Homogenization of periodic metamaterials by field averaging over unit cell boundaries: use and limitations

Equation (10) means that circulation of a static $E$ field around any closed contour is zero. Let us consider the contour $ABCDA$ depicted in figure 2(a). It consists of two curves $AB$,$CD$ and two parallel line segments $BC$, $DA$. The initial points $A$, $B$ as well as the curve $AB$ are all chosen arbitrarily on the bottom face of the cell. The other two points $C$, $D$ and corresponding curve $DC$ are obtained by translating points $A$, $B$ as the curve $AB$ along the $z$-axis by the distance $a$. Choosing the direction of traversing $ABCDA$ as indicated in figure 2(a) and splitting the integral in (10) into four integrals,

$$\begin{eqnarray*}&&\oint_{ABCDA} {{\bf E} \cdot {\rm d}{\bf l}} = \int_{AB} {{\bf E} \cdot {\rm d}{\bf l}} + \int_{BC} {{\bf E} \cdot {\rm d}{\bf l}} + \int_{CD} {{\bf E} \cdot {\rm d}{\bf l}} + \int_{DA} {{\bf E} \cdot {\rm d}{\bf l}} = 0, \end{eqnarray*}$$

we note that

$$\begin{equation} \int_{BC} {{\bf E} \cdot {\rm d}{\bf l}} + \int_{DA} {{\bf E} \cdot {\rm d}{\bf l}} = 0, \end{equation} \tag{ 15 }$$

since

$$\begin{eqnarray*} &&\int_{AB} {{\bf E} \cdot {\rm d}{\bf l}} + \int_{CD} {{\bf E} \cdot {\rm d}{\bf l}} = 0, \end{eqnarray*}$$

the latter is due to the field periodicity (14). After changing the direction of integration in the second integral in (15) and, simultaneously, the sign before the integral, equation (15) reads

$$\begin{equation} \int_{BC} {{\bf E} \cdot {\rm d}{\bf l}} = \int_{AD} {{\bf E} \cdot {\rm d}{\bf l}} . \end{equation} \tag{ 16 }$$

Equation (16) tells us that the line integral of the $E$ field along a line segment parallel to the $z$-axis does not depend on the position of the segment within the unit cell. In other words, the line integral between two corresponding points $A$ and $D$ has the same value for any choice of the points:

$$\begin{equation} \int_{L_z } {{\bf E} \cdot {\rm d}{\bf l}} = {\rm{const}}\quad {\rm{(}}L_z \parallel z). \end{equation} \tag{ 17 }$$

We may, therefore, use any such segment—lying either on an edge, a face or inside the cell, see figure 2(b)—to calculate the line integral $\int_{L_z } {{\bf E} \cdot {\rm d}{\bf l}}$.

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The coordinate form of (17) can be obtained by noting that ${\bf E} \cdot {\rm d}{\bf l} = E_l \,{\rm d}l$ (only the longitudinal component $E_l$ of ${\bf E}$ along $L_z$ contributes to the integral). Thus, we have

$$\begin{equation} \int_{L_z } {E_z \,{\rm d}z} = {\rm{const}} \equiv C_z . \end{equation} \tag{ 18 }$$

Since there is nothing special about the $z$-direction in (10), by considering closed contours with their line segments oriented in either the $x$- or $y$-directions, we can obtain

$$\begin{equation} \int_{L_i } {E_i \,{\rm d}l_i } = C_i , \end{equation} \tag{ 19 }$$

where $L_i$ denotes line segments parallel to $i$th-axis ($i = x,y,z$) and $C_i$ are constants.

Now we can use the constancy of the line integrals to reduce the volumetric averages $\left\langle {E_i } \right\rangle _V$ of $E_i$ components to their line averages $\left\langle {E_i } \right\rangle _L$. Reducing the triple integral appearing in the definition of $\left\langle {E_i } \right\rangle _V$ to iterated integrals

$$\begin{eqnarray*} &&\int\! \!\!\!{\int \!\!\!\!{\int _V{ E_i (x,y,z)\,{\rm d}V} } } = \int\! \!\!\!\int_{S_i} {\rm d}S_i \int_{L_i } {E_i (x,y,z)\,{\rm d}l_i } \end{eqnarray*}$$

and accounting for (19) and $V = a^3$, we may rewrite $\left\langle {E_i } \right\rangle _V$ as

$$\begin{equation} \left\langle {E_i } \right\rangle _V = \frac{1}{{a^3 }}\int_{L_i } {E_i \,{\rm d}l_i } \int\! \!\!\!\int_{S_i } {\rm d}S_i = \frac{1}{a}\int_{L_i } {E_i \,{\rm d}l_i } \equiv \left\langle {E_i } \right\rangle _{L_i } . \end{equation} \tag{ 20 }$$

According to (20), volumetric average of the $i$th component of a static and periodic $E$ field over the volume of the unit cell equals to the line average of that component calculated over a line segment $L_i$ of length $a$ parallel to the $i$th-axis. The arbitrariness of the location of $L_i$ means that PHRS requirement that $L_i$ must be one of the edges of a unit cell is excessive.

