13. Anisotropic Representation for Spatially Dispersive Periodic Metamaterial Arrays

It can be proven using Lu’s book [2, Eq. (2.16) on p. 23, Eq. (5.11) on p. 57, and the equation in exercise 4 on p. 28] that sufficient conditions for (13.1) to hold are that \(\epsilon (\omega _c)\) and \(\mu ^{-1}(\omega _c)\) are analytic and single-valued (that is, holomorphic) functions in the upper half (\(\omega _i>0\)) of the complex \(\omega _c=\omega +i\omega _i\) plane having boundary values equal to \(\epsilon (\omega )\) and \(\mu ^{-1}(\omega )\) [that is, \(\lim _{\omega _i\rightarrow 0} \epsilon (\omega _c) = \epsilon (\omega )\) with \(\omega _i\) in the upper half plane, and similarly for \(\mu ^{-1}(\omega _c)\)] such that these boundary values are Hölder continuous, and \(\epsilon (\omega _c)-\epsilon _0\) and \(\mu ^{-1}(\omega _c)-\mu ^{-1}_0\) both \(\sim 1/|\omega _c|^\alpha \), \(\alpha >0\), as \(|\omega _c|\rightarrow \infty \) for \(\omega _i\ge 0\). These sufficient conditions for the Kramers–Kronig relations also imply (with ordinary continuity replacing the stronger Hölder continuity) the corresponding time-domain causality relations by means of the Cauchy theorem for continuous functions [2, p. 64; 3, p. 347]. For example \(\int _{-\infty }^{+\infty }[\epsilon (\omega )-\epsilon _0]e^{-i\omega t} \text{d}\omega = 0\) for \( t\;<\;0 \).

The theorem in the Appendix or in [4, Sect. 123] can be invoked to prove that \(\epsilon ^{-1}(\omega )-\epsilon _0^{-1}\) or \(\mu ^{-1} (\omega )-\mu _0^{-1}\) is causal if \(\epsilon (\omega )-\epsilon _0\) or \(\mu (\omega )-\mu _0\) is causal (and vice versa), provided \(\epsilon (\omega )\) and \(\mu (\omega )\) satisfy the passivity conditions in (13.2). Nonetheless, we show the Kramers–Kronig relations in (13.1b) for \(\mu ^{-1}(\omega )-\mu _0^{-1}\) rather than for \(\mu (\omega )-\mu _0\) because when spatial dispersion is taken into account later by introducing an arbitrary real plane-wave propagation vector \({{\varvec{\beta }}}\), we find that the passivity conditions in (13.2) are coupled and do not necessarily hold separately where the spatial or temporal dispersion is large. Thus, in general, for spatially dispersive macroscopic permittivity and permeability, only \(\epsilon ({{\varvec{\beta }}},\omega )-\epsilon _0\) and \(\mu ^{-1}({{\varvec{\beta }}},\omega )-\mu _0^{-1}\) satisfy causality at each fixed \({{\varvec{\beta }}}\) for electric charge-current defined polarizations.

The Bohr–van Leeuwen theorem, which states that a classical material composed of free charges in thermal equilibrium cannot be effected by a magnetic field because the magnetic field does no work on moving charges [9, Sects. 24–27], is often used to argue that classical electromagnetics cannot describe diamagnetism. This theorem assumes that all the energy at thermal equilibrium is in the kinetic and potential energy of the charge carriers. However, by using a conducting metamaterial inclusion as a model for a molecule, an additional inductive magnetic-field energy is introduced such that the Bohr–van Leeuwen theorem no longer applies and one is able to classically describe diamagnetic materials and metamaterials.

In Refs. [6] and [11], inequalities for \(\epsilon (\omega )\) and \(\mu (\omega )\) in a spatially nondispersive dipolar continuum that were derived from an energy-conservation inequality also imply the inequalities in (13.3) under the same assumptions. Although not stated explicitly in Refs. [6] and [11], the energy-conservation inequality that was used does not hold, in general, for diamagnetic \(\mu (\omega )\) produced by induced magnetic dipole moments on molecules (or inclusions) that have no permanent magnetic dipole moments; that is, the energy-conservation theorem in [6] and [11] assumes that the macroscopic \(\mu (\omega )\) is produced entirely by the alignment of initially randomly oriented permanent magnetic dipoles.

