# Impact of the inherent periodic structure
on the effective medium description

of left-handed and related meta-materials

###### Abstract

We study the frequency dependence of the effective electromagnetic parameters of left-handed and related meta-materials of the split ring resonator and wire type. We show that the reduced translational symmetry (periodic structure) inherent to these meta-materials influences their effective electromagnetic response. To anticipate this periodicity, we formulate a periodic effective medium model which enables us to distinguish the resonant behavior of electromagnetic parameters from effects of the periodicity of the structure. We use this model for the analysis of numerical data for the transmission and reflection of periodic arrays of split ring resonators, thin metallic wires, cut wires as well as the left-handed structures. The present method enables us to identify the origin of the previously observed resonance/anti-resonance coupling as well as the occurrence of negative imaginary parts in the effective permittivities and permeabilities of those materials. Our analysis shows that the periodicity of the structure can be neglected only for the wavelength of the electromagnetic wave larger than 30 space periods of the investigated structure.

###### pacs:

41.20.Jb, 42.25.Bs, 42.70.Qs, 73.20.Mf## I Introduction

Recent progress in studies of left-handed meta-materials (LHM)Veselago68 confirmed that the fabrication of structures with negative effective permittivity and permeability, and their application in technical praxis is possible. The most promising structures are based on the combination of periodic arrays of metallic split ring resonators (SRR) and thin metallic wires, a design proposed theoretically by Pendry et al. Pendry96b ; Pendry98 ; Pendry99 and experimentally verified by Smith et al. Smith00 ; Shelby01a ; Shelby01b .

It is assumed that in a well defined frequency interval both effective permittivity and permeability of LHM are simultaneously negative. Consequently, also the refractive index is negative SK . This theoretical prediction was supported experimentally by measurements of the transmission of the electromagnetic (EM) wave through the LHM: A transmission peak was observed in the frequency region where the LH band is expectedSmith00 ; Shelby01a . Negativeness of the index of refraction was verified experimentally by the Snell’s law experimentShelby01b and confirmed later by other experiments Parazzoli03 ; Houck03 . Numerical simulations were performed which also observed a transmission peak in the resonant frequency intervalMarkos02 ; Markos02a ; Markos02b . Effective electromagnetic parameters were calculatedSmith02b by comparison of numerically obtained transmission and reflection amplitudes of the LHM with theoretical formulas for a homogeneous slab. The obtained results confirmed that the refractive index of the LHM is indeed negative in the resonant frequency interval. Moreover, the obtained frequency dependence of the effective permittivity and permeability also agreed qualitatively with theoretical predictions. In particular, the effective magnetic permeability shows a resonant behavior,

(1) |

typical for lattice of SRR Pendry99 in the vicinity of the magnetic resonance frequency . The effective permittivity is determined by the electric response of the array of thin wiresSigalas95 ; Pendry98 ; Sarychev ; Pokrovsky02b ,

(2) |

and is negative if the frequency is smaller than the plasma frequency . Transmission data, obtained using either the transfer matrix method Markos02a or commercial software Weiland01 was analyzed to find the dependence of the resonance frequency on the structural parameters of the SRR and on the design of the unit cell of the LHM structure.

Further progress in numerical methods brought more accurate data and strong evidence that the effective parameters of the LHM differ considerably from the theoretical prediction (1, 2). Although the main properties – resonant behavior of the magnetic permeability at and negativeness of the effective permittivity – are clearly visible in the data, the effective medium picture is spoiled by partially very significant anomalies:

Resonance/anti-resonance coupling. We expect the electric and magnetic response of the discussed meta-materials to be independent from each other. However, whenever there is a resonance in , we simultaneously observe an anti-resonant behavior in OBrien02 ; Smith02b ; Markos03a ; OBrien04 ; Chen04 and vice versaKoschny03b .

The anti-resonant structures in the real part are accompanied by an negative imaginary partMarkos03a ; Koschny03b .