The current density ${\bf J}$ in (11) consists, generally speaking, of two terms

$$\begin{eqnarray*} &&{\bf J} = {\bf J}_{{\rm{ext}}} + {\bf J}_{{\rm{ind}}}, \end{eqnarray*}$$

${\bf J}_{{\rm{ext}}}$ and ${\bf J}_{{\rm{ind}}}$ are the densities of external (impressed) and induced currents. The former is assumed to be zero, ${\bf J}_{{\rm{ext}}} = 0$, see section 1. The latter is related to the local value of the total electric field through Ohm's law

$$\begin{eqnarray*}&&{\bf J}_{{\rm{ind}}} = \sigma {\rm{ }}{\bf E},\end{eqnarray*}$$

$\sigma$ is the local value of the conductivity of the inclusion material. In the static case considered here, electric field ${\bf E}$ cannot induce steady currents flowing through inclusions of finite size. Thus, ${\bf J}_{{\rm{ind}}} = 0$, ${\bf J} = 0$, and (11) now reads

$$\begin{equation} \oint_L {{\bf H} \cdot {\rm d}{\bf l}} = 0, \end{equation} \tag{ 21 }$$

which is similar to (10) for $E$ field. Then, a consideration completely similar to that given in the previous section for the $E$ field (or formal replacement $E \to H$ in (20)) gives

$$\begin{equation} \left\langle {H_i } \right\rangle _V = \left\langle {H_i } \right\rangle _{L_i } . \end{equation} \tag{ 22 }$$

Equation (13) means that the total flux of a static $B$ field through any closed surface is zero. Let us consider the surface $S$ enclosing a part of the volume of a unit cell as shown in figure 3. The surface consists of the bottom face $S_{\rm bott}$ of the unit cell, four segments $S_{{\rm{side, }}j}$ ($j = \overline {1,{\rm{ }}4}$) of the side faces of the cell, and a segment $S_{{\rm{mid}}}$ of an intermediate plane $z = {\rm{const}}$ located between the bottom $S_{\rm bott}$ and upper $S_{{\rm{upp}}}$ faces of the cell.

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Representing the total flux $\Phi ^B$ through $S$ by the sum of the respective partial fluxes $\Phi _{\rm bott}^B$, $\Phi _{{\rm{mid}}}^B$ and $\Phi _{{\rm{side, }}j}^B$, we can write (13) as

$$\begin{equation} \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\nolimits_S {{\bf B} \cdot {\rm d}{\bf S}} = \int\! \!\!\!\int _{S_{\rm bott} } {\bf B} \cdot {\rm d}{\bf S} + \int\! \!\!\!\int_{S_{{\rm{mid}}} } {\bf B} \cdot {\rm d}{\bf S} + \sum\limits_{j = 1}^4 {\int\! \!\!\!\int_{S_{{\rm{side, }}j} } {\bf B} \cdot {\rm d}{\bf S}} = 0 \end{equation} \tag{ 23 }$$

or

$$\begin{eqnarray*}&&\Phi ^B = \Phi _{\rm bott}^B + \Phi _{{\rm{mid}}}^B + \sum\limits_{j = 1}^4 {\Phi _{{\rm{side, }}j}^B } = 0. \end{eqnarray*}$$

In (23), ${\rm d}{\bf S} \equiv {\bf n}\,{\rm d}S$ with ${\bf n}$ being the unit vector of the external (relative to the enclosed volume) normal to the surface element ${\rm d}S$. For example, on the surfaces $S_{\rm bott}$ and $S_{{\rm{mid}}}$

$$\begin{eqnarray*}&&\begin{array}{ll} \displaystyle {\rm d}{\bf S}_{\rm bott} \equiv {\bf n}_{\rm bott} \,{\rm d}S_z = - {\hat{\bf z}}\,{\rm d}x\,{\rm d}y, \\ \displaystyle {\rm d}{\bf S}_{{\rm{mid}}} \equiv {\bf n}_{{\rm{mid}}} \,{\rm d}S_z = {\hat{\bf z}}\,{\rm d}x\,{\rm d}y \end{array} \end{eqnarray*} \noindent$$

with ${\hat{\bf z}}$ being the unit vector in the $z$-direction, and ${\rm d}S_z \equiv {\rm d}x\,{\rm d}y$ is a surface element normal to the $z$-axis.

Due to the periodicity of the $B$ field in a lattice, the field vectors at corresponding points on opposite faces of a unit cell are equal (see figure 3), which leads to

$$\begin{eqnarray*}&&\sum\limits_{j = 1}^4 {\Phi _{{\rm{side, }}j}^B } = 0. \end{eqnarray*}$$

Then, (23) is reduced to

$$\begin{equation} \int\! \!\!\!\int_{S_{\rm bott} } {\bf B} \cdot {\rm d}{\bf S} + \int\! \!\!\!\int_{S_{{\rm{mid}}} } {\bf B} \cdot {\rm d}{\bf S} = 0 \end{equation} \tag{ 24 }$$

or, in a coordinate form

$$\begin{eqnarray*} &&\int\! \!\!\!\int_{S_{\rm bott} } B_z {\rm d}S = \int\! \!\!\!{\int {_{S_{{\rm{mid}}} } B_z {\rm d}S} } . \end{eqnarray*}$$

Accounting for the arbitrariness of $S_{{\rm{mid}}}$, this gives

$$\begin{equation} \int\! \!\!\!\int_{S_z } B_z \,{\rm d}S = {\rm{const}} \end{equation} \tag{ 25 }$$

with $S_z$ being the arbitrary cross section of the unit cell whose normal is in the $z$-direction. Following an identical procedure (or formally replacing the $z$ index in (25) by $x$ or $y$), one can obtain two more equations that can be written, along with (25), in a general form

$$\begin{equation} \int\! \!\!\!\int_{S_i } B_i \,{\rm d}S = {\rm{const}} \end{equation} \tag{ 26 }$$

the constants being, generally speaking, different for different $i$ values ($i = x,y,z$).