Landau and Lifshitz introduced the use of the single polarization vector for “natural optical activity” in Sect. 83 of the first edition (1957) of their book, Electrodynamics of Continuous Media [5]. Later, Silin and Rukhadze [16] and Agranovich and Ginzburg [17] continued to use this single-polarization approach in their general formulation of spatial dispersion. In the formulation of spatial dispersion for continua given in Sect. 103 of the second edition of the Landau and Lifshitz (and Pitaevskii) book [5], unlike in the formulations of [16] and [17], they do not show an applied electric current density explicitly in Maxwell’s equations and yet they assume arbitrary real values of the propagation vector \({{\varvec{\beta }}}\) and frequency \(\omega \). This is understandable because they show only the space-time (\(\mathbf{r},t\)) Maxwell equations in which the applied source densities can be localized. Nonetheless, one must assume in the Landau–Lifshitz–Pitaevskii formulation that applied electric current density is present in the (\({{\varvec{\beta }}},\omega \)) representation.

Bianisotropic constitutive relations for both applied magnetic and electric plane-wave \(e^{i({{\varvec{\tiny \beta }}}\cdot \mathbf{r}-\omega t)}\) dependent current exciting the inclusions of the array are expressible as \(\mathbf{D}_{b} = \overline{{\boldsymbol{\epsilon }}}_{b}\cdot \mathbf{E}_{b} +\overline{{\varvec{\tau }}}_{b}\cdot \mathbf{B}_{b}\) and \(\mathbf{H}_{b} = \overline{{\varvec{\mu }}}^{-1}_{b}\cdot \mathbf{B}_{b} +\overline{{\boldsymbol{\nu }}}_{b}\cdot \mathbf{E}_{b}\) with finite bianisotropic constitutive parameters \((\overline{{\boldsymbol{\epsilon }}}_{b},\overline{{\varvec{\tau }}}_{b},\overline{{\varvec{\mu }}}^{-1}_{b},\overline{{\boldsymbol{\nu }}}_{b})\), which, like the fields, are functions of \({{\varvec{\beta }}}\) and \(\omega \). Then if the applied magnetic current is set equal to zero so that \(\omega \mathbf{B}={{\varvec{\beta }}}\times \mathbf{E}\) (now omitting the subscripts “\({b}\)” on the fields) as in (13.11a), the corresponding anisotropic permittivity dyadic and inverse transverse (to \({{\varvec{\beta }}}\)) permeability dyadic are given by \(\overline{{\boldsymbol{\epsilon }}}=\overline{{\boldsymbol{\epsilon }}}_{b} + \overline{{\varvec{\tau }}}_{b}\cdot \overline{{\varvec{\beta }}}/\omega \) and \(\overline{{\varvec{\mu }}}_\mathsf{tt }^{-1} =\overline{{\varvec{\mu }}}^{-1}_{{b}\mathsf{tt }} - \omega \overline{{\boldsymbol{\nu }}}_{{b}\mathsf {tt} }\cdot \overline{{\varvec{\beta }}}/|{{\varvec{\beta }}}|^2\) with \(\overline{{\varvec{\beta }}}\cdot \) denoting the antisymmetric dyadic equivalent to \({{\varvec{\beta }}}\times \) and given explicitly in (13.75) below. From the results in [22] and [23], it can be shown that as (\({{\varvec{\beta }}}\rightarrow 0\), \(\omega \rightarrow 0\)) the magnetoelectric parameters, \(\overline{{\varvec{\tau }}}_{b}\) and \(\overline{{\boldsymbol{\nu }}}_{{b}{\mathsf{tt}} }\), have a lattice contribution that is generally proportional to \(\omega {{\varvec{\beta }}}\) plus a bianisotropic-inclusion contribution that approaches a constant. (A third contribution to \(\overline{{\boldsymbol{\nu }}}_{b}\) proportional to \(\omega \) can be produced by the contribution from the electric dipole moment of an inclusion to its magnetic dipole moment [1, Eq. (2.9); 25]. However, the origin of the unit-cell integration can always be chosen with respect to the inclusion to make this contribution go to zero as \({{\varvec{\beta }}}\rightarrow 0\) such that it becomes proportional to \(\omega {{\varvec{\beta }}}\) like the lattice contribution. It should be noted that although helical metal inclusions can produce bianisotropic effects, this geometrically produced bianisotropy vanishes at low enough frequencies \(\omega \) for inclusions electrically isolated from one another—an assumption made throughout.) Therefore, only if the bianisotropy at the inclusion level does not become negligible at the low spatial and temporal frequencies (\({{\varvec{\beta }}}\rightarrow 0\), \(\omega \rightarrow 0\)), does the anisotropic constitutive parameters \(\overline{{\boldsymbol{\epsilon }}}({{\varvec{\beta }}},\omega )\) or \(\overline{{\varvec{\mu }}}^{-1}_\mathsf{tt }({{\varvec{\beta }}},\omega )\) exhibit strong temporal or spatial dispersion as \(\omega /|{{\varvec{\beta }}}|\rightarrow 0\) or \(|{{\varvec{\beta }}}|/\omega \rightarrow 0\), respectively. Nevertheless, both \(\overline{{\boldsymbol{\epsilon }}}({{\varvec{\beta }}},\omega )\) and \(\overline{{\varvec{\mu }}}_\mathsf{tt }({{\varvec{\beta }}},\omega )\) generally remain finite (nonsingular) as \(|{{\varvec{\beta }}}|/\omega \rightarrow 0\).

