**2. Theoretical scheme of metamaterial absorbers**

We start by reviewing general theories that explain the origin and underlying physics for achieving perfect absorption in metamaterials. The first theory is by designing both electric and magnetic resonances in a metamaterial, so that the effective permittivity and permeability of the metamaterial can be tailored for achieving impedance matching with free space [45, 46]. In such a case, no reflection occurs at the interface and the entire incident energy has a chance to be absorbed inside the metamaterial absorber. The other theory is based on the destructive interference of multipleorder reflections due to the multiple inner reflections inside the dielectric substrate.

#### **2.1. Impedance matching theory**

A metamaterial absorber is typically a sandwiched structure, consisting of an array of certain metallic patterns on one side of a substrate and backed with a highly conductive metallic ground plane. The electric permittivity and magnetic permeability of the metamaterial are ε = *ε*<sup>0</sup> *ε<sup>r</sup>* (*ω*) and *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *μr* (*ω*), respectively. Here, *ε*<sup>0</sup> and *μ*<sup>0</sup> are the free space permittivity and permeability. *ε<sup>r</sup>* (*ω*) and *μr* (*ω*) are the frequency-dependent relative permittivity and permeability of the medium, which are unitless and normalized with respect to the values of free space. Due to the presence of the ground plane, no transmittance could be found on the other side of the metamaterial. This allows us to focus only on the reflection from the metamaterial.

According to the Fresnel formula of reflection, the reflectivity (*R*) from the metamaterial is [29]

$$R\_{\rm rE} = \|r\_{\rm rE}\|^2 = \left|\frac{\mu\_r \cos \theta - \sqrt{n^2 - \sin \theta}}{\mu\_r \cos \theta + \sqrt{n^2 - \sin \theta}}\right|^2 \tag{1}$$

$$R\_{\rm TM} = \|r\_{\rm TM}\|^2 = \left|\frac{\varepsilon\_r \cos \theta - \sqrt{n^2 - \sin \theta}}{\varepsilon\_r \cos \theta + \sqrt{n^2 - \sin \theta}}\right|^2,\tag{2}$$

where the subscripts TE and TM refer to transverse electric (TE) and transverse magnetic (TM) polarized waves, *θ* is the angle of incidence, and *<sup>n</sup>* <sup>=</sup> <sup>√</sup> \_\_\_\_ *εr μr* is the effective refractive index of the metamaterial. For the case of normal incident, we have *θ* = 0° , so that these equations reduce to:

$$\mathbf{R} = \left| \frac{\mathbf{Z} - \mathbf{Z}\_0}{\mathbf{Z} + \mathbf{Z}\_0} \right|^2 = \left| \frac{\sqrt{\mu\_r} - \sqrt{\varepsilon\_r}}{\sqrt{\mu\_r} + \sqrt{\varepsilon\_r}} \right|^2 \tag{3}$$

with *<sup>Z</sup>* <sup>=</sup> <sup>√</sup> \_\_\_\_ *<sup>μ</sup>*/*ε* being the impedance of the metamaterial and *Z*<sup>0</sup> <sup>=</sup> <sup>√</sup> \_\_\_\_\_ *μ*0 /*ε*0 being the impedance of free space. Since the metallic ground leads to zero transmissivity, the absorptivity arrives:

$$A = \mathbf{1} - \mathbf{R} = \mathbf{1} - \left| \frac{Z - Z\_0}{Z + Z\_0} \right|^2 = \mathbf{1} - \left| \frac{\sqrt{\mu\_r} - \sqrt{\varepsilon\_r}}{\sqrt{\mu\_r} + \sqrt{\varepsilon\_r}} \right|^2. \tag{4}$$

The above equation indicates that impedance matching, *<sup>Z</sup>* <sup>=</sup> *<sup>Z</sup>*<sup>0</sup> or *μr* <sup>=</sup> *<sup>ε</sup><sup>r</sup>* , is a critical condition for achieving perfect absorption. It is worth noting that, to achieve impedance matching in a metamaterial absorber, simultaneous electric and magnetic resonances are required. For a metamaterial with single resonance, either electric or magnetic resonance, its impedance will be strong mismatched with that of free space. As a consequence, no perfect absorber would be found.

