Metamaterials, artificially structured systems with peculiar properties not found in nature, have received considerable attention in recent years (see Ref.[1] for a recent review). The periodic array of infinitely long metal wires, the wire medium that supports an artificial plasma, is one of the earliest and most famous examples[2, 3]. For a certain polarization, the effective permittivity (ε) of such a wire medium possesses a dispersion relation that is similar to that of a plasma, and is negative in the frequency range below the effective plasma frequency. Designing a metamaterial that exhibits a strong magnetic response is very difficult, especially in the optical domain. The split-ring resonator (SPR) is one example that is known to provide magnetic resonance in the microwave and radio frequency (RF) range[4]. Although several previous designs have been proposed[5, 6, 7], achieving a magnetic response in the optical domain is more challenging, mainly because the optical properties of metals are very different from those in the RF domain.

The structural unit of a wire medium is an infinitely long metal wire. When excited by a TM wave, that is, a wave with the only electric field component parallel to the wire, the induced current on the wire is related to the local electric field E _loc by I=α_eE _loc, where α_e is in general a complex value with the imaginary part much larger than the real part, indicating an almost π/2 phase lag between I and E _loc. The electric polarization possesses another π/2 phase lag with respect to the current, and thus is anti-parallel to the local electric field. Such a phase lag is responsible for the negative effective permittivity of an artificial dielectric assembled from an array of metal wires. Now consider a dual situation, i.e. an infinitely long cylindrical structure (the specific structure to be discussed later) under a TE-wave excitation (the only magnetic field component is parallel to the cylindrical axis). If it produces a scattered magnetic field for which the phase and modal shape are similar to those of the scattered electric field from an infinite metal wire under a TM-wave excitation, its behavior can be ascribed to an induced effective magnetic current that also possesses a π/2 phase lag with respect to the incident magnetic field. Thus, by duality, an artificial magnetic plasma can be supported by an array of such structures (refer to Fig. 1(a)). Here we show a design for a cylindrical plasmonic structure that yields a strong, negative magnetic response in the optical domain.

To achieve this goal, we designed an artificial “flower” with alternating “petals” of a plasmonic material (ε _r1<0) and a conventional dielectric (ε _r2>0). The structure is invariant along its axis, and the cross section reveals the flower structure, as we see in Fig. 1(b) and 1(c). An intuitive understanding of this structure is obtained with an effective optical circuit. Previous researchers [8, 9] have shown how to model a particle in an external electromagnetic field as a lumped circuit element when the free space wavelength is much larger than the dimension of the particle, which is the usual situation in the RF and microwave domain. In the optical frequency range, based on the phase lag between the displacement current and the electric excitation, plasmonic particles can be modeled as inductors while conventional dielectric particles behave as capacitors[8]. When the TE mode is represented as cylindrical harmonics, the electric field of the fundamental mode (the 0^th order mode) has a circular shape (Fig. 1(b)). Thus, the flower structure may be approximated by the equivalent circuit shown in Fig. 1(c), in which the voltage sources in the loop circuit are provided by the electric field in the incident TE₀ mode. The resistors represent the ohmic loss in the plasmonic material. The circuit goes to resonance at an operating frequency determined by the inductance L and capacitance C as ω₀=1/√LC when the material loss (R) is negligible. The current in the equivalent circuit (the displacement current in the structure) is maximized at the resonance frequency, emitting strong radiation (the scattering) from the flower structure. The corresponding radiated field from the induced current is also a TE₀ mode, the same modal shape that would be induced by an infinitely long effective “magnetic current” flowing along the cylinder axis. Thus, the entire structure may be modeled as an induced magnetic current that goes to resonance, experiencing a phase shift of π with respect to the incident radiation. Above the resonance, the induced effective magnetic current is anti-parallel to the incident magnetic field, and is thus able to provide a negative effective permeability, in analogy to the wire medium that provides a negative effective permittivity under TM excitation.

 

Fig. 1. (a) An artificial medium composed of flower structures. (b)A single flower with TE0 incident radiation. (c) Left: the cross section of a flower. Dark: plasmonic petals; light: dielectric petals. Right: the equivalent RLC circuit for the structure. Each side represents one period in the structure. The capacitors (C) model the dielectric sectors, the inductors (L) together with the resistors (R) model the lossy plasmonic sectors, and the voltage sources (V) represents the incident electric field.