Except where explicitly stated otherwise in the contexts of Footnote 6 and Eq. (13.52), we avoid introducing hypothetical applied magnetic currents in addition to the applied electric currents into the microscopic Maxwellian equations because this would require either a macroscopic bianisotropic representation [21, 23] or a much more complicated macroscopic anisotropic formulation for the fundamental modes of the arrays in terms of applied sources that would involve macroscopic bianisotropic constitutive parameters. Also, of course, magnetic currents or charges on the right-hand side of (13.4a) or (13.4c) have never been observed experimentally.

The externally applied fields satisfy the Maxwell time-domain equations given by and the corresponding spectral-domain Maxwellian equations given by Since the externally applied fields hold in free space with no induced sources, only the fundamental Floquet mode is nonzero for the applied spectral-domain fields. Taking the dot product of \({{\varvec{\beta }}}\) with (13.14b) shows that \({{\varvec{\beta }}}\cdot \mathbf{E}_{a}={{\varvec{\beta }}}\cdot \mathbf{J}_{a}/(i\omega \epsilon _0)\) and thus the longitudinal (parallel to \({{\varvec{\beta }}}\)) component of \(\mathbf{E}_{a}\) diverges as \(1/\omega \) for \(\mathbf{J}_{a}\) equal to a nonzero value as \(\omega \rightarrow 0\). Therefore, for some numerical solutions, it may be beneficial to choose the longitudinal component of \(\mathbf{J}_{a}\) proportional to \(\omega \).

All the Floquet modal coefficients, in addition to the fundamental ones, are found from the space-frequency fields simply by replacing \({{\varvec{\beta }}}\) in the integrand of (13.15) or (13.17) by \({{\varvec{\beta }}}_{lmn}={{\varvec{\beta }}}+\mathbf{b}_{lmn}\) to giveand similarly for the other fields and sources.

One might object that the induced magnetization current \(\nabla \times \varvec{\mathcal{{M}}}_\omega (\mathbf{r})\) is not included in addition to the induced equivalent electric current \(\varvec{\mathcal{{J}}}^p\!\!\!_\omega (\mathbf{r})\) in the decomposition (13.18–13.23) since this Amperian magnetization current is also an equivalent electric current in the microscopic Maxwell equations as given by (13.4). The reason it is not included is that if \(\nabla \times \varvec{\mathcal{{M}}}_\omega (\mathbf{r})\) were included with \(\varvec{\mathcal{{J}}}^p\!\!\!_\omega (\mathbf{r})\) in (13.20), one finds that it would produce no additional electric polarization in (13.23a), an additional magnetization in (13.23b) equal toand an additional electric quadrupole term in (13.23c) equal toThat is, it would merely produce extra terms of order \({{\varvec{\beta }}}\) higher than the \(\mathbf{P}^e({{\varvec{\beta }}},\omega )\) and \(\mathbf{M}^e({{\varvec{\beta }}},\omega )+\mathbf{M}({{\varvec{\beta }}},\omega )\) terms in (13.28) produced by keeping \(\nabla \times \varvec{\mathcal{{M}}}_\omega \) separate from \(\varvec{\mathcal{{J}}}^p\!\!\!_\omega \) in (13.18–13.23), and these extra terms would become negligible as \({{\varvec{\beta }}}\rightarrow 0\) and the generalized averages in (13.23) become ordinary averages. Moreover, the electric quadrupole term in (13.25) and the magnetic dipole integral term in (13.24) cancel when they are inserted into the electric quadrupole and magnetic dipole contributions in (13.22).

Sections 13.2.1–13.2.1.1 are not essential to the development in the rest of the chapter.