The constancy (26) of the surface integrals of the $B$ field can now be used to reduce the volumetric averages $\left\langle {B_i } \right\rangle _V$ of the $B$ components to their surface averages $\left\langle {B_i } \right\rangle _S$, the procedure is analogous to that used in the derivation of (20). In this way, we obtain

$$\begin{equation} \left\langle {B_i } \right\rangle _V = \frac{1}{{a^2 }}\int\! \!\!\!\int_{S_i } B_i \,{\rm d}S \equiv \left\langle {B_i } \right\rangle _{S_i } \end{equation} \tag{ 27 }$$

with $S_i$ not necessarily located on a face of the unit cell, in contrast to what is required within the PHRS approach.

Care needs to be exercised while replacing $\left\langle {D_i } \right\rangle _V$ by $\left\langle {D_i } \right\rangle _S$. Proceeding in full analogy with the above derivation for $B$ field, we could start from Gauss's theorem for $D$ field

$$\begin{equation} \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\nolimits_S {{\bf D} \cdot {\rm d}{\bf S}} = Q_{{\rm{free}}} \end{equation} \tag{ 28 }$$

written as

$$\begin{equation} \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\nolimits_S {{\bf D} \cdot {\rm d}{\bf S}} = \Phi _{\rm bott}^D + \Phi _{{\rm{mid}}}^D + \sum\limits_{j = 1}^4 {\Phi _{{\rm{side, }}j}^D } = Q_{{\rm{free}}} \end{equation} \tag{ 29 }$$

or, accounting for zero contribution of the side fluxes $\Phi _{{\rm{side, }}j}^D$ due to $D$ field periodicity,

$$\begin{equation} \Phi _{\rm bott}^D + \Phi _{{\rm{mid}}}^D \equiv \int\! \!\!\!\int_{S_{\rm bott} } {\bf D} \cdot {\rm d}{\bf S} + \int\! \!\!\!\int_{S_{{\rm{mid}}} } {\bf D} \cdot {\rm d}{\bf S} = Q_{{\rm{free}}} , \end{equation} \tag{ 30 }$$

where $Q_{{\rm{free}}}$ is the total free charge enclosed by the surface $S$. Unlike the respective equation (24) for the $B$ field, (30) has a non-zero right-hand side. Therefore, we cannot conclude that $D$ flux in the $z$-direction is always conserved within the unit cell. For lattices of dielectric inclusions, $D$ flux is indeed conserved due to $Q_{{\rm{free}}} = 0$, which automatically ensures that $\left\langle {D_i } \right\rangle _V = \left\langle {D_i } \right\rangle _{S_i }$. For metal inclusions, the argumentation based on conservation of $D$ flux does not work. However, there are numerical evidences (see section 3.3) that the use of the surface average $\left\langle {D_i } \right\rangle _{S_i }$ instead of volumetric one gives correct static values of $\varepsilon _{\rm eff}$ for lattices of metal inclusions, too. Hence, a different approach needs be used to decide whether $\left\langle {D_i } \right\rangle _V = \left\langle {D_i } \right\rangle _{S_i }$ or not.

We can validate the replacement from the definition of the $D$ field

$$\begin{eqnarray*}&&{\bf D} = \varepsilon _0 {\bf E} + {\bf{P}}\end{eqnarray*}$$

and by using Gauss's theorem (12) for the $E$ field. In this way, $\left\langle {D_i } \right\rangle _V$ is expressed in terms of $E$ and $P$ averages:

$$\begin{equation} \left\langle {D_i } \right\rangle _V = \varepsilon _0 \left\langle {E_i } \right\rangle _V + \left\langle {P_i } \right\rangle _V . \end{equation} \tag{ 31 }$$

The average $\left\langle {E_i } \right\rangle _V$ can, in turn, be expressed in terms of $E$ flux through a proper face of the unit cell. For instance, for the $z$-component one has

$$\begin{equation} \left\langle {E_z } \right\rangle _V = \frac{1}{V}\int_0^a {{\rm d}z} \int\! \!\!\!\int_{S_z (z)} E_z \,{\rm d}S = \frac{1}{V}\int_0^a {\Phi _z^E (z)\,{\rm d}z} , \end{equation} \tag{ 32 }$$

where

$$\begin{eqnarray*}&&\Phi _z^E (z) \equiv \int\! \!\!\!\int_{S_z (z)} E_z \,{\rm d}S \end{eqnarray*}$$

is $E$ flux through the cross section $S_z (z)$ of the unit cell defined by the plane $z = {\rm{const}}$, see figure 4. The flux $\Phi _z^E (z)$ is related to the flux $\Phi _z^E (0)$ through the bottom face $S_{\rm bott} \equiv S_z (0)$ of the unit cell via

$$\begin{equation} \Phi _z^E (z) = \Phi _z^E (0) + \frac{{Q(z)}}{{\varepsilon _{\rm{0}} }}, \end{equation} \tag{ 33 }$$