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The simplified picture of the equivalent circuit provides valuable insight into the system, the actual situation is much more complicated. A rigorous analysis requires numerical tools. For a TE incidence, the magnetic field can be expanded into the cylindrical harmonics as [10]

and the scattered field can be similarly expanded as

An e ^-iωt convention is assumed through out the paper. J_n(·) is the Bessel function, and H ⁽¹⁾ _n (·) is the Hankel function, of the first kind of order n. 𝓗 ₀ is a normalization constant with a unit of magnetic field. γ_mn is the coefficient connecting the contribution to the n ^th order mode in the scattered field from the m ^th order mode in the incident field, which is dimensionless and is completely determined by the geometry and material properties of the structure. For our design, the coefficient γ ₀₀ dominates the behavior of the structure and the other γ _mn-s can be neglected. The 0^th order mode with magnitude γ ₀₀ a ₀ in the scattered field can be considered to result from an induced effective magnetic current. An infinitely long magnetic current I_m (with units of volts) radiates a TE₀ mode magnetic field with magnitude -ωε ₀ I_m/4, where ε₀ is the permittivity of the host medium[11]. It is easy to show that a ₀=H _loc/𝓗 ₀, where H _loc is the local magnetic field exciting the structure (i.e. H_iz at the origin in Eq.(1)). Thus, we can model the structure as an induced magnetic current I_m=α_mH _loc, with α_m=-4γ ₀₀/(ωε ₀) completely determined by γ ₀₀.

In the following, we first study the change of γ ₀₀ as we change the value of ε _r1 for a given geometry, and fix the operating wavelength to be 20 times the outer radius. This is not a frequency-dependent property study, but nevertheless will yield an improved understanding of this problem and will provide guidelines for the design of such a structure using realistic material properties. The structure studied here has 6 angular periods, i.e. β ₁ +β ₂=60° as we see in Fig. 1(c). Other periodicities are in general similar. The dielectric portion is defined with ε _r2=2.2, which can be SiO₂. We assume the plasmonic material is lossless at this stage, so that the expected resonance can be clearly seen. The material loss will be considered later by adding an imaginary term to the relative permittivity of the plasmonic material. The structure is simulated using the commercially available finite element method software COMSOL^™. We calculate the scattered field for a given incident field of known distribution, and γ ₀₀ can then be extracted.

The magnitude of the scattering coefficient γ ₀₀ is shown in Fig. 2 (bold solid line) for a design for which the angles of the two sections in one period are identical to each other, i.e. β ₁=β ₂=30°. A case when β ₁≠β ₂ is discussed later. We can see that γ ₀₀ displays a resonance as ε _r1 varies. Specifically, the magnitude of γ ₀₀ goes to the maximum possible value |γ ₀₀|=1 as ε _r1 goes to -4.07. The phase of γ ₀₀, which determines the phase difference between the induced effective magnetic current and the excitation, is also plotted in Fig. 2 (red bold dashed line). Notice that it indeed flips from approximately -π/2 (indicating an effective magnetization in phase with H _loc) to approximately π/2 as |γ ₀₀| achieves the maximum value. In other words, the effective magnetization of the structure can be either in phase or 180° out of phase with H _loc. Thus, in analogy to the metal wire under TM excitation, such a structure can support an artificial magnetic plasma. Figure 3(a) displays the instantaneous scattered magnetic field (i.e. the total field minus the incident field for the region both inside and outside the particle) distribution, for the case of ε _r1=-4.07. The magnitude of the scattered magnetic field is much higher than that of the incident field because of the resonance. The plot is a snapshot of the field distribution at a phase of π. At this instant, the scattered field inside the structure achieves its maximum value when the incident field is at the negative maximum. This verifies that the scattered field is anti-parallel to the incident excitation.

 

Fig. 2. Magnitude (solid lines) and phase (red dashed lines) of γ ₀₀, for the lossless (bold) and the lossy (thin) case.

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In reality, the plasmonic material is always lossy, which may influence the resonance feature. Quantitatively, such an influence is reflected in the magnitude of the γ00 coefficient at resonance: it is smaller than 1 when loss exists. To see this influence, we calculate γ ₀₀ again, assuming an imaginary part of the relative permittivity with a magnitude 1% that of the real part for the plasmonic material. Such a ratio of the real to imaginary parts of the relative permittivity is common for many plasmonic materials in a wide frequency range of the infrared and/or visible spectra[12]. The magnitude of the scattering coefficient |γ ₀₀| for different values of Re(ε _r1) for the same structure is also shown in Fig. 2 (solid light line). Notice that it still exhibits a clear resonance feature despite the existence of material loss, at almost the same value of Re(ε _r1) of the lossless case. This is most easily seen in the phase plot (thin dashed line in Fig. 2, red online). The maximum magnitude of |γ ₀₀| is lower than that of the lossless case. Our studies show that geometries with different β ₁ and β ₂ values actually exhibit different material loss tolerance. In fact, when we use a larger plasmonic sector together with a smaller dielectric sector (β ₁>β ₂) but keep β ₁+β ₂=60°, the magnitude of γ ₀₀ at resonance is larger, indicating a stronger resonance, although we are using similarly lossy plasmonic material (Im(ε _r1)=0.01Re(ε _r1)). Of course, the value of Re(ε _r1) at which γ ₀₀ is resonant is also different when β ₁ and β ₂ take values different than 30° as that in Fig. 2. An example is described in detail below.

