Equal-potential interpretation of electrically induced resonances in metamaterials

We propose a general description of electrically induced resonances (EIR) in metamaterials (MMs) comprising subwavelength unit cells. Based on classical electrodynamics, we found that EIR is governed by an equal-potential effect. Our theory accounts for the EIR phenomena and can give a renewed definition of the effective electric field and hence effective permittivity for MMs made of either dielectrics or metals as well as combinations thereof. The EIR, inherent to the periodic structures, may be the unifying origin of recently observed anomalous electromagnetic phenomena, e.g. the enhanced transmission, the suppressed transmission and the enhanced absorption by a variety of metal film structures in the terahertz range.


Introduction
The scalar potential is a key concept that has allowed physicists and engineers to understand and interpret a variety of electromagnetic (EM) phenomena [1], yet its strength in the context of metamaterials (MMs) remains unrevealed. In the design and fabrication of MMs, metallic structures are successfully being applied in the microwave range [2]- [4]. At the same time, rodtype structures made of dielectrics with a large permittivity have also been suggested as another route potentially applied in the terahertz and optical range [5]- [7], although some metallic structures are still being applied in the terahertz range [8,9]. Although these two classes of MMs can realize similar macroscopic EM responses such as anomalous effective permittivity, they do not seem connected due to the apparent different theory bases and design methodologies, which indicates that the common physics behind the MMs still needs to be revealed. In order to clearly advance our fundamental understanding of those EM phenomena in MMs, a unifying theoretical description is essential. The scalar potential offers an attractive solution to this challenge.
To appreciate the common physics of the two classes of MMs, consider first the quasistatic EM response of mixtures of sub-wavelength dielectric particles. An external electric field E s will polarize the particles, see figure 1(a). The effective permittivity is normally accounted for by the classical Maxwell Garnett (MG) or Clausius-Mossotti (CM) theory [10,11], with the inherent assumption of only a weakly localized electric field and the absence of resonances [12]. However, in MMs, resonances in the unit-cell structure are critical in making the whole composite possess peculiar EM responses, such as negative and µ [2]- [4], [13]. In the presence of resonances, a strongly localized field is excited inside the single unit-cell, thus potentially challenging the use of MG or CM theory. As an alternative, more elaborate field-averaging techniques have been used widely to extract effective material parameters [3,5], [14]- [18], but methods of this kind may lack a solid foundation beyond the static limit and in particular in resonances, i.e. the fields vary rapidly on the scale of the MM unit-cell.
In this paper, we employ an electrical potential approach and propose the electrically induced resonances (EIR) as the unifying concept of MMs made of either dielectrics or metals.

Theory
Metallic and dielectric particles may seem to have quite different EM properties, yet their polarizations are similar. In general, for an isolated particle much smaller than the wavelength of light, its polarization is dipole like. The macroscopic dipole moment of particles with positive or negative permittivity ( p ) is similar if [1]: (i) the operation frequency is far away from the inclusions' plasma frequency, i.e. p −1 or p 1, or (ii) no magnetic resonance occurs, see figure 1 Consider next an MM made of sub-wavelength structures, generically shown in figure 1(c). All the unit-cell structures are assumed identical and periodically arranged, with incident field (E i ) along one axis of the unit cell (z-direction). In resonances, strongly localized electric and magnetic fields are excited inside the unit cell. Although the polarized charges and currents may occupy the entire unit cell and make the problem complex, ∇ · B = 0 holds everywhere in space. The Maxwell equations may thus conveniently be rewritten in terms of the scalar potential φ and the vector potential A [1], i.e. B = ∇ × A and E = −∇φ + iω A. We emphasize that the distribution of electric and magnetic fields is identical inside each unit cell of an ordered MM, which follows from the Bloch theorem in the k 0 a 1 limit. Conceptually, this allows us to focus attention on a single unit-cell [2].
Suppose that a fictitious electron moves from point X (on the bottom surface of the unit cell) to point Y (on the top surface) along path s, see figure 1(c). From energy conservation, the work done by the EM force equals the change in the electron's potential energy and Because the E term is of capacitive nature while the A term (determined by the magnetic field) has an inductive origin, we may phenomenologically consider a simplified L-C series equivalent circuit for the MMs, see figure 1(c). To account for MMs made of lossy material, an extra series resistor may be included in the circuit. Note that for a given MM, different polarizations of the incident wave may make the L and C different; thus the resonant frequency shifts.