where $Q(z)$ is the total induced charge within the volume $V(z)$ enclosed by the boundaries of the unit cell and the plane $z = {\rm{const}}$ (see figure 4). Equation (33) is a consequence of Gauss's theorem (12) applied to the volume $V(z)$ and accounting for the field periodicity. Now, accounting for (33) as well as $\Phi _z^E (0) = {\rm{const}}$, $V = a^3$, and that $\Phi _z^E (0)/a^2$ is nothing but $\left\langle {E_z } \right\rangle _{S_{\rm bott} }$, we can write (32) as

$$\begin{equation} \left\langle {E_z } \right\rangle _V = \left\langle {E_z } \right\rangle _{S_{\rm bott} } + \frac{1}{{\varepsilon _{\rm{0}} V}}\int_0^a {Q(z)\,{\rm d}z} . \end{equation} \tag{ 34 }$$

Similar equations for the $x$ and $y$ components can be obtained by replacing $z \to x$ or $z \to y$ so we may write

$$\begin{equation} \left\langle {E_i } \right\rangle _V = \left\langle {E_i } \right\rangle _{S_i } + \frac{1}{{\varepsilon _{\rm{0}} V}}\int_0^a {Q(x_i )\,{\rm d}x_i } \quad (x_i = x,y,z), \end{equation} \tag{ 35 }$$

$S_i$ being the face of the unit cell defined by the plane $x_i = 0$. Note, as an insight, that (35) proves that $\left\langle {E_i } \right\rangle _V \ne \left\langle {E_i } \right\rangle _{S_i }$, i.e. $E$ field cannot be surface averaged in a flux-like manner.

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Now, let us show that the contribution to $\left\langle {D_i } \right\rangle _V$ due to the integral term appearing in (35) is completely compensated by $\left\langle {P_i } \right\rangle _V$. From the definition of the average polarization

$$\begin{eqnarray*} &&\left\langle {\bf{P}} \right\rangle _V \equiv \frac{1}{V}\int\! \!\!\!{\int \!\!\!\!{\int _V{ {\bf{P}}\,{\rm d}V} } } = \frac{{{\bf{p}}^{{\rm{total}}} }}{V},\end{eqnarray*}$$

where ${\bf{p}}^{{\rm{total}}}$ is the total dipole moment within the unit cell, we have

$$\begin{eqnarray*}&&\left\langle {P_z } \right\rangle _V = \frac{{p_z^{{\rm{total}}} }}{V} = \frac{1}{V}\int\! \!\!\!{\int \!\!\!\!{\int _V{ \rho ({\bf{r}})z\,{\rm d}V} } } = \frac{1}{V}\int_0^a {z\,{\rm d}z} \int\! \!\!\!\int_{S(z)} \rho ({\bf{r}})\,{\rm d}S. \end{eqnarray*}$$

The term ${\rm d}z\int\!\!\!\int_{S(z)} \rho ({\bf{r}})\,{\rm d}S$ is nothing but the total charge ${\rm d}Q(z)$ of the layer of thickness ${\rm d}z$ bounded by the planes $z = {\rm{const}}$ and $z + {\rm d}z = {\rm{const}}$ (see figure 4). Therefore,

$$\begin{eqnarray*}&&\left\langle {P_z } \right\rangle _V = \frac{1}{V}\int_0^a z \,{\rm d}Q(z). \end{eqnarray*}$$

Integrating the right-hand side by parts and accounting for $Q(a) = 0$ (recall that we consider lattices of initially uncharged inclusions) gives

$$\begin{equation} \left\langle {P_z } \right\rangle _V = - \frac{1}{V}\int_0^a {Q(z)} \,{\rm d}z. \end{equation} \tag{ 36 }$$

Similar expressions for the $x$ and $y$ components are obtained from this one by replacing $z \to x$ or $z \to y$ thereby giving

$$\begin{equation} \left\langle {P_i } \right\rangle _V = - \frac{1}{V}\int_0^a {Q(x_i )} \,{\rm d}x_i . \end{equation} \tag{ 37 }$$

By combining equations (31), (35) and (37), we finally obtain

$$\begin{equation} \left\langle {D_i } \right\rangle _V = \varepsilon _0 \left\langle {E_i } \right\rangle _{S_i } \equiv \left\langle {D_i } \right\rangle _{S_i } . \end{equation} \tag{ 38 }$$

This result holds true for inclusions of arbitrary materials, either dielectric or metal.

It follows from the above derivation that the surface $S_i$ appearing in (38) is the cell face defined by $x_i = 0$. Due to the field periodicity, $S_i$ may be chosen on either of the two faces of the unit cell that are perpendicular to $i$th-axis. Unlike the derivation for the $B$ field, we cannot, generally speaking, choose $S_i$ located anywhere between the faces. However, in case of dielectric inclusions one may still select $S_i$ arbitrarily since $D$ flux along $i$th-axis is, like $B$ flux, conserved in this case.