Misshapen, truncated resonances. The divergence in appears to be cut-off at the edges of the first Brillouin zone and, in particular, the negative regions of the magnetic resonance in and cut-wire resonance in do not return from large negative real part but seem to saturate in a rather shallow behavior. The corresponding absorption peak in the imaginary parts is misshapen and highly asymmetric tooKoschny03b .

Discrepancy between and about the positions of the resonances. We expect the peaks (or zeros) in the index of refraction and the impedance to appear exactly at the resonance frequency. From the simulations, however, we find different frequencies from and , respectively. This leads, for instance, to an “internal structure” of the magnetic resonance as shown in Figs. 5 and 6. This structure can not easily be explained within the assumed effective medium pictureKoschny03b .

Additional spectral structures. Apart from structures around the anticipated contributions of the meta-material’s constituents, we observe a lot of additional structure, especially at higher frequency, which can not be accounted for.

The above described observation, especially the negativeness of the imaginary part of effective permittivity or permeability, raised objections Depine04 ; Efros04b of other groups. EfrosEfros04b argued that the LHM can not be approximated by a homogeneous system because of the periodicity of the meta-materialPokrovsky02b ; Efros04a .

In this paper, we show that the observed artifacts in the homogeneous effective approximation are quite generic. They are given by the periodic structure of the investigated meta-materials. The periodic structure becomes important when the wavelength of the electromagnetic wave is comparable with the lattice structure of the materialKoschny03b . We proposed a more general description of the LHM, based on the concept of a periodic effective medium (PEM). This method enables us to distinguish between the resonant frequency dependence corresponding to Eq. 1 and effects of the periodicity of the structure. We apply the PEM method for the analysis of numerical data obtained by the transfer matrix method (TMM).

The paper is organized as follows.

In Section II we first explain basic ideas of the homogeneous effective medium (HEM). Special attention is given to the correction of the phase of the EM wave at the interfaces, which is crucial for any retrieval procedure.

In Section III we define and analyze one dimensional periodic structures. The analyzed medium consists of thin slabs of homogeneous LH material separated by slabs of vacuum. We show that the approximation of such periodic medium by a homogeneous one give us effective parameters and which possess unusual frequency dependences, similar to those observed when we approximate meta-materials by a homogeneous medium. This proves that the periodicity of meta-material must be taken into consideration in the analysis of the effective parameters.

The periodic effective medium is analyzed in two different formulations: continuous (Section III.1) and lattice (Section III.2). The latter is more relevant for the analysis of numerical data since all known numerical algorithms use spacial discretization.

In Section IV we analyze transmission data, observed from numerical simulations of periodic lattices of SRR, LHM and cut wires. We map these structures to periodic effective media which consist of homogeneous slabs separated by vacuum. In this formulation, and of the homogeneous slabs are free from any modifications of the resonant behavior. To show the role of the periodicity of the meta-materials more clearly, we also analyzed a lattice of SRR in which we filled the gaps of the SRR by a dielectric with very strong dielectric permittivity. This decreases the magnetic resonant frequency so that the wavelength of the incident EM wave is 25 times larger than the lattice period. We show that effective parameters again do not possess any deviations from resonant formula (1).

A discussion of the applicability of various proposed models to the analysis of transmission data is given in Section V. We discuss how the periodicity and anisotropy of the structure influence the transmission amplitudes and, subsequently, the effective parameters of the meta-materials. Final conclusions are given in Section VI.

## Ii Homogeneous effective medium

For the one-dimensional plain wave scattering problem at a homogeneous finite slab it is straightforward to obtain the scattering formulae. For the transfer matrices for a single slice of vacuum and for a single slice of homogeneous material with the thickness we find in wave-representationTMMDEF

with the elements

(3) | |||||

(4) |

In the continuum formulation and for normal incidence the momentum inside the slab is related to the momentum in the vacuum by the index of refraction , the impedance is defined by for the TE and TM mode, respectively. Here, and denote the frequency-dependent complex permeability and permittivity of the homogeneous medium. On the lattice, ie. when we are going to compare with TMM simulation results, we have to take the modified dispersion relations in the vacuum and inside the slab into account. Then we have a modified which gets noticeable at higher frequencies. Using the interrelation between the transfer matrix and the scattering matrix which defines the transmission () and refection () amplitudes,

