The possibility of designing structure and arrangement of the elements in electromagnetic artificial materials permits to achieve electromagnetic properties different from the properties of constituents [1]. Most fully this possibility realizes in metamaterials, in which the interaction of the electromagnetic fields with artificial structural elements is of resonant nature or at least the solenoidal part of the electromagnetic field plays a dominant role in this interaction. As a consequence, metamaterials exhibit advantageous and unusual electromagnetic properties (see e.g. in [2,3]). Nevertheless, it is still desirable to describe the metamaterials as homogeneous ones introducing effective constitutive parameters. Assignment of conventional effective parameters (permittivity, permeability, chirality parameter) to metamaterial samples often produces the values of the parameters whose properties differ from one of any physically possible homogeneous medium. Firstly, the retrieved parameters may depend on the sample size and surrounding environment (see e.g [4,5].). Secondly, this approach may result in the nonzero imaginary parts of ε and μ in the absence of real dissipation (see e.g [6].). Thirdly, the sign of these imaginary parts may contradict to general passiveness of the system [7–9] and fourthly, the frequency dispersion of material parameters may violate the causality principle [7–9]. To fix the problems additional constitutive material parameters are often introduced. This enlargement may be determined by new physical phenomena (anisotropy, artificial permeability, chirality etc.) or by the peculiarities of the homogenization scheme [10–12]. In the latter case additional parameters not obviously have clear physical meaning.

Below we show that the key moment of the problems is the boundary conditions. We believe that modification of the boundary conditions by introduction of additional (“excess”) surface currents returns the conventional permittivity and permeability of metamaterials their usual physical properties.

Introduction of effective parameters is called homogenization, because in this approach a heterogeneous structure is replaced by a homogeneous one (see [1]). Usually, homogenization procedure assumes that inside an inhomogeneous medium the homogenized fields are governed by the material Maxwell equations and that at any interface the Maxwell boundary conditions should be used. Most of the theories are of phenomenological type and differ in definition of homogenized fields and in the number of introduced effective parameters [13–16]. In the experiment the procedure of homogenization is substituted for measuring of far fields (scattering matrix) instead of measuring microscopic picture of field distribution inside and around the body. In the simplest case when the sample is a parallel plate (this case is of prime importance for metamaterials) the scattering matrix reduces to the pair reflection coefficient r and transmission coefficient t. Minimizing the difference between these values and those for the slab made of hypothetic homogeneous material one retrieves the bulk parameters of the hypothetic material. Below this method is referred to as the rt-retrieval method [4,17–21]. Really, in the rt-retrieval method the boundary conditions relate the measured far fields analytically extended to the interface with homogenized (smooth) fields ignoring both near fields obviously excited in the vicinity of boundaries and microstructure of the real fields inside the slab [22–24]. There is no reason to expect that homogenized fields inside the composite material and far fields outside should be connected by Maxwell’s boundary conditions. These mean fields may exhibit jumps of tangential components. How to take these jumps into account? It is worth emphasizing that the jumps are reflections of existence of real near fields. In [37] the jumps are considered as a consequence of the homogenization procedure, the excess currents are considered as nonphysical and to fix the problem the author changes the homogenization scheme.

The jumps of the tangential component of macroscopic $\overrightarrow{E}$ and $\overrightarrow{H}$ fields at the composite material boundaries can be considered from the microscopic point of view by introduction of the transition layers [25–32]. The transition layer may have different origin appearing due to averaging procedure when the volume of averaging includes the sample boundary [27,29–32] or due to different response of inclusion on the boundary and inside the material [26] or even it can appear due to cloak of magnetization in the theory [33,34]. There is no unique recipe how to construct a transition layer. It is inconvenient for treatment of the experimental data as usually there is no information of the material internal structure. Nevertheless, if one properly guesses the structure of the transition layer [31,32] the results of retrieval procedure contradict neither causality nor passiveness principles.

The introduction of a transition layer can be reduced to introduction of excess surface currents [25]. The introduction of the excess current has been used previously to describe mesoscopic systems like grids [22,35] or rough surface [36]. Moreover, sometimes the introduction of additional bulk effective parameters can be replaced by the introduction of excess surface currents [12,34]. For example, the introduction of the magneto-electric coupling resulting from the phase shift of the wave per one asymmetric unit cell [12] can be replaced by choosing the symmetrical representation and adding excess surface currents in the boundary conditions. Thus the introduction of excess surface currents seems to be a universal (flexible and suitable) tool in solution the problem of the rt-retrieval method. Additional parameters can be retrieved from usual rt-measurements caring out measurements for samples of different thickness.