• 1.  
  So far, we almost always treated the quantities $\left\langle {D_i } \right\rangle _V$ and $\left\langle {B_i } \right\rangle _V$ as if the local values ${\bf D}({\bf{r}})$ and ${\bf B}({\bf{r}})$ are known. (The only exception was made in the preceding subsection.) However, electrodynamic solvers only usually provide information on local $E$ and $H$ fields, not $D$ and $B$. Therefore, the question arises, how should ${\bf D}({\bf{r}})$ and ${\bf B}({\bf{r}})$ be calculated?One might calculate ${\bf D}$ and ${\bf B}$ from the known values of the local $E$ and $H$ fields, respectively, by using either multiplicative or additive relations [34]

  $$\begin{equation} {\bf D} = \varepsilon {\bf E},\quad {\bf B} = \mu {\bf H} \end{equation} \tag{ 39 }$$

  or

  $$\begin{equation} {\bf D} = \varepsilon _0 {\bf E} + {\bf{P}},\quad {\bf B} = \mu _0 {\bf H} + {\bf{M}}, \end{equation} \tag{ 40 }$$

  where $\varepsilon \equiv \varepsilon ({\bf{r}}) = \varepsilon _0 \varepsilon _r ({\bf{r}})$, $\mu \equiv \mu ({\bf{r}}) = \mu _0 \mu _r ({\bf{r}})$; $\varepsilon _r ({\bf{r}})$ and $\mu _r ({\bf{r}})$ are local values of the relative permittivity and permeability at point ${\bf{r}}$; ${\bf{P}} = {\bf{P}}({\bf{r}})$ and ${\bf{M}} = {\bf{M}}({\bf{r}})$ are local polarization and magnetization. Surprisingly, despite being often considered equivalent, equations (39) and (40) predict different results for $\varepsilon _{\rm eff}$ and $\mu _{\rm eff}$, the results from (39) may even be paradoxical. To illustrate this, let us consider effective permittivity of a lattice of metal inclusions.From (39) it follows that

  $$\begin{eqnarray*}&&\left\langle {\bf D} \right\rangle _V = \varepsilon _0 V^{ - 1} \int\! \!\!\!{\int \!\!\!\!{\int _V{ \varepsilon _r ({\bf{r}}){\bf E}({\bf{r}})\,{\rm d}V} } } . \end{eqnarray*}$$

  Splitting the volume integrals into two parts corresponding to the region of the inclusion of a unit cell, $V_{{\rm{incl}}}$, and the region of the cell exterior to the inclusion, $V_{{\rm{ext}}} = V - V_{{\rm{incl}}}$, gives

  $$\begin{equation} \left\langle {\bf D} \right\rangle _V = \varepsilon _0 V^{ - 1} \left( \int\! \!\!\!\int \!\!\!\!\int _{V_{{\rm{incl}}} } \varepsilon _r {\bf E}\,{\rm d}V + \int\! \!\!\!\int \!\!\!\!\int_{V_{{\rm{ext}}} } {\bf E}\,{\rm d}V \right), \end{equation} \tag{ 41 }$$

  where it was accounted that $\varepsilon _r = 1$ in $V_{{\rm{ext}}}$ (vacuum or air outside the inclusions).For a metal inclusion in a static field, ${\bf E} = 0$ inside the inclusion. Therefore,

  $$\begin{eqnarray*}&&\left\langle {\bf D} \right\rangle _V = \varepsilon _0 V^{ - 1} \int\! \!\!\!\int \!\!\!\!\int _{V_{{\rm{ext}}} } {\bf E}\,{\rm d}V ,\quad \left\langle {\bf E} \right\rangle _V = V^{ - 1} \int\! \!\!\!\int \!\!\!\!\int_{V_{{\rm{ext}}} } {\bf E}\,{\rm d}V. \end{eqnarray*}$$

  Then, the diagonal components of the effective permittivity of the lattice should be

  $$\begin{eqnarray*}&&\left( {\varepsilon _{\rm eff} } \right)_{ii} \equiv \frac{{\left\langle {D_i } \right\rangle _V }}{{\left\langle {E_i } \right\rangle _V }} = \varepsilon _0 . \end{eqnarray*}$$

  The obviously incorrect result $\varepsilon _{\rm eff} = \varepsilon _0$ has already been mentioned (yet for a particular case of thin metal inclusions) in [6].In contrast, the additive relations (40) which, in fact, are definitions of ${\bf D}$ and ${\bf B}$, lead to correct results. Direct averaging of (40) over the volume of a unit cell yields

  $$\begin{eqnarray*}&&\left\langle {\bf D} \right\rangle _V = \varepsilon _0 \left\langle {\bf E} \right\rangle _V + \left\langle {\bf{P}} \right\rangle _V ,\quad \left\langle {\bf B} \right\rangle _V = \mu _0 \left\langle {\bf H} \right\rangle _V + \left\langle {\bf{M}} \right\rangle _V, \end{eqnarray*}$$

  the average polarization and magnetization are related to the total electric and magnetic dipole moments of the unit cell, ${\bf{p}}^{{\rm{total}}}$ and ${\bf{m}}^{{\rm{total}}}$:

  $$\begin{eqnarray*}&&\left\langle {\bf{P}} \right\rangle _V \equiv \frac{1}{V}\int\! \!\!\!{\int \!\!\!\!{\int _V{ {\bf{P}}\,{\rm d}V} } } = \frac{{{\bf{p}}^{{\rm{total}}} }}{V},\quad \left\langle {\bf{M}} \right\rangle _V \equiv \frac{1}{V}\int\! \!\!\!{\int \!\!\!\!{\int _V{ {\bf{M}}\,{\rm d}V} } } = \frac{{{\bf{m}}^{{\rm{total}}} }}{V}.\end{eqnarray*}$$