This criterion of “\(\ll \)1” can sometimes be relaxed to “<1” [8], and in the special case of homogeneous inclusion material that occupies nearly the entire volume of the unit cell, the criterion can be relaxed to values greater than \(1\). Although such a nominally periodic array can be considered a continuum over spatial and frequency bandwidths, \(|{{\varvec{\beta }}}d|\) and \(|k_0d|\), greater than \(1\), it is a relatively uninteresting case that we shall ignore in the discussion of continua and their boundary conditions in this section and the following Sect. 13.2.1.1.

Delta functions \(\delta (n)\) and their derivatives in the polarizations as represented by \(\varvec{\mathcal{{J}}}_{\!\delta }(\mathbf{r},t)\) and \(\varvec{\mathcal{{K}}}_\delta (\mathbf{r},t)\) may exist if these polarizations are proportional to unusually high spatial derivatives of the fields, that is, if they display strong enough spatial dispersion. In that case, the boundary conditions in (13.43) may have to be modified. However, significant polarization proportional to unusually high spatial derivatives of the fields generally indicates the presence of higher-order multipoles. For example, \(\varvec{\mathcal{{P}}}^e_{0\text{ave}}\) proportional to the second spatial derivative of \(\varvec{\mathcal{{E}}}_{\text{ave}}\) would indicate the presence of magnetic dipoles or electric quadrupoles. Similarly, \(\varvec{\mathcal{{M}}}^e_{0 \text{ave}}\) or \( \overline{\varvec{\mathcal{{Q}}}}^e_{0\text{ave}}\) proportional to the second or third spatial derivatives of \(\varvec{\mathcal{{B}}}_{\text{ave}}\) or \(\varvec{\mathcal{{E}}}_{\text{ave}}\) respectively, would indicate the presence of octopoles.

The field symbols on the left-hand sides of Eqs. (13), (15), and (16) in [38] should be boldface, and the \(\nabla \) symbol one line below Eq. (12) of [38] should be \(\nabla _{\!s}\).

At unit-cell resonant frequencies (\(\omega = \omega _{{\text{uc}}}({{\varvec{\beta }}})\)) of lossless arrays, the \(\mathbf{E}({{\varvec{\beta }}},\omega _{{\text{uc}}})\) and \(\mathbf{B}({{\varvec{\beta }}},\omega _{{\text{uc}}})\) can be zero and the permittivity and inverse permeability can have poles. Thus, (13.84b) does not necessarily hold at these unit-cell resonant frequencies. These unit-cell singularities disappear for real values of \({{\varvec{\beta }}}\) and \(\omega \) if a small loss is inserted into the material of the array inclusions.

One could conjecture that the induced fundamental Floquet-mode electric field could cancel the applied electric field (for all applied current excitations) such that \(\varvec{\mathcal{{E}}}_0(\mathbf{r},t)=0\) for \(t\;<\;t_0\) where \(t_0>0\). Then \(\varvec{\mathcal{{D}}}_0(\mathbf{r},t)\) would begin before \(\varvec{\mathcal{{E}}}_0(\mathbf{r},t)\) and \(\overline{{\boldsymbol{\epsilon }}}_e({{\varvec{\beta }}},\omega )\) would not be causal. To see that such a cancellation is not possible, consider an applied field with the time dependence of a delta function at \(t=0\) (uniform frequency spectrum). If there is any loss in the inclusions, the frequency spectrum of the induced fields will decay as \(|\omega | \rightarrow \infty \) and thus the induced time-domain fields will be finite for all time including \(t=0\) and unable to cancel the delta function in the applied fields. Even if the inclusions are lossless and a delta function in the induced fields exactly canceled the applied-field delta function (an extraordinarily unlikely occurrence), the reactance of the inclusions would not allow the induced fields to be zero in the finite interval \(0<t<t_0\). A similar argument shows that the total time-domain fundamental Floquet-mode magnetic field \(\varvec{\mathcal{{B}}}_0(\mathbf{r},t)\) turns on with the applied magnetic field.

Some authors reserve the name “Clausius–Mossotti” for the formulas with quasi-static polarizabilities and use the name “Maxwell-Garnett” for the formulas with polarizabilities that are functions of frequency \(\omega \). In the rest of the section, we shall omit the label “Maxwell-Garnett” and simply refer to these formulas by the name “Clausius–Mossotti.”

If the real part of \(\chi (x)\) has a \(\pm 1/x\) singularity at \(x=0\) the change in \(\phi \) over both the negative and positive \(x\) axis is \(\mp \pi /2\) to yield a \(\Delta \phi = \mp \pi \). However, Eq. (7.22) on page 76 of [2] shows that with a \(1/x\) singularity, the formula in (13.133) changes toand thus we still find \(N_0=0\).