Broadband absorber with dispersive metamaterials

2.1 Ideal absorbing layer

Consider an EM wave from the air incident to the planar interface of an absorber attached to a metal surface (Figure 1). For simplicity, we only consider normal incidence and assume that the metal is a perfect electric conductor (PEC) and that the absorber (thickness d _L ) has the complex refractive index ε L = n + i κ . For a monochromatic wave of wavelength λ incident from the air, the reflected wave arises from the mismatched impedance with the reflection coefficient r:

where k ₀, Z ₀(k ^L , Z ^L ) are the wavenumber and impedance, respectively, in a vacuum and in the layer. Complete absorption occurs when r is zero, for which the refractive index satisfies

Figure 1:

Light incident from the air to a unit cell consisting of an ideal absorber on a perfect electric conductor (PEC).

Since equation (2) is complex with the complex refractive index n + ik, we separate the real and imaginary parts of the equation and solve them for n, k. We note that n, k are functions of the dimensionless quantity k ₀ d _L up to integer -m branches. Although analytic solutions are not available, we can solve this numerically within the region of interest for each branch. The results in Figure 2 show that index n is anomalously dispersive, i.e., it reduces as the frequency increases. Index n grows for higher branch number m because it supports more nodes inside the layer. From this point, we restrict our focus to the lowest branch (m = 0).

Figure 2:

Dispersion of the refractive index branches for an ideal absorber: (a) real and (b) imaginary parts of the refractive index.

2.2 Anomalous dispersion and oscillator fitting

Anomalous dispersion is not commonly observed in natural materials, arising only within a narrow frequency band near resonance and accompanied by strong absorption. Within the region of interest, such as the X-band in Figure 3(a), anomalous dispersion can be modeled using a multipole Lorentz oscillator model of a dielectric constant that satisfies the physical causality constraint. In our case, we fit the numerically calculated dispersion with a two-pole Lorentz oscillator model by adjusting the parameters of the Lorentz oscillators as follows:

were ɛ _inf is the relative permittivity for the large frequency limit, and ω _p , ω _o , and Γ are the plasma frequency, resonance frequency, and damping factor, respectively. The fitting parameters for various thicknesses of the absorber are summarized in Table 1.

Figure 3:

Anomalous refractive index for an ideal X-band absorber of 4 mm thickness with (a) Lorentz model fitting and (b) metamaterial fitting.

To realize the permittivity of a two-pole Lorentz oscillator, we consider a double-layered, square-loop metamaterial. Square-loop metamaterials exhibit resonance features that can be effectively described using an equivalent circuit model [5]. Our double-layered metamaterial is also described using an RLC parallel circuit and lumped elements [17] in Figure 4. These lumped elements are related to the structural parameters of the metamaterial, in particular the period of the unit cell p, the width of the square loop strip w _strip, and the slot spacing between the square loops. Even though the lumped element circuit model describes the transmission line accurately, it agrees with the propagation properties of the metamaterial as an effective continuous medium only for the limit λ ≫ p ≫ w _strip. Here, we only note the appearance of two distinct resonances in the double-layered, square-loop metamaterial but do not pursue this further. Rather, instead of relying on a circuit-model–based approach, we design a metamaterial via a direct optimization process, as described in the next section.

Figure 4:

Equivalent circuit diagram for a double-layer metamaterial.

To confirm the validity of our direct optimization process, we use the Nicolson–Ross–Weir (NRW) parameter retrieval method [18, 19] to obtain the effective refractive index for the designed double-layered, square-loop metamaterial. A comparison between the dispersion of an ideal absorber and the NRW-retrieved dispersion of the metamaterial is presented in Figure 3(b).

2.3 Design of the double-layered, square-loop metamaterial

The metamaterial absorber consists of two layers of a copper square loop attached to a silicon rubber substrate (Figure 5). To balance damping, the copper square loop is divided into four pieces that are connected by resistor elements. Using the particle swarm algorithm, we optimize the structural parameters to minimize the average reflectance over the X-band range. Using copper (which is assumed to be a PEC) and silicon rubber (dielectric constant = 3.2), we optimize the structural parameters for layer 1 (the PEC side) and layer 2 (the air side) in terms of the unit cell size (p ₁ = p ₂ = 14.4 mm), the side of the copper square (d ₁ = 7.3 mm d ₂ = 6.8 mm), the width of the copper loop (c ₁ = 1.3 mm c ₂ = 0.7 mm), and the thickness of the rubber substrate (t ₁ = 1.5 mm t ₂ = 2.5 mm). The frequency range is 8–12 GHz. For the resistor, the lumped elements built into the simulation program CST Studio Suite are selected. Simulations are performed with three resistor pairs: set 1 (R ₁ = 40 Ω R ₂ = 150 Ω), set 2 (R ₁ = 30 Ω R ₂ = 155 Ω), and set 3 (R ₁ = 45 Ω R ₂ = 140 Ω). The unit cell size (p ₁, p ₂) of metaatom is relatively big compare to the wavelength (30 mm) of center frequency (10 GHz). However, it is still within the range where reflection and absorption are governed by the zeroth order of diffraction and the effective refractive index description works. The results are summarized in Figure 9.

Figure 5:

Schematic diagrams of (a) the unit cell and (b) the two metamaterial layers on a PEC.

3.1 Sample fabrication

The double-layered metamaterial is prepared so that each layer contains 13 × 13 unit elements. The layers are fabricated using conductive ink, printing film from Nova Centrix, and a silicon rubber substrate. The overall size of the absorber is 187.2 mm × 187.2 mm × 4 mm. Details of the unit elements are presented in Figures 6 and 7. The resistor elements and conducting wires in the unit elements are fabricated from conductive silver and carbon ink. The thickness of the silicon rubber (refractive index = 1.8), substrate between the PEC and the first layer is 1.5 mm and that between the first and second layers is 2.5 mm. We discover that a reasonable resistance lies between 30 Ω and 155 Ω. We control the resistance by adjusting the ratio of silver to carbon ink and changing the length of the resistor units. The shape of the resistor units is presented in Figure 7(b). By changing the length of the resistor units (l _resistor) from 0.8 mm to 2.5 mm, the resistance can be adjusted from 30 Ω to 180 Ω.

Figure 6:

Design of the unit cell for the experimental samples.

Figure 7:

Design of the unit cell for the experimental samples: (a) circuit layer close to the PEC, (b) circuit layer close to the air, and (c) resistor unit design. The resistance is adjustable depending on length l _resistor.

3.2 Measurement

A schematic diagram of the setup used to measure the reflectance (R) is presented in Figure 8(a). The system consists of a pair of X-band horn antennas, a sample holder, an anechoic chamber, an HP 8920D network analyzer, and a computer for S-parameter data acquisition. The reflectance is calculated from the S-parameters after calibration. The S-parameters of the chamber background and a metal plate of the same size as the metamaterial sample are measured. The reflection signal of the metamaterial and the metal plate excluding the reflection signal of the chamber background are used for the calibration. The frequency range between 8 and 12 GHz is scanned, and the measurement results are presented in Figure 9.

Figure 8:

Experiment setting. (a) Measurement setup. (b) Metamaterial absorber sample. (c) Metamaterial absorber sample.

Figure 9:

Comparison of simulation and measurement results for reflectance with different resistor designs: (a) R ₁ 40 Ω, R ₂ 150 Ω, (b) R ₁ 30 Ω, R ₂ 155 Ω, and (c) R ₁ 35 Ω, R ₂ 140 Ω.