(5) |

we can calculate the transmission and reflection amplitudes for a sample composed of a left vacuum slice of length , followed by homogeneous unit cells of length in propagation direction, and terminated by a right vacuum slice of length ,

(6) | |||||

(7) |

In order to relate to the simulated scattering amplitudes computed numerically by the TMM by decomposition of the EM waves in the vacuum right of the sample with respect to the vacuum wave base left of the sample, it is convenient to introduce the normalized scattering amplitudes and which, after unit cells, take the form

(8) | |||||

(9) |

In the continuum the scattering amplitudes of the homogeneous slab are typically defined from interface to interface of the sample, ie. assuming . In the numerical simulation this is not possible because of the lattice: we always have to make vacuum-transfermatrix step from the last left vacuum slice into the sample and another vacuum-transfermatrix step out of the sample onto the first right vacuum slice. Therefore, the TMM scattering amplitudes and are related to the normalized and involving an additional vacuum-phase compensation and .

Now we can resolve the above scattering formulae with given amplitudes and obtained from the simulation (or measurement) of a meta-material with respect to the material parameters impedance and index of refraction . If the solutions are (virtually) independent on the length of the sample those parameters define the homogeneous effective medium (HEM) representation (or approximation) of the respective meta-material. Then we haveSmith02b

(10) | |||||

(11) |

with . Note that we obtain and from the scattering amplitudes only up to a common sign and the real part of the effective index of refraction, only as a residue class. The former issue can be resolved by imposing additional physical requirements, for instance (causality). The problem of the residue class for can be addresses by considering different length . Then we obtain a system of linear congruences, the solution of which – if any – is a reduced residue class modulo given by the greatest common divisor of the lengths . Since due to the inherent periodic structure of real meta-materials in simulations and experiments the lengths of the sample can only be integral multiples of the unit cell’s length, the minimum possible ambiguity for will be a residue class modulo where is the length of a single unit cell. For physical reasons we can assume a smooth frequency dependence between resonances which enables us to obtain as the corresponding residue class of piecewise continuous functions. The correct branch then has to be chosen exploiting additional physical information or assumptions of the model like the behavior of at the plasma-frequency, in resonance induced transmission gaps an periodicity induced band gaps (discussed later). For known and the effective permeability and permittivity can be defined as

(12) | |||||

(13) |

respectively.

Results for the effective parameters of the HEM approximation of simulated meta-materials like arrays of SRR or cut-wires, LHM and even multi-gap SRR have been published by several authorsShelby01b ; Markos02b ; Smith02b ; Koschny03b ; OBrien02 ; OBrien04 ; Yen04 . They all expose details which are in conflict with the simple effective medium behavior in term of a resonant and a plasmonic , originally proposed by Pendry, even under the assumption of an additional electric response of the SRR. Typical examples are also shown in Figs. 5, 6, 9, and 10. All results show resonance/anti-resonance coupling in and accompanied by negative imaginary parts, apparently different resonance frequencies for and , the cut-off of the expected resonant positive (SRR) or negative (LHM) index of refraction, a misshapen, strongly asymmetric anticipated magnetic resonance in for the SRR and LHM or electric resonance in for the cut-wire, and finally a lot of unexplained additional structure (erratic stop-band and passbands) at higher frequency.

Our extensive numerical simulations suggested that common cause for all these problems has to be sought in the inherent periodicity, always present in the artificial meta-materials as they are composed of a repetitions of a single unit cell. To prove that the behavior is generic and really independent on the details of the unit cell, and that we can reproduce each of the effect above purely as a consequence of periodicity in the propagation direction, we investigated the most simple model for an effective medium with a non-trivial periodicity.