#### **2.2. Interference theory**

compact sizes and enhanced directionalities [17–19]. It is well known that one of the main obstacles toward practical engineering applications is the inevitable intrinsic loss in metamaterials. A significant amount of effort has been devoted to achieving low-loss devices through optimizing structural geometries [20–22]. Loss compensation using gain elements [23–25] is

On the other hand, absorption is also highly desired in many applications, such as energy harvesting [26], scattering reduction [27], as well as thermal sensing [28]. By utilizing the full usefulness of loss, metamaterials with nearly uniform absorption are achievable through properly engineering the electric and magnetic resonances [29–35]. Due to the resonance nature, the first reported metamaterial-based perfect absorber is of narrow bandwidth and polarization sensitive, which restrict its usefulness in practical applications [29]. Great efforts have been devoted to expanding the bandwidth of metamaterial absorbers. Metamaterials with multiband absorption have later been developed using multiple resonant unit cells and combining them through a co-plane arrangement [36, 37]. When these resonances are closed to each other in frequency, broadband absorption is achievable [38]. Broadband absorption can also be achieved in metamaterial absorbers with multi-layer structures [39, 40] or using vertically standing nanowires [41, 42]. Moreover, by incorporating active mediums, the absorptivities and frequencies of metamaterial absorbers could be adjusted via external biases [43, 44].

In this chapter, we present a brief review on the fundamental theories and recent evolutions in the research field of electromagnetic metamaterial absorbers, whose operating frequencies cover from microwave, THz, infrared, to visible regimes. The rest of this chapter is organized as follows. Section 2 introduces the general theories on the design of metamaterial absorbers, where impedance matching theory and multiple interference theory are introduced. Section 3 reviews the narrowband metamaterial absorbers of various structures, all of which have nearly uniform absorptivities. Next, the technologies for broadening the bandwidths of metamaterial absorbers are presented in Section 4. Metamaterial absorbers with tunable absorption properties are reviewed in Section 5. Moreover, the coherent control of the metamaterial absorber's absorptivity through phase modulation is discussed in Section 6. Finally, the conclusion is given.

We start by reviewing general theories that explain the origin and underlying physics for achieving perfect absorption in metamaterials. The first theory is by designing both electric and magnetic resonances in a metamaterial, so that the effective permittivity and permeability of the metamaterial can be tailored for achieving impedance matching with free space [45, 46]. In such a case, no reflection occurs at the interface and the entire incident energy has a chance to be absorbed inside the metamaterial absorber. The other theory is based on the destructive interference of multiple-

A metamaterial absorber is typically a sandwiched structure, consisting of an array of certain metallic patterns on one side of a substrate and backed with a highly conductive metallic

order reflections due to the multiple inner reflections inside the dielectric substrate.

another scheme which requires external excitation sources.

134 Metamaterials and Metasurfaces

**2. Theoretical scheme of metamaterial absorbers**

**2.1. Impedance matching theory**

A metamaterial absorber can be regarded as a coupled system and, particularly, its magnetic resonance is induced due to the anti-parallel currents between the front and back metallic layers. However, we may also independently consider the functionalities of the front meta-layer and the ground plane on the other side [47]. The front layer with certain metallic patterns functions as a partial reflection surface, which can be utilized to modify the complex reflection and

oblique with an angle *θ*, the propagation length inside the dielectric substrate becomes longer.

The first metamaterial absorber was theoretically investigated in 2006, consisting of an array of split ring resonators (SRRs)-backed with a resistive sheet [49]. The incident wave is parallel to the SRR plane with magnetic field being perpendicular to the SRR array. Such an SRR array is placed on a resistive sheet having a resistance of 377 Ω for impedance matching with free space, similar to a Salisbury screen. Both the reflection and transmission are below −20 dB at the vicinity of 2 GHz as numerically found. This is due to the strong resonances in this structure, where nearly perfect absorption was achieved at this frequency. However, due to the standing arrangement of the SRR array, this structure is not of low profile as compared with planar structures, which also increase its complexity in the manufacture. The bandwidth of absorption is also very limited. Nevertheless, the design of this metamaterial absorption

In 2008, Landy et al. [29] proposed a planar sandwiched structure that consists of electric ring resonators and cut wires separated by an FR-4 substrate, as shown in **Figure 2**. This is the first-reported metamaterial absorber with a planar and deeply subwavelength structure. The absorptivity observed is as high as 96% at 11.65 GHz in simulation and 88% at 11.5 GHz in experiment. The relative bandwidth of full width half maximum (FWHM) is around 4%. The front electric ring resonators couple strongly to the incident electric field and contribute electric response, while the circulating flow of antiparallel surface currents at the front and back metallic layers contributed a magnetic response. The absorption intensity and frequency could be controlled by adjusting the geometric parameters of electric ring resonators or the

**Figure 2.** (a) Unit cell of the first planar metamaterial absorber, (b) simulated reflection, transmission, and absorptance

<sup>~</sup> <sup>=</sup> <sup>√</sup> \_\_ *ε<sup>d</sup> k* 0 *d*′

Electromagnetic Metamaterial Absorbers: From Narrowband to Broadband

, where *d*′ = *d*/cos *θ*′

http://dx.doi.org/10.5772/intechopen.78581

can be obtained

is

137

Therefore, the propagation phase delay should be modified as *β*

*<sup>ε</sup><sup>d</sup>* sin *<sup>θ</sup>*′ <sup>=</sup> sin*θ* [48].