As illustration of the approach we consider a bounded 1D periodical inhomogeneous system with optically small period [4, 5]. Both the theory [6] and direct application of the rt-retrieval method to the computer simulation data [4] produce unsatisfactory results.

S. M. Rytov [6] theoretically considered an infinite periodic system with elementary cell made of two layers with thickness $d_1$ and and derived an exact dispersion equation for the effective refractive index

S.M. Rytov also defined the effective impedance as a ratio of the averaged $\overrightarrow{E}$ and $\overrightarrow{H}$ fields (over the period) ${\zeta }_{eff}^{Ryt}=〈E(z)〉/〈H(z)〉$.Knowing the effective refractive index $n_{eff}^{Ryt}=k_{Ryt}/k_0$ and effective impedance ${\zeta }_{eff}^{Ryt}$ it is easy to retrieve effective permittivity ${\epsilon }_{eff}^{Ryt}=n_{eff}^{Ryt}/{\zeta }_{eff}^{Ryt}$ and permeability ${\mu }_{eff}^{Ryt}=n_{eff}^{Ryt}{\zeta }_{eff}^{Ryt}$. At this point the theory gives rather strange result. Though the refractive index is real (we consider the frequencies far below the first band gap) both the effective permittivity and permeability have imaginary parts. Moreover the signs and values of the parts are different and depend on the cell presentation.

For bounded layered systems we still have troubles [4,5]. Below we consider a regular system whose elementary cell consists of two layers of the same thickness d and permittivity ${\epsilon }_1=2$ and ${\epsilon }_2=3$ respectively. The rt-retrieval method produces the effective impedance depending on the system thickness. This is the reflection of the fact that the input impedance of the elementary cell depends on the position of the sample surface planes.

It is a priory clear that an asymmetric system, which contains integer number N of elementary cells, cannot be adequately presented as a uniform layer (always symmetric) described by effective permittivity and permeability (or by $n_{eff}$ and ${\zeta }_{eff}$). The inadequacy of this homogenization results in the oscillation of the retrieved impedance versus L, though the refractive index tends to the Rytov value versus L (Fig. 1(a) ). Moreover, the imaginary part of ${\zeta }_{eff}$ is not negligible.

 

Fig. 1 $\mathrm{Re}{\zeta }_{eff}-{\zeta }_{stat}$ (solid lines) and $\mathrm{Re}{n}_{eff}-n_{eff}^{Ryt}$ (dashed lines) for asymmetric system (a) and symmetric system (b) as functions of system thickness L. The normalized frequency is $k_{0}d=0.05$. In (a) and (b) the $\mathrm{Re}{n}_{eff}$ tends to the Rytov value given by Eq. (1): $n_{eff}^{Ryt}\approx 1.58114$.

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For a symmetric system, which comprises $N+1/2$ elementary cells and has the same layer at both interfaces, the common retrieval procedure gives effective permittivity and permeability with poles in thickness dependence (see Fig. 1(b)). The poles’ positions are equal to the half-integer number of the effective wavelength $L=(N+1/2)d=\pi l/k_{Ryt}$, $l=1,2,\mathrm{...}$

The T-matrix of a uniform layer with refractive index n, impedance $\zeta =1/n$ and thickness L is: $T_{11}^{ul}=T_{22}^{ul*}=\cos n{k}_{0}L+i0.5\left(\zeta +1/\zeta \right)\sin n{k}_{0}L$, $T_{12}^{ul}=T_{21}^{ul*}=0.5i\left(1/\zeta -\zeta \right)\sin n{k}_{0}L$.The T-matrix $T^{odd}$ for a symmetric system is equal to the product $T^{odd}=T_1{T}_{cell}^N$, where $T_{cell}$ is the T-matrix of the elementary cell and $T_1$ is the T-matrix of the layer with permittivity ${\epsilon }_1$. It is implied that the system is sandwiched with the layers of the first type. Equating the entries of the $T^{ul}$ and $T^{odd}$ we can find out the expressions for constitutive parameters:

We can see that the impedance ζ has a singularity at $n_{eff}^{Ryt}{k}_{0}L=\pi m$.