## Iii Periodic effective medium

To study the impact of the periodicity, or more precise the reduced translational symmetry of the sample in propagation direction, we consider a sample composed of a repetition of the unit cell shown in Fig. 1, finite in direction of propagation and infinite perpendicular to it. The unit cell consists of a thin homogeneous core of thickness characterized by arbitrary and , sandwiched by two slabs of vacuum with thickness and which break translational invariance. is the length of one unit cell, the number of unit cells in propagation direction. To make a connection to our meta-materials we choose a simple Lorentz-type resonant form of and/or to represent the magnetic and cut-wire response of the SRR. To model the LHM we would add a plasmonic term in to account for the response of the continuous wires. Now we can calculate the scattering amplitudes for this model and subject them to the HEM inversion discussed in the previous section. The description (or approximation) of the scattering amplitudes for a given meta-material in terms of the effective parameters of such a periodic medium as defined in Fig. 1 will be denoted an "periodic effective medium" (PEM).

The following results will show that this periodic medium can expose all the problematic effect discussed above. In a subsequent section we shall then demonstrate that this also applies to the simulated real meta-material. Their effective behavior can be decomposed into a "well-behaving" effective response of the resonances and a contribution of periodic structure described by the PEM.

### iii.1 Continuum formulation

With the transfer matrices and introduced above, we can express the total transfer matrix of a finite slab of the periodic effective medium defined in Fig. 1 in the form

As expected from the -inversion symmetry both transfer matrices and are unimodular, obviously is and a short calculation yields . Therefore we can easily calculate the -th power of the unimodular 2x2 matrix above by diagonalizing it and computing the -th power of its eigenvaluesBorn-Wolf-Principles-of-Optics . Using the interrelation between the transfer matrix and the scattering matrix, we obtain the transmission and reflection amplitudes corresponding to those computed numerically by the TMM,

(14) | |||||

(15) |

Here, the are the Chebyshev polynomials of the second kind, , taken at the argument

The wave vector and the impedance refer to the homogeneous core of the unit cell. For the normalized scattering amplitudes and after unit cells we find

(17) | |||||

(18) |

Now we shall discuss what happens if we try to approximate the explicitly periodic medium discussed above by a homogeneous effective medium. This basically corresponds to our previous attempts to describe the periodic meta-materials by an homogeneous effective medium. We have two options: First, we could simply consider the analytic scattering amplitudes (17, 18) derived above to be those of a homogeneous system and try to solve for effective material parameters and . This has the advantage that the approximation can deal with a possible residual length dependence of the approximate homogeneous medium, leaving an explicit possibility to assess the quality of the approximation. The disadvantage is that we have to handle the rather complicated structure of the formulae arising from the Chebyshev polynomials. The second approach is to assume that an exact correspondence of the periodic effective medium to an homogeneous effective medium exists. This assumption is supported by the length-independence (after appropriate phase compensation) of the conventionally inverted simulation data. If there is such a homogeneous effective medium we can write the transfer matrix of the periodic medium in terms of the transfer matrix for the homogeneous slab,

which implies in particular for a system length of only a single unit cell

(20) |

Since for a homogeneous slab the identity holds, finding a that satisfies equation (20) in turn implies length independence of the homogeneous effective medium description. Note that has only two independent elements, because is anti-symmetric and the determinant is fixed, such that we can calculate the matrix elements and from , . The assumption (20) imposes a restriction on the boundaries of the periodic medium in propagation direction. The off-diagonal elements of are anti-symmetric but on the left side of equation (20) this symmetry is broken by the phase factors and introduced in the off-diagonal elements by the two vacuum slabs. As a consequence the description as a homogeneous medium is only possible for . Besides choosing a symmetric unit cell in the first place we may alternatively compensate the factor in the reflection amplitude (which works simultaneously for all lengths, see equation 18), effectively redefining the boundaries of the system such that the slab is centered in the unit cells. In terms of the (normalized) scattering amplitudes and , for the single unit cell we then have the conditions

(21) | |||||

(22) |

We already know how to invert the right side of these equations, this is just what we did in the retrieval procedure for the HEM in the previous section. Defining renormalized scattering amplitudes and , we could apply the same procedure to the left side. Note that the possibility that we can always solve and for and guarantees a solution of equation (20). In other words, there is always an exact, length independent description of the periodic effective medium as a homogeneous effective medium characterized by and . There is no freedom to chose the boundaries of the homogeneous medium relative to the periodic medium. As shown above, we get the full information about the homogeneous effective medium which describes a given periodic effective medium characterized by , and the geometry , already from the first unit cell. Inserting the renormalized transmission and reflection amplitudes (21, 22) for a single unit cell into the inverted scattering formulae above we obtain

where and are the parameters of the material slab in the middle of the unit cell of the periodic effective medium. With the defined in equation (III.1) and we can write simpler