  The moments ${\bf{p}}^{{\rm{total}}}$ and ${\bf{m}}^{{\rm{total}}}$ are due to charges and currents induced in the inclusion of a unit cell. It is these charges and currents that are responsible for the deviation of $\varepsilon _{\rm eff}$, $\mu _{\rm eff}$ from the corresponding free space values $\varepsilon _{\rm{0}}$, $\mu _{\rm{0}}$.To summarize, the multiplicative constitutive relations, equations (39), should not be used in calculation of $\varepsilon _{\rm eff}$ and $\mu _{\rm eff}$ of lattices of metal particles since they do not properly account for the contributions to ${\bf D}$ and ${\bf B}$ due to the charges and currents induced in a unit cell.
• 2.  
  The two averaging procedures—the line averaging for ($E,H$) fields and surface averaging for ($D,B$)—stem from different properties of these fields: the properties of $E$ and $H$ fields are governed by contour integrals (10) and (11), while those of $D$ and $B$ fields are governed by surface integrals (12) and (13). Note that the averaging formula for the $D$ field is a consequence of Gauss's theorem for the $E$ field rather than $D$ itself. The reason for this is explained in section 2.2.4.
• 3.  
  Periodicity of the total fields plays a crucial role in validity of the averaging formulae. Maxwell's equations (10)–(13) themselves cannot guarantee that (volumetric) averages are equivalent to either line or surface averages. It is the field periodicity (14) which makes the equivalence possible.
• 4.  
  Magnetic response of lattices of non-magnetic conducting inclusions (such as made of Cu, Al, Ag, Au, GaAs, etc), being due solely to induced conduction currents, is only possible in a time-varying magnetic field that generates the currents by Faraday's law. In the purely static case we considered so far, the response is zero. Therefore, the approach based on Maxwell's integral equations for static fields cannot capture the artificial magnetism of such lattices. For an extension of this approach to time-varying fields, see the next subsection.

Let us first note that in calculating averages of $E$ and $H$ fields, we actually are not bound to integration over line segments, see (6) and (7). In fact, integration over any surface within the unit cell that is parallel to the respective axis will give the same result. To show this, let us consider the volumetric average of $z$ component of $E$ field

$$\begin{equation} \left\langle {E_z } \right\rangle _V \equiv \frac{1}{V}\int\! \!\!\!{\int \!\!\!\!{\int _V{E_z \,{\rm d}V} } } = \frac{1}{{a^3 }}\int_0^a {{\rm d}x} \int_0^a {{\rm d}y} \int_0^a {{\rm d}z} \,E_z . \end{equation} \tag{ 42 }$$

In section 2.2.1, we calculated this average as

$$\begin{eqnarray*}&&\left\langle {E_z } \right\rangle _V = \frac{1}{{a^3 }}\int_0^a {E_z \,{\rm d}z} \int\! \!\!\!\int_{S_z } {\rm d}x\,{\rm d}y = \frac{1}{a}\int_0^a {E_z \,{\rm d}z} \equiv \left\langle {E_z } \right\rangle _{L_z } . \end{eqnarray*}$$

However, this choice of calculation is not unique: by using the constancy (18) of the line integral of $E$ field, we can combine the three line integrals in (42) in two additional ways: either

$$\begin{equation} \left\langle {E_z } \right\rangle _V = \frac{1}{{a^3 }}\int\! \!\!\!\int_{S_x } E_z \,{\rm d}y\,{\rm d}z \int_0^a {{\rm d}x} = \frac{1}{{a^2 }}\int\! \!\!\!\int _{S_x } E_z \,{\rm d}S \equiv \left\langle {E_z } \right\rangle _{S_x} \end{equation} \tag{ 43 }$$

or

$$\begin{equation} \left\langle {E_z } \right\rangle _V = \frac{1}{{a^3 }}\int\! \!\!\!\int_{S_y } E_z \,{\rm d}x\,{\rm d}z \int_0^a {{\rm d}y} = \frac{1}{{a^2 }}\int\! \!\!\!\int_{S_y } E_z \,{\rm d}S \equiv \left\langle {E_z } \right\rangle _{S_y }. \end{equation} \tag{ 44 }$$

The surfaces $S_x$ and $S_y$ do not necessarily coincide with the corresponding faces of the unit cell and form a set of planes parallel to the $z$-axis, see figure 5. Thus, they will be labeled as $S'_z$. The prime here is used to prevent confusing $S'_z$ with the faces $S_z$ whose normal is parallel to the $z$-axis. The latter appear in the PHRS averages of $D$ and $B$ fields. Now, both equations (43) and (44) can uniformly be written as

$$\begin{equation} \left\langle {E_z } \right\rangle _V = \frac{1}{{a^2 }}\int\! \!\!\!\int_{S'_z } E_z \,{\rm d}S \equiv \left\langle {E_z } \right\rangle _{S'_z } \end{equation} \tag{ 45 }$$

and for arbitrary component of $E$ field we have

$$\begin{equation} \left\langle {E_i } \right\rangle _V = \left\langle {E_i } \right\rangle _{S'_i } \quad (i = x,y,z). \end{equation} \tag{ 46 }$$