\_\_

**3. Narrowband metamaterial absorbers**

motivates further research in these kinds of absorbers.

at microwave frequency. Reproduced from [29] with permission.

following Snell's law √

the modified propagation length inside the substrate and the refractive angle *θ*′

**Figure 1.** Multiple reflections and interference model of metamaterial absorber. The yellow dashed line refers the resonator array. Reproduced from [47] with permission.

transmission coefficients. On the other hand, the highly conductive ground plane works as a perfect reflector, offering a phase delay of 180° to the electromagnetic wave reflecting on it.

As shown in **Figure 1**, the front meta-layer resides at the interface between air and the dielectric substrate. The incident electromagnetic wave is partially reflected back to air with a reflection coefficient *r* ~ <sup>12</sup>(*ω*) = *r*12(*ω*)*ei<sup>ϕ</sup>*12(*ω*) and partially transmitted into the substrate with a transmission coefficient *t* ~ <sup>12</sup>(*ω*) = *t* 12(*ω*)*ei<sup>θ</sup>*12(*ω*) . The transmitted wave will propagate further until reaching the metallic ground plane. The complex propagation constant inside the dielectric substrate is *β* <sup>~</sup> <sup>=</sup> *<sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>i</sup> <sup>β</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup> \_\_ *<sup>ε</sup><sup>d</sup> <sup>k</sup>*<sup>0</sup> *<sup>d</sup>*, where *k*<sup>0</sup> is the wavenumber of free space, *d* is the thickness of the substrate, *β*<sup>1</sup> represents the propagation phase, and *β*<sup>2</sup> refers to the absorption in the dielectric substrate. At the ground plane, a total reflection occurs with a reflection coefficient of −1. After direct mirror reflection and an additional propagation phase delay *β* <sup>~</sup>, partial reflection and transmission occur again at the front interface. The corresponding reflection and transmission coefficients are *r* ~ <sup>21</sup>(*ω*) = *r*21(*ω*)*ei<sup>ϕ</sup>*21(*ω*) and *t* ~ <sup>21</sup>(*ω*) = *t* 21(*ω*)*ei<sup>θ</sup>*21(*ω*) , respectively. It is worth noting that multiple reflections and transmissions exist inside the dielectric substrate, and the totally output energy at the left side of the metamaterial is the superposition of reflections of all orders:

$$\widetilde{r}(\omega) = \widetilde{r}\_{12}(\omega) \cdot \frac{\widetilde{t}\_{12}(\omega) \cdot \widetilde{t}\_{21}(\omega) \ e^{2i\widetilde{\beta}}}{1 + \widetilde{r}\_{21}(\omega) \cdot e^{2i\widetilde{\beta}}} \tag{5}$$

where the first term in the right is the reflection directly from the meta-layer, and the second term is the contribution of the superposition of the multiple higher-order reflections. As long as we know the total reflection *r* ~, the absorption spectrum of the metamaterial absorber could be obtained by *A*(*ω*) = 1 −|*r* ~(*ω*)| 2 . The interference theory can well explain the features observed in those metamaterial absorbers with metallic grounds and also provide an alternative understanding of the origin and underlying physics of metamaterial absorbers.

It is worth noting that the above analysis is fully based on the assumption that the incident wave is normal to the metamaterial. For the case when an electromagnetic wave incident is oblique with an angle *θ*, the propagation length inside the dielectric substrate becomes longer. Therefore, the propagation phase delay should be modified as *β* <sup>~</sup> <sup>=</sup> <sup>√</sup> \_\_ *ε<sup>d</sup> k* 0 *d*′ , where *d*′ = *d*/cos *θ*′ is the modified propagation length inside the substrate and the refractive angle *θ*′ can be obtained following Snell's law √ \_\_ *<sup>ε</sup><sup>d</sup>* sin *<sup>θ</sup>*′ <sup>=</sup> sin*θ* [48].