(24) |

with . The problem with the signs of and , as well as with the ambiguity of is similar, and can be resolved the same way as for the case of the homogeneous slab discussed above. Analogously we can express the impedance of the effective homogeneous medium in terms of the and of the homogeneous core as

(25) |

where and

The parameters of an effective homogeneous medium describing the periodic material from Fig. 1, which have been obtained from the formulae (24) and (25), are shown in Fig. 2 for a concrete example of SRR-type and . For the homogeneous core in the middle of the unit cell we have chosen

(27) | |||||

(28) |

with model functions

(29) |

and

(30) |

to emulate the anticipated magnetic and electricKoschny04a resonances of the SRR. For a LHM-type behavior we have to add the plasmonic response of the continuous wire in the permittivity,

(31) |

According to a simple effective medium picture, we would expect that we can approximate a homogeneous unit cell characterized by and by concentration the magnetic and electric polarizations into the homogeneous core of the periodic medium. Fig. 2 shows the actual effective impedance and index of refraction obtained via the HEM inversion of the periodic medium. Comparing with the expected effective medium behavior (dashed lines) we clearly see the typical anomalies in the shape and positions of the resonances, the same qualitative behavior as observed for real SRR meta-materials in the literature and our own previous work. The effective parameters of the HEM approximation of our periodic medium model show the resonance/anti-resonance coupling in and together with the negative imaginary part around the magnetic resonance frequency , and also a very involved behavior close to the cut-wire resonance . The effective index of refraction is cut-off at the edge of the Brillouin zone which corresponds to the appearance of additional band gaps origination from the periodicity rather than from the underlying material properties. The qualitative behavior presented in Fig. 2 is generic for a wide range of parameters , , , of the resonances and , of the geometry. If the periodic medium model is used with only the electric resonance or with an additional plasmonic term in , it qualitatively reproduces the observed deviations from the expected plain effective medium behavior published for the array of cut-wires and the LHM, respectively.

Although the curves show most of the discussed abnormalities in the HEM description of the SRR, the analytic description matches the simulation and inversion results for the real meta-material present in literature not in all aspects. Clearly, there are problems very close to the resonance frequencies. Instead of the divergence in the effective index of refraction being virtually cut off at the upper edge of the first Brillouin zone as observed in the simulations of the actual SRR meta-material, the analytic description produces a series of consecutive band gaps at the boundaries of the first and higher Brillouin zones and a lot of structure in the imaginary part of . The same holds for the analytic description applied to the periodic effective medium model of the LHM (not shown). Here, we particularly miss the cutoff at the bottom of the negative region. In either case the underlying lattice in the simulation starts to become visible. Since the lattice has a finite lattice constant it cannot support arbitrarily large momenta, such that we expect additional effects if the continuum momentum reaches the order of . In order to understand also the details of the retrieved HEM parameters in our simulation of real SRR and LHM meta-materials we have to take the discretization lattice of the employed TMM into consideration. To see the modification of the continuum results by the discretization lattice we have to derive the scattering formulae for the periodic medium model on the lattice.