Similarly one can obtain the respective formula for the $H$ field average

$$\begin{equation} \left\langle {H_i } \right\rangle _V = \left\langle {H_i } \right\rangle _{S'_i } \quad (i = x,y,z). \end{equation} \tag{ 47 }$$

Hence, the averages of static $E$ and $H$ fields in a lattice can be calculated in three different ways—either as volumetric, surface or line averages,

$$\begin{equation} \langle{E_i }\rangle_V = \langle {E_i } \rangle _{S'_{i}} = \langle {E_{i}} \rangle _{L_{i}}, \quad \langle {H_{i}} \rangle _V = \langle{H_{i}} \rangle _{S'_{i}} = \langle {H_{i}}\rangle _{L_{i}}. \end{equation} \noindent \tag{ 48 }$$

The value of equations (46) and (47) is in that they provide an alternative recipe for calculation of the average $E$ and $H$ fields. In the purely static case, it makes no difference: the equivalence $\left\langle \cdots \right\rangle _V = \left\langle \cdots \right\rangle _{S'_i } = \left\langle \cdots \right\rangle _{L_i }$ of the three averages holds strictly true. Beyond the static case, one of the averaging methods may be more advantageous.

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In a non-static regime, the phase advance of an incident wave leads to a spatial variation of its electric and magnetic fields across unit cells, see figure 6(a). While the surface averaging introduced by PHRS for $D$ and $B$ fields can account (within certain accuracy) for the phase variation of these fields, the line averaging introduced for $E$ and $H$ fields cannot. The latter uses only information on the local field values at the points with the same coordinate along the wave propagation direction ${\hat{\bf k}}$. Therefore, the integrals over two different lines—e.g. $L_1$ and $L_2$ in figure 6(b)—will give different average field values, even for empty space.

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In contrast, integration over the surfaces $S'_i$ that are parallel to ${\bf k}$ (see $S'_1$, $S'_2$ and $S'_3$ in figure 6(b)) does account for the field variation. In the case of an incident plane wave, such $S'_i$ are not necessarily the cell's faces: due to constancy of the plane wave fields in the directions ${\bot}{\bf k}$ (see figure 6(a)), averaging over any cross section of the cell that is parallel to the faces (e.g. $S'_2$ in figure 6(b)) returns the same result.

Based on this observation, the optimal averaging of $E$ and $H$ fields should employ integration of the local field values over surfaces that are parallel to the $i$th-axis and the wave vector ${\bf k}$. Such surfaces form a subset of the previously introduced surfaces $S'_i$ appearing in (46)–(48) and shown in figure 5. To differ these specific surfaces from $S'_i$, we add a subscript ${\bf k}$ to $S'_i$ thus denoting them as $S'_{i,{\bf k}}$. In this way, formulae of optimal averaging of $E$ and $H$ fields take the form

$$\begin{equation} \fl \left\langle {E_i } \right\rangle = \frac{1}{{a^2 }}\int\! \!\!\!\int_{S'_{i,{\bf k}} } E_i \,{\rm d}S \equiv \left\langle {E_i } \right\rangle _{S'_{i,{\bf k}} } ,\quad \left\langle {H_i } \right\rangle = \frac{1}{{a^2 }}\int\! \!\!\!\int_{S'_{i,{\bf k}} } H_i \,{\rm d}S \equiv \left\langle {H_i } \right\rangle _{S'_{i,{\bf k}} } \end{equation} \tag{ 49 }$$

and the formulae for effective parameters now read

$$\begin{equation} \left( {\varepsilon _{\rm eff} } \right)_{ij} = \frac{{\left\langle {D_i } \right\rangle _{S_i } }}{{\left\langle {E_j } \right\rangle _{S'_{j,{\bf k}} } }},\quad \left( {\mu _{\rm eff} } \right)_{ij} = \frac{{\left\langle {B_i } \right\rangle _{S_i } }}{{\left\langle {H_j } \right\rangle _{S'_{j,{\bf k}} } }}. \end{equation} \tag{ 50 }$$

Expressions (50) are new definitions of $\varepsilon _{\rm eff}$ and $\mu _{\rm eff}$. They are reminiscent of the respective expressions of the PHRS homogenization method. However, due to different definitions of the average $E$ and $H$ fields used here and by PHRS, definitions (50) involve only surface averages of all the fields. Yet, the surfaces to average over are different for $(E,H)$ and $(D,B)$ fields: the averages of the $i$th component of latter fields are computed over a plane that is normal to the $i$th-axis (flux-like surface averages); rather, the averages of $E$ and $H$ components are calculated over a plane that is along the $i$th-axis (in-plane or tangential averages).

Formulae (50) for $\varepsilon _{\rm eff}$, $\mu _{\rm eff}$ together with the field averaging formulae (27), (38) and (49) are the basis for our method of homogenization of periodic materials. The advantages and the accuracy of this method are discussed below.

We considered a lattice of spheres made of a non-magnetic material with $\varepsilon _{\rm{r}} = 8$, $\mu _{\rm{r}} = 1$. Figure 9 shows our numerical results for both the real and imaginary parts of the effective permittivity of the lattice. The choice of $l$ value and whether the correction has been applied are both indicated in the legend. Since non-magnetic dielectric spheres possess a weak magnetic response, below we do not analyze their permeability.