With the insight gained above, we now examine designs using realistic material parameters. We use silver with ε _r1 at different operating frequencies given by Ref.[12] and SiO₂ with ε _r2=2.2. A structure similar to Fig. 1 but with β ₁=55° and β ₂=5° is used, to achieve a strong resonance despite the existence of material loss in silver. For a structure with a=45.2nm, calculations show that the structure has a resonance at λ₀=904nm, or 20 times the diameter of the cylindrical structure, as we see in the plot of γ ₀₀ as a function of optical frequency in Fig. 3(b) (light solid line). To create an artificial magnetic plasma, we arrange the structures into a square lattice with a period d=135.6nm, or 0.15λ₀ (refer to Fig. 1(a)). According to the duality principle, the formula for ε_r of the artificial electric plasma composed of metal wires[13] can be revise to calculate µ_r for our medium. The result is

where M=0.5275 is a constant determined by the lattice shape[13] and

where C=0.5772 is the Euler constant and α_m=-4γ ₀₀/(ωε ₀) as discussed before. By using this formula, the response of the cylindrical structure is assumed to come from the induced magnetic current only, thus the relative permittivity is assumed to be 1, similar to the case of the wire medium where the relative permeability is assumed to be unity[13]. Equation (3) should not be interpretted as a Drude (or Drude-Lorentz) dispersion because A is a function of ω. For the lossless case, we have Re(α ^-1 _m)=1/(4ωε ₀), determined by energy conservation[13], so that Eq. (3) yields a real-valued µ_r. The real and imaginary parts of µ_r are shown in Fig. 3(b) as bold lines. As we expected, the real part of the effective permeability enters the negative region at frequencies above the resonance, when the phase of the induced effective magnetic current with respect to the incident magnetic field is shifted by π. Thus, the artificial medium behaves as a magnetic plasma in this frequency range. In contrast to the wire medium, here µ_r displays an obvious resonant feature. The real part of µ_r reverses sign at the resonance, when the imaginary part is maximized. Far below the resonance frequency, the structure shows obvious magnetism in this optical frequency range, when Re(µ_r) is greater than 1 with negligible Im(µ_r). This is the result of the strong scattering of the structure under TE incidence that is equivalent to an induced effective magnetization in phase with the excitation.

 

Fig. 3. (a)The distribution of the instantaneous scattered magnetic field (normalized to the the incident magnetic field at the origin) at resonance, for lossless plasmonic sectors. (b) γ ₀₀ (thin solid) and γ ₁₁ (thin dashed) of the cylindrical structure made of silver and SiO₂; and Re(µ_r) (bold solid), Im(µ_r) (bold dashed) for the artificial material composed of an array of such cylindrical structures.

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One assumption in making the analogy between the structure described here and the wire medium is that the scattering of the fundamental mode (TE₀ mode) dominates the behavior of the structure, as in the case for metal wires. This is indeed the case, as revealed by the numerical calculations. To show this, we plot the magnitude of γ ₁₁ in Fig. 3(b) as dashed thin lines. Over most of the frequency range we studied, γ ₁₁ is much smaller than γ ₀₀. Since γ ₁₁ mostly contributes to the effective permittivity, even in the frequency range when |γ ₁₁| is comparable with |γ ₀₀|, it may not have a qualitative influence on the results. Because the cross section of the structure is very small compared to the free space wavelength, the higher order scattering coefficients (γ_nn for n>1) are orders of magnitude smaller than γ ₁₁, as are the off-diagonal scattering coefficients (γ_mn for m≠n). This justifies the modeling of the structure as an induced effective magnetic current.

The current proposal differs from the former studies (such as Ref.[5]) in the aspects that dielectric segments are provided and placed side by side with the plasmonic segments, so that the two parts are strongly coupled with each other. Thus the current structure assembles the closed circuit model (Fig. 1(c)) better, and may lead to a further miniaturized geometry of deep sub-wavelength size, which is the topic currently under study. The intuitive understanding of the structure shown in Fig. 1 was prompted by the concept of the equivalent optical circuit with elements shown in the figure. This picture enabled us to qualitatively predict that such a structure would have a magnetic resonance, but quantitative analysis required detailed numerical modeling. A major feature omitted in the equivalent circuit model is the coupling to the surrounding medium. Also, away from the center of the cylinder, the magnetic field possesses some angular variation(Fig. 3(a)) that implies an electric field component along the radial direction, indicating that the interface between each pair of sectors is not strictly equi-potential as assumed in the equivalent circuit. Further study is also required to gain a deeper understanding of related metamaterial designs, for example, a sphere with alternating plasmonic and dielectric sectors around the axis. Different geometries are currently under study.