From these two examples we see that the mesoscopic behavior is an inherent property of the retrieved impedance of any layered structure (there is no such problem with the refractive index that always tends to the Rytov value Eq. (1) when the thickness L grows). Introduction of excess surface currents takes all these facts into account and modifies the Maxwell boundary conditions on the left and right sides of the sample as follows:

Employing the new boundary conditions Eq. (2) it is easy to get the new T-matrix $T^{ulc}$ for a uniform layer with surface currents:

We have exactly simulated r and t coefficients for different L and extract six unknown material parameters minimizing the discrepancy

Below we characterize two symmetric and two asymmetric variants of the structure. In two symmetric variants the front and rear layers are of the same dielectric, either ${\epsilon }_1$ or ${\epsilon }_2$. Depending on this we refer to them as (1,1) or (2,2)-systems. Asymmetrical systems contain integer number of cells where the wave impinges either the layer ${\epsilon }_1$ or the layer ${\epsilon }_2$. Depending on this we refer to them as to either (1,2) or (2,1)-systems, correspondingly. Minimizing the discrepancy we have found six coefficients$n_{eff}$, ${\zeta }_{eff}$, $s_E^{(right)}$, $s_E^{(left)}$, $s_M^{(right)}$, and , which are practically independent on L. The summation in Eq. (3) is made for sample lengths from $i=20000$ to $40000$ periods ($N_{real}=2\cdot {10}^4$). The discrepancy between the exact and model values of r and t is equal to ${\delta }_{r,t}=\sqrt{\delta /(2N_{real})}$. We have performed computer simulation of the aforementioned four systems and get at frequency $k_{0}d=0.05$ the results tabulated in Table 1 .

Table 1. Material Parameters are Shown with Accuracy ${10}^{-7}$

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Since for such thick systems the refractive index achieves the Rytov value we have always consider $n_{eff}=n_{eff}^{Ryt}$. The effective values of ${\zeta }_{eff}$ practically coincide with the static value. The surface susceptibilities strongly depend on the type of the sample but do not depend on the sample thickness. They are proportional to the difference ${\epsilon }_b-〈\epsilon 〉$ where ${\epsilon }_b$ is the permittivity of the layer located at the given sample boundary. Thus, we obtain $s_E{}^{(left)}=s_E{}^{(right)},s_M{}^{(left)}=s_M{}^{(right)}$ for the symmetric case that agree with the theoretical result coming from equating the entries of the $T^{ulc}$ and $T^{odd}$. For $k_{0}d<<1$ the theory yields $\frac{1}{{\zeta }_{eff}}=\sqrt{\frac{{\epsilon }_1+{\epsilon }_2}{2}}+\frac{{\left({\epsilon }_1-{\epsilon }_2\right)}^2{\left(k_{0}d\right)}^2}{96\sqrt{\left({\epsilon }_1+{\epsilon }_2\right)/2}}$, $s_E=i\frac{{\epsilon }_2-{\epsilon }_1}{4}{k}_{0}d$, $s_M=0$ with accuracy $O\left({\left(k_{0}d\right)}^3\right)$.

For asymmetric cases we get $s_E{}^{(left)}=-s_E{}^{(right)},s_M{}^{(left)}=-s_M{}^{(right)}$. Averaging of the results for (1,1)-system with (2,2)-system as well as that of the (1,2)-system with (2,1)-system gives symmetric conductivities. The discrepancy between the exact and model values of and t significantly increases: ${\delta }_{r,t}\approx {\text{10}}^{-2}\sim k_{0}d$. Introduction of the averaged values of surface susceptibilities weakly influences the values of effective impedance and refractive index. With accuracy of $k_{0}d$ the latter averaged retrieval procedure can be considered as that producing effective parameters independent not only on the system size but also on its type. Thus, introduction of the excess currents removes the mesoscopy of the retrieved impedance (as well as its imaginary part for asymmetric systems) by price of physically sound material parameters – electric and magnetic susceptibilities of the sample surfaces. These new parameters depend on the permittivity of surface layers and not on the structure thickness.

When this paper has been in preparation there appeared a preprint [38] in the arxiv web-site where the authors consider introduction of the excess surface currents for a 3D metamaterial made of Mie-resonant magneto-dielectric spheres.