### iii.2 Lattice formulation

We follow the TMM introduced for the Maxwell equations by PendryPendry92a ; Pendry92b ; Pendry94 ; Pendry96 in the formulation described by Markoš and SoukoulisMarkos02a . The electric and magnetic field, together with the spatially dependent material relative constants and which define the meta-material, are discretized on the bonds of mutually dual lattices and . With the renormalized material constants and , used throughout this section, we can write the transfer matrix equations for a stratification in -direction for the two independent components of the electromagnetic field. Using (quasi-)periodic boundary conditions in the plane we can introduce a Fourier representation of the fields with respect to this plane defining an in-plane momentum . To derive a scattering formula corresponding to the continuum case considered in the previous section we restrict ourself to the most simple case of normal incidence, ie. zero in-plane momentum . Then the transfer matrix for normal incidence takes the form

The generally -dependent matrices and reduce to simple off-diagonal form, with the product diagonal,

(33) | |||||

(34) |

such that the transfermatrix (III.2) factorizes, reordering the electromagnetic field vector in the form , into a two-fold degenerated block-diagonal structure

Without loss of generality we can restrict ourself to consider just the first polarization. We denote the single-polarization transfer matrix for the modes in the last equation . It is expedient to introduce the decomposition

(36) | |||||

(37) |

Further we can factorize the into a vacuum and a material contribution, related to the polarization for the magnetic and analog for the electric field step. Note the renormalized vacuum permittivity and permeability . As expected, is unimodular. Now we can easily find the eigensystem; the eigensystem of the vacuum transfer matrix defines the plain wave basis on the lattice which we use to define the scattering formalism. Because of the unimodularity the two eigenvalues are mutually reciprocal and for the propagating modes we are interested in on the unit circle, ie. is real. We get the characteristic polynomial , hence . The two signs of correspond to the right- and left-moving waves. Note that and implicitly contain the dependence. To obtain the scattering matrix on the lattice we need the wave-representation of the total transfer matrix of a unit cell. The right and left eigenvectors of are distinct, and , and satisfy the orthogonality relation . Note that we applied the common normalization to the left-eigenvectors in order to normalize the electric field component of all right-eigenvectors to one. This is required for a clean definition of the scattering amplitudes analog to the continuum case. Further, the two right- and the two left-eigenvectors are linearly independent, respectively. Therefore we may group the two right- and the two left-eigenvectors of the vacuum transfer matrix into the matrices

(38) |

(39) |

where the eigenvalues satisfy the vacuum dispersion relation for the vacuum wave vector , and use the projector to obtain the wave-representation of the total transfer matrix of the finite system as

(40) |

Then we get the usual definition of the scattering amplitudes from the correspondence between the scattering and the transfer matrix given by equation (5).

#### The homogeneous slab.

Now we have to consider the total transfer matrix of our meta-materials. The most simple case is just a homogeneous slab of finite length. On the lattice, the composition of the total transfer matrix depends on the material discretization. We compute the total transfer matrix by starting from a right-eigenvector of the vacuum base at the last vacuum site just before one side of the sample and apply successively the single-step transfer matrices until we reach the first site right of the sample for which the is a vacuum step again. We have material layers inside the sample but transfer matrix step which depend on the material parameters , of the sample. Since we only have to consider a single polarization, we drop in the following the -indices in and in order to improve readability. Because in the discretized Maxwell equations the electric and magnetic fields live on mutual dual lattices, we distinguish three different single step inside the sample instead of only one, as one would expect for a homogeneous slab. depends on and . Therefore the first step inside the sample sees only the electric response but no magnetic response of the material. The subsequent steps see both, and , and are constant across the bulk of the sample. The last step back into the vacuum behind the slab is special again. Both steps across the boundaries of the sample depend on the chosen material discretization. Here we adopt a symmetric material discretizationKoschny04c which respects the -isotropy such that the the steps into and out off the sample become equal. Then we may calculate the wave representation as

where is the eigenbase of the vacuum transfer matrix step with the eigenvalues as before, but now denotes the eigenbase of the transfer matrix step inside the homogeneous medium with the eigenvalues . We made use of the afore mentioned identity . The symmetric material discretization introduces the averaged at the material’s surface. As shown above, the wave vector in the vacuum and inside the homogeneous slab satisfy the dispersion relations and . Since the matrix in Eq. III.2 is diagonal, we basically have to calculate the matrix . After some algebra we obtain for the homogeneous slab

with the diagonal and

where

(43) | |||||

(44) | |||||

(45) |

with and consequently . Further we have and . Note the anti-symmetry of the off-diagonal elements. Using again the definition of the scattering matrix (5), we find the transmission and reflection amplitudes as

(46) | |||||

The non-vacuum factor of the lattice transfer matrix (III.2) appears to have the same symmetries as the transfermatrix of the homogeneous slab in the continuum: the off-diagonal terms are anti-symmetric, the diagonal terms are mutual complex conjugates if and are real.

