Modeling standing-wave plasmonic resonance for split-ring resonators

Forty years ago, the physicist V. G. Veselago proposed the concept of a left-handed material (LHM) in which the electric and magnetic permeability are simultaneously negative. An LHM would have a negative refractive index and, thus, have optical phenomena (such as negative refraction, reversed Čerenkov radiation, and inverse Doppler effect) opposite to those found in natural materials.¹ However, the lack of negative magnetic permeability in nature made these negative-refractive-index media (NRIM) scientific fiction for decades.

In the late 1990s, Sir Pendry and colleagues proposed artificially constructed subwavelength metamaterials that promised strong and even negative electric and magnetic responses, respectively.^2,3 Using those materials, Shelby and colleagues successfully demonstrated negative refraction in the microwave regime⁴ through an NRIM composed of plasmonic wires² and split-ring resonators (SRRs).³ These metamaterials also offer many possibilities in optics, including super/hyperlenses,⁵ invisibility cloaks,⁶ and other devices.

The SRR was the earliest and most frequently used metamaterial, with unprecedented negative magnetic permeability³ and high-frequency magnetism.⁷ In 2004, we demonstrated the first artificial magnetic response beyond the terahertz gap by a planar SRR array (see Figure 1). This indicated the feasibility of applying 2D structures to achieve effective macroscopic responses from those subwavelength structures. This, in turn, would allow researchers to take advantage of mature lithographic techniques to fabricate smaller metamaterials and integrated systems. Yet a few challenges of operating the SRR remain. For example, it is conventionally excited by the external magnetic field normal to its plane (i.e., grazing-angle incidence is preferred),³ but that is impractical for the tiny samples used in nanophotonics applications. In addition, there is no quantitative model to describe the electric and magnetic responses of the SRR under normal incidence. Normal incidence is better for nanophotonic applications, but the resonator is excited by the external electric (E) field instead.

Figure 1. (a) Illustration of planar split-ring resonator (SRR) arrays, which were operated in the terahertz (THz) regime. (b) (top) Ratios of reflectance under s-polarized (r_s) and p-polarized (r_p) excitation from three-different-dimension SRRs, respectively. The corresponding real—μ′ (ω)—and imaginary—μ′′ (ω)—magnetic permeability indicate the artificial magnetism beyond the THz gap.⁷ E: Electric field. H: Magnetic field. K: Wave vector of the excitation. ω: Frequency.

Figure 2. (a) Scanning-electron micrographs of the SRRs and (b) measured absolute (abs.) reflectance under various polarized angles.⁸ ω₁, ω₂, ω₃: Frequencies 1, 2, and 3. θ: Angle between the electric polarization and the X axis.

To solve these challenges we established a model of standing-wave plasmonic resonance (SWPR).^8,9 Under normal incidence the SRR presents multimode plasmonic resonance. This results in electric dipoles as the external E field is symmetric to the SRR (i.e., along the X direction), magnetic dipoles as the E field is asymmetric to the SRR (i.e., along the Y direction), and both kinds at arbitrary polarization angles because of the superimposed contribution from two individual orthogonal electric excitations (see Figure 2). We also quantified the relationship between the multimode plasmonic resonance in the SRR, the resonance wavelength and mode, and the local refractive index, which verified the standing-wave model (see Figure 3). The concept of the SWPR can also be applied to multiple split-ring resonators (MSRRs),¹¹ a variation of the SRR with a structure that usually functions at higher frequencies than those in SRRs based on the same size unit cells. Applying this model to both resonators, we can clarify the nature of the excited electric or magnetic dipoles, predict the corresponding working frequencies, and even control the polarization-dependent permittivity.⁹

Figure 3. The reciprocal values of multiple resonance wavelengths from experiment (solid squares) and simulation (hollow triangles) are plotted versus the mode numbers, where n_∥are the modes excited by perpendicular polarization (E_⊥), and n_⊥ are the modes excited by parallel polarization, E_∥(and n∊[1, 8]). A clear linear relationship (green line) verifies the model of standing-wave plasmonic resonance.⁹

Using the SWPR model, we can manipulate the operating frequency of the SRR by simply designing its proper length accordingly, paving the way for other electromagnetic applications. More recently, we proposed mimicking electromagnetically induced transparency in quantum mechanics, but in a classical analog. By introducing asymmetric coupling resonance (ACR) into two asymmetric SRR nanostructures, which resonate at similar frequencies but with different quality (Q) factors, an additional allowed band emerges—see Figure 4(a). This contrasts with the symmetric planar SRR pairs shown in Figure 4(b).¹⁰ The ACR stems from the interaction between the sub- (low-Q) and superradiant (high-Q) modes to suppress the induced currents for radiation. Such a coupled system with a sharp peak could be an excellent refractive-index sensor for chemical and biological detection because of its great figure of merit.

Figure 4. (a) Asymmetric and (b) symmetric planar SRR pairs and their corresponding transmission spectra. Note an additional allowed peak in (a), due to the asymmetrically coupled resonance (ACR), to exhibit metamaterial-induced transparency.¹⁰

In SRR-based metamaterials, we recently demonstrated a NRIM operated at various incident angles.¹² We also demonstrated a preliminary, tunable, coupler- and label-free SRR-based sensor beyond conventional surface-plasmon-resonance sensors, which can even be applied in a label-free biological imaging system. In addition, we are working on freezing photons and external cloaking based on the ACR effect.