The electrostatic case
In solving (1), we first address the electrostatic limit before discussing the general case. For ω → 0, the gradient of the scalar potential clearly relates only to the electric field. The integration path s could be arbitrary, but the effective electric field on s defined as E eff =ŝ(φ X − φ Y )/S with S being the total length from X to Y will depend on the choice of s. For the given 4 polarization shown in figure 1(c), the MM's macroscopic electric response is then determined by the potential difference between X and Y . In particular, if φ X = φ Y , the unit cell will behave as an equal-potential body, hence supporting no effective electric field in total. Of course, this is strictly possible only inside a perfect electric conductor (PEC). However, for sufficiently good conductors we may still have φ X ≈ φ Y , which turns out to be a central requirement for the existence of the equal-potential effect.

The general electromagnetic case
Now, we proceed to the more general case of a non-zero frequency. Since the change in energy can be accounted for by considering the potential difference experienced by the fictitious charge, the macroscopic effective electric field should still be expressed as Meanwhile, the effective electric displacement in a single unit-cell can also be found by taking the polarization into account, i.e. D eff = 0 E eff + P eff . We emphasize that zero value of φ = φ X − φ Y (and consequently E eff ) may also arise in the presence of a finite vector potential's contribution, which often occurs in MMs. The strong dispersion of effective permittivity of MMs occurs close to those frequencies at which the equal-potential condition is satisfied, which we show below. Basically, the macroscopic property of a bulk sample made of the MM specified above can be equivalently described through an effective permittivity with V being the volume of the unit cell, ẑ · P = 1 V Vẑ · P dv being the z-component of average polarization and φ = φ L being the average scalar potential difference across the whole unit cell. Again, a strongly dispersive permittivity is apparent when φ X ≈ φ Y , and hence anomalous EM responses occur, e.g. in the form of anomalous reflection (R = |S 11 | 2 ), transmission (T = |S 21 | 2 ) and enhanced absorption (A = 1 − R − T ). Here, S i j are the usual components of the scattering matrix.
We should note that here we assume the incident wave to be polarized along the z-direction, which allows us to estimate eff,z . The MM we study here is in general anisotropic and the other components of¯ eff can also be estimated through the same procedure but under some other excitation condition assumption.

Formulation
To further illustrate our points, we consider the example of a periodic cylindrical-rod array, see figure 1(c), with rods of finite length l. In the limit where l = L, the rod is effectively infinitely 5 long. The internal field of the rods E int can conveniently be expressed by the superposition of different-order cylindrical waves [19]. Due to the sub-wavelength scale, high-order cylindrical waves are hardly excited inside a single unit-cell and we have E int (ρ) ≈ẑ E 0 J 0 (k p ρ) with k p = √ p (k) 0 being the wavenumber inside the rod. Since the scalar potential is uniform on both the bottom and top surfaces, the potential difference can be conveniently determined by (1) with an integration path connecting the bottom center and the top center, i.e. O-O in figure 1(d). Four points are critical in determining the integration, i.e. O, O , K and K , and The three terms can be worked out by interpreting the electrodynamics inside the unit cell. Since the polarization current flows along the cylinder but the polarization charge mainly distributes on the two ends, it is natural to assume that the scalar potential inside the cylinder, i.e. close to the straight line K -K , is related to both the electric and magnetic fields. However, close to the straight segments O-K and K -O , only the instantaneous electric field is involved, since A will be completely canceled by the non-instantaneous component of E in (1) in a source-free case. Having calculated the potential difference, we apply (2) and, after some tedious but straightforward algebra, we finally arrive at with f = πr 2 l a 2 L being the volume fraction, φ = φ L being the average scalar potential difference and Here, R = a/ √ π and (d) = r 0 J 0 (k p ρ) √ ρ 2 +d 2 ρ dρ. Note that our theoretical analysis is general and can equally well describe the effective permittivity of MMs made of either dielectrics ( p > 0) or metals ( p < 0). Resonances are expected when φ ≈ 0.

