrays of two elements. This dispersive scheme was used to calculate the parameters within the metamaterial.

TRANSVERSE MAGNETIC MODES IN METAMATERIALS
The simulations done by Ziolkowski [4] for transverse electric mode in y direction was modified to transverse magnetic (TMz) mode. For normal mode the reflection coefficient 0 R  and transmission coefficient 1. T  The general index of refraction of the medium (index i) through the wave is propagating is .
Consider the case when 1 n  , then With the source distance and the slab depth equal, the foci of the beam in the metamaterial slab and in the background medium beyond the metamaterial slab coincide for 1 n  at a distance equal to the depth of the slab. That is, the focal plane of the beam occurs at the rear face of the slab. This means that the metamaterial slab should ideally reproduce the beam at its rear face as it exists from the metamaterial. Therefore the metamaterial slab in this case acts as an acoustic lens.
For the refractive index 6, n  the simulations for z-component of electric field z E or equivalently the pressure P of the acoustic wave at time steps 1000, 3000 and 5000 are presented in figures 6, 7 and 8 respectively. The highest values are represented by red colour and the lowest values by blue. When the beam interacts with the matched metamaterial slab with 6 n  , little focusing is observed. The negative angles of refraction dictated by Snell's law are shallower for this higher magnitude of the refractive index. Rather than a strong focusing, the medium channels power from the wings of the beam towards its axis, hence, maintaining its amplitude as it propagates into the metamaterial medium. The fact that the wings of the beam feed its center portion can be perceived by the converging wave fronts shown in Figure 6 at the edges of the beam in the metamaterial slab.  DNG slab is shown by the black rectangle.

CONCLUSIONS
In order to illustrate the characteristics of an acoustic metamaterial network, the finite difference time dependent (FDTD) method and Drude model in transverse magnetic mode was adopted and the results were simulated. These models incorporate terms such as, induced electric and magnetic currents, magnetic and electric plasma frequencies. For acoustic metamaterial parameters, such terms are not useable. To avoid confusions, the derivations and simulations were carried on for the electromagnetic metamaterials, to show the final output and then to compare the results for the acoustic metamaterials as the results are completely identical. Namely under the variable exchange the behaviour of magnitude of z-component of electric field z E is analogous to the behaviour of magnitude of pressure P in an acoustic metamaterial slab. Also the boundary condition for electromagnetic metamaterial slab is that the normal component of z E is continuous and in acoustic metamaterials, the normal component of pressure is continuous.
The phenomenon known as negative refraction is visible in these simulations. The propagation of sound is reverse to the incident waves within the metamaterial. For the negative refractive index of one, the sub-wavelength imaging is apparent, as the sound waves from source diverges into two sources, one inside and the other outside the slab. In the case of negative refractive index of six the reverse propagation is much significant than for the refractive index of minus one and shows that the group velocity is much larger inside the metamaterial by the closely packed wave front and the wave pattern continuation is more significant though the metamaterial in to the normal space. In all the metamaterial cases the beam appears to diverge significantly once it leaves the metamaterial slab. The properties of the metamaterial medium hold the beam together as it propagates through the slab. If the metamaterial slab focuses the beam as it enters, the same physics will cause the beam to diverge as it exits. The rate of divergence of the exiting beam will be determined by its original value and the properties and size of the metamaterial medium. The beam focusing into a metamaterial slab generating a diverging beam within the slab and a converging beam as it exits from the slab is confirmed with the FDTD simulator.