#### The periodic medium.

Knowing the transfer matrix of the finite slab it is now easy to obtain the transfer matrix for a sample of multiple unit cells of the homogeneous as well as the periodic medium with the unit cell corresponding to Fig. 1. We can reduce the wave representation of the total transfer matrix to a product involving the wave representation of the homogeneous core we already know and some additional vacuum transfer matrix steps for the free space in the unit cell. We assume the measures , and in Fig. 1 to correspond to , and layers on the lattice. Then we get for the total transfer matrix of unit cells of the periodic medium using and consequently

with the defined in equation (III.2). Since the phase factors and introduced by the two vacuum slabs in the bracket on the last line of equation (III.2) do explicitly break the anti-symmetry of the off-diagonal elements that is present for the single homogeneous slab in the continuum and, in the symmetric material discretization, also on the lattice, we can obtain a representation of the periodic medium by a homogeneous medium only for the case . As already explained for the continuum case this is not a real restriction but instead just fixes the definition of the effective boundaries of the periodic medium. In the numeric simulation we have to explicitly compensate the corresponding vacuum phases in the scattering amplitudes. We can use the Chebyshev formula to explicitly calculate the -th power such that we get the transmission and reflection amplitudes for the periodic medium after unit cells in propagation direction as

where the argument of the Chebyshev polynomials is given by

As for the continuum formulation, we actually get all the information about the meta-material from the single unit cell. Comparing the scattering amplitudes (III.2, III.2) on the lattice with the normalized scattering amplitudes for homogeneous slab in the continuum tells us how to do the phase compensation for the lattice-tmm results: and . The condition for arises from the additional vacuum step into the slab on the lattice, the compensation in results from the symmetric definition of the boundary of the unit cell which is required to describe the periodic by a homogeneous medium as explained above.

#### Continuum HEM inversion.

Again we ask whether the model periodic medium from Fig. 1 can be represented by an effective homogeneous medium. Here we have two choices: () we can compare the scattering amplitudes of the lattice periodic medium with the scattering formulae (21, 22) derived for the homogeneous slab in the continuum, or we can () compare with the lattice scattering formulae for the homogeneous slab derived in this section. Moreover, we have to decide which material discretization to use. In this paper we will concentrate on comparing the lattice scattering results to the continuum scattering formulae for the homogeneous slab, as we previously did with the standard inversion procedure to obtain effective and from the meta-material simulations.

Analytically, the effective material parameters obtained from the HEM inversion for the lattice formulation of the model periodic media used in the last section to emulate the SRR and LHM meta-materials are shown in Fig. 3 and Fig. 4, respectively. As expected, the qualitative behavior is very similar to that found with the continuum formulation. All the problematic effects seen in the previously published simulations, like resonance/anti-resonance coupling, negative imaginary parts, deformed resonances, bad gaps and so on, are present. The major difference to the continuum formulation becomes visible around the resonances. Where we previously found a series of tiny periodicity band gap around the resonances, in the lattice formulation we obtain a much simpler structure with basically one gap before each resonance. This is in excellent agreement with the numerical simulations, hence, expectedly, the lattice formulation compares much better to numerical simulations also obtained via discretization of the Maxwell equations than the continuum formulation. The discussion of further details we shall defer to a dedicated section below.