Numerical evaluation
In the following, we numerically explore the consequences of (3). Firstly, we consider dielectric cylinders. Figure 2(a) shows the effective permittivity calculated with the aid of (3), the parameters of which are explained in the figure caption. As seen, the MM composed of ILR possesses a strongly dispersive effective permittivity, in full agreement with results in [5]- [7]. When turning to rods of finite length, the EIR is shifted towards higher frequencies and the effective permittivity exhibits almost no frequency variation in the considered frequency range, see the dashed line in figure 2(a). Due to the absence of resonance in the close frequency vicinity, the nearly non-dispersive eff can here equally well be interpreted by MG or CM theory.
Secondly, we turn to cylinders made from a dispersive media whose permittivity obeys a Lorentz model in the terahertz region, i.e. p = 1 − ω 2 p /(ω 2 − ω 2 0 ) with plasma frequency ω p /2π = 600 THz and binding frequency ω 0 /2π = 200 THz. The dimensions of the MM unitcell are now scaled down to nano size, as specified in the caption of figure 2(b). As seen from the results, the MM made of infinitely long cylinders will preserve the binding frequency while having an effective plasma frequency much smaller than that of the original plasma media, as previously predicted [2]. However, for the MM comprising cylinders of finite length, the effective permittivity is strongly dispersive with both the effective plasma frequency and the binding frequency are modified. Clearly, figures 2(a) and (b) display the same qualitative Lorentz-like behavior (compare blue and green curves), thus emphasizing the common physics of the two MM classes.

The gold rod array
As shown above, our theory can successfully be used to describe the MMs, both dielectric and metallic. However, its application is not restricted to infinite stacks of MM unit-cells. If only a single layer of MM unit-cells exists, equation (2) (3). Since equation (3) is basically not applied to this one-layer composite, hence, the two curves shown cannot perfectly match. In the simulation, the incident plan wave propagates along thex-direction, with polarization along theẑ-direction, see figure 1(c). Only a single unit-cell is placed into the simulation site, with periodic boundary conditions being applied in the in-plane (ŷ andẑ) direction.
still exhibits EIR behavior, which is also plasmon like in the terahertz range, i.e. anomalous reflection together with strong absorption. To illustrate this, we take an MM made of finite gold rods as an example. The unit-cell is quadratic with lattice constant L = 100 nm and the gold rods have radius r = 15 nm and length l = 80 nm. For the dielectric constant of gold, see [20].
In figure 3(a), a full-wave simulation of the one-layer MM clearly reveals a resonance around 460 THz at which strong reflection and absorption occur and consequently the transmission is low. Figure 3(b) shows corresponding results for an effective medium slab with a permittivity derived from (3). Note the qualitative agreement between panels (a) and (b).  offering an equal-potential description. However, our theory applies to MMs beyond the rodtype structures emphasized so far. In fact, the EIR discussed here occurs in most MMs comprising periodically arranged sub-wavelength structures, including fish-net structures and arrays of sub-wavelength holes in thin metal films. As an example, figure 4(a) considers a semitransparent 30 nm thick (comparable to the skin depth at 460 THz) gold film with a periodic array of holes. The detailed parameters employed in the simulation are listed in the caption. The film investigated resembles the inverse structure of the rod array studied in figure 3(a), and from Barbinet's principle, we expect it to exhibit similar EM behavior. Formally, the film's qualitative EIR behavior may also be addressed by our equal-potential description, but analytical calculations resembling (3) are too complex and results are not included. However, our full-wave simulations clearly show the appearance of the anticipated resonance appearing around ω 0 /2π = 460 THz, see figure 3(c). For comparison, figure 3(b) shows the corresponding results for a closed gold film, where the absence of holes prevents extraordinary transmission (accompanied by suppressed reflection) below ω 0 and suppressed transmission above ω 0 . 9

Conclusion
In conclusion, a new description of EIR has been established with the help of the scalar potential. Originating from classical electrodynamics, our theory accounts well for MMs made of either dielectrics or metals and could be expanded to describe an arbitrary composite of subwavelength structures. The theory presented establishes the relationship between the equalpotential effect and the EIR, which also helps in interpreting recently reported phenomena, such as extraordinary transmission, suppressed transmission and enhanced absorption by MMs and periodic structures [21]- [25]. Our unified potential description may stimulate more use of dielectric structures to mimic desired properties readily achieved with more lossy metallic counterparts.