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As seen from the figure, the results from all the seven methods converge to each other in the long-wavelength limit. This confirms the conclusion we made in section 2 on equivalence, in the static regime, of the PHRS method, volumetric averaging and our method. Conversely, as the unit cell size increases, the results diverge. Note, first of all, that both the initial PHRS method and its corrected version (see, respectively, the black dashed and solid curves in panels (a) and (b)) give oscillating results that are far from those predicted by volumetric averaging (red dashed curves with open squares). The use of the optimal location of $L$ (with $l = a/2$) smoothes the curves out (see the blue dashed curves with open circles) and returns the correct value of the imaginary part of permittivity, $\varepsilon ''_{\rm eff} = 0$. Yet, the monotonic decrease in the real part $\varepsilon '_{\rm eff}$ remains to be a problem. Further correction of this 'optimized' PHRS method (see the blue solid curves with open circles) leads to qualitatively plausible behavior of $\varepsilon '_{\rm eff}$, although the value of $\varepsilon '_{\rm eff}$ deviates substantially from those predicted by either volumetric averaging, our method or TLEM (see the inset in panel (a)).

To quantify the difference between the methods we consider, we have calculated a relative deviation $\Delta \varepsilon '_{\rm eff} \equiv |\varepsilon '_{\rm eff} - \varepsilon _{\rm eff}^{{\rm{vol}}} |/\varepsilon _{\rm eff}^{{\rm{vol}}}$ of the values $\varepsilon '_{\rm eff}$ calculated by different methods from the value $\varepsilon _{\rm eff}^{{\rm{vol}}}$ calculated by volumetric averaging. The deviation versus unit cell size is shown in panel (c) for the most accurate methods presented in the inset. As seen from figure 9(c), at $a/\lambda _0 = 0.35$ our method provides a substantially more accurate value of the effective permittivity when compared to the optimized corrected version of PHRS method ($\Delta \varepsilon '_{\rm eff} = 6\%$ versus 12%). Also, up to $a/\lambda _0 \approx 0.25$ our proposed method returns effective permittivity values that are nearly identical (with $\Delta \varepsilon '_{\rm eff} < 0.5\%$) to those obtained by volumetric averaging.

The results presented in figure 9 suggest that PHRS method is highly sensitive to the choice of the integration path location not only in the trivial case of empty cells (see the preceding subsection) but also in the case of non-empty lattices. Moreover, the initial (with $l = 0$) version of this method and its corrected modification both provide, similar to the case of empty cells, complex valued effective permittivity, despite the fact that the inclusions we consider here are lossless.

Furthermore, it turns out (the results are not shown in this paper) that the first three of the above considered modifications of PHRS method predict implausible behavior of both $\varepsilon _{\rm eff}$ and $\mu _{\rm eff}$ for inclusions of various sizes, shapes and materials. Based on this, we remove these modifications from further discussion and shall keep only the corrected PHRS method which employs the optimal choice $l = a/2$.

Finally, it is worth noting that, according to figure 9, the effective permittivity of dielectric spheres disperses monotonically upward in the region $a/\lambda _0 < 0.35$. In terms of frequency dependence of the effective index of refraction $n_{\rm eff} (\omega ) = [\varepsilon _{{\rm{eff, r}}} (\omega )\mu _{{\rm{eff, r}}} (\omega )]^{1/2}$ this means (accounting for $\mu _{\rm eff} \approx 1$) that the lattice has normal dispersion at $\omega < 0.35 \times (2\pi c/a) \approx 2.2{\rm{ }}c/a$. Normal dispersion of $\varepsilon _{\rm eff}$, $\mu _{\rm eff}$ and $n_{\rm eff}$ of lossless lattices beyond resonance regime is well known (see e.g. [39]) and has been confirmed numerically by different methods (e.g. in [37, 40]). In contrast, Alù [11] calculated effective parameters of dielectric spheres within his homogenization theory and obtained a different behavior of $\varepsilon _{\rm eff}$ with anomalous dispersion at low frequencies, see figure 3 in [11].

Lattices of particles made of magnetic non-conductive materials such as ferrites, magnetic polymers, magnetodielectric composites and spin glasses may have a substantial magnetic response. Under proper tuning of their effective permittivity and permeability, such lattices can be used in the design of left-handed metamaterials [35, 41]. In figure 10, calculation results are presented for effective permeability of an SC lattice of spheres having $\varepsilon _{\rm{r}} = 3$, $\mu _{\rm{r}} = 4$, at lattice fill factor $f = 0.2$. The results were obtained by using the four most accurate methods mentioned in the preceding subsection.

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One can see from figure 10(a) that all the four methods predict the same static value of the effective permittivity. As the unit cell size increases, our proposed method, TLEM and volumetric averaging give values of $\mu '_{{\rm{eff, r}}}$ that are qualitatively close to each other. In contrast, the value calculated with the optimal PHRS method diverges substantially from these three. For example, at $a/\lambda _0 = 0.3$ the relative deviation $\Delta \mu '_{\rm eff} \equiv |\mu '_{\rm eff} - \mu _{\rm eff}^{{\rm{vol}}} |/\mu _{\rm eff}^{{\rm{vol}}}$ is less than 2% for our method and about 12% for the PHRS method, see figure 10(b).

Based on figures 9 and 10 we may conclude that for lattices of non-conducting particles our proposed method provides better results than any modification of the PHRS method does.