HEM | Homogeneous Effective Medium, a homogeneous medium characterized by and which, substituted for a finite meta-material slab, length-independently reproduces (or approximates) the given scattering amplitudes. Here always used in continuum formulation. Finding a HEM for given , is called HEM inversion (if exact) or HEM approximation. |
---|---|

PEM | Periodic Effective Medium, a most simple periodic model-medium defined by , and a geometry shown in Fig. 1 which length-independently reproduces (or approximates) given scattering amplitudes. Also used a priori with given , to demonstrate effects of the periodicity. Here used in lattice formulation. |

HEM(PEM) | The HEM which reproduces the scattering amplitudes calculated analytically from a given PEM. |

## Iv Simulation results

In this section we now present actual TMM simulation results for real SRR and off-plane LHM meta-materials. All numerical simulation are done using an implementation of the TMM method described by Markoš and SoukoulisMarkos02a . The meta-materials are uniformly discretized on a cubic lattice using a symmetric material discretization. The dimensions of the unit cell are 6x10x10 mesh steps, the single-ring SRR is a square ring of 7x7 mesh steps with a gap in the top side one mesh step wide. Propagation is for all cases along the SRR plane with the polarization of the incident plain wave such that the electric field is parallel to the two continuous sides of the SRR. Therefore we have only magnetic coupling to the magnetic resonance of the SRRSmith00 ; Katsarakis04a . Periodic boundary conditions apply to both directions perpendicular to the direction of propagation. For the off-plane LHM we add a one mesh-step thick continuous wire in front of the SRR such that the position of the wire is symmetric in the middle between two periodic repetitions of the SRR plane and centered with respect to the gap in the SRR. The direction of the wire is parallel to the continuous sides of the SRR, thus parallel to the incident electric field. All components of the meta-materials, the ring ring of the SRR and the continuous wire, are made from metal characterized by a constant relative permittivity of and . Note that the results do not depend much on as long as it does not fall below a certain thresholdMarkos02a . The chosen value is reasonable to emulate metals like Cu, Ag, Au in the range of GHz to a few THz. The rest of the unit cell is vacuum, there are no dielectric boards. The special geometry of the unit cell has been carefully chosen to preserve the inversion symmetry of the unit cell in the two directions perpendicular to the direction of propagation. This allows us to consider the scattering for only one polarization as it avoids complications by cross-polarization terms in the scattering amplitudesKoschny04c . In this paper, we concentrate our consideration on the region around the magnetic resonance frequency , where we expect to become transitionally negative, for two reasons: first this is the region of interest for any left-handed application, and second, this is the frequency window for which simulation data is typically shown in the literature. A more detailed investigation of the higher frequency region, particularly the vicinity of the of the electric cut-wire response of the SRR and the intermediate periodicity band gaps will be published elsewhere.

In the following we show HEM inversion results for the scattering data numerically obtained for the meta-material with the TMM. After the correct vacuum-phase compensation described above the inverted HEM scattering formulae (10, 11) are applied to the simulated and for meta-material slabs with a thickness of one, two and three unit cell in propagation direction. We shall denote the results as and or and , correspondingly. This approach is the same as chosen in the literature. Then we find the PEM approximation for the simulated meta-material using the lattice formulation for the analytic scattering formula of a model periodic medium consisting of a homogeneous core which is a single discretization mesh-step thick and located the unit cell in the plain of the SRR gaps and the LHM continuous wire. This constitutes the lattice equivalent of a single scattering plain in the continuum. A model periodic medium, characterized by effective material constants and of the homogeneous core, which reproduces the simulated and independent on the system length is called a periodic effective medium. The numeric inversion of the lattice scattering formulae (III.2, III.2) is applied to the simulated and for the firsts unit cell of each meta-material, providing us with effective material constants and for the homogeneous core of the PEM approximation. From the core parameters we can derive two further sets of effective parameters. First, we calculate the HEM inversion of the PEM scattering data obtained from the retrieved and and compare the results with the HEM inversion of the direct simulation data to assess the quality of the PEM approximation. We denote this as HEM(PEM). Second, we introduce the material parameters and of a homogeneous unit cell that would correspond to the PEM approximation in the effective medium limit, equating the total electric and magnetic polarizations of the respective unit cells,

(51) |

and

(52) |

The idea of this definition is to obtain parameters which we can compare with those of the HEM inversion, becoming equivalent with the latter if we can truly neglect the periodicity of the material. This allows us, to some degree, to consider the meta-material’s electromagnetic response as being composed of an actual contribution of the internal geometry of the meta-materials constituents and an explicit contribution of the periodic arrangement.