## Abstract

An approach for designing a wide-angle perfect absorber at infrared frequencies is proposed. The technique is based on a perfectly impedance-matched sheet (PIMS) formed by plasmonic nanostructure. It is shown that the effective impedance is more physical meaningful and beneficial than effective medium in describing the electromagnetic properties of metamaterial absorber. As a specific implementation of this technique, a wide-angle polarization-independent dual-band absorber is numerically demonstrated at frequencies of 100THz and 280THz with absorption close to 100% simultaneously. Circuit models are utilized to describe the impedance property of localized plasmon modes and the results show good agreement with that retrieved from reflection coefficient at normal incidence.

©2011 Optical Society of America

## 1. Introduction

Metamaterial (MM) absorbers have attracted much attention due to their unique ability to achieve narrowband near-unity absorption in wide frequency region ranging from microwave to terahertz and optical frequencies [1–13]. The sub-wavelength size of unit cell allows many relevant applications including focal plane array (FPA) detectors for imaging [1], thermal emitters [7], and plasmonic sensor [13].

Generally, MM is characterized by its effective permittivity and permeability in the framework of effective medium theory (EMT) [1–3,9,12,14]. However, MM absorber doesn’t exactly satisfy the homogeneous-effective limit as it is not uniform and periodic in the direction of wave propagation [15]. In addition, surface plasmons (SPs) can also bring on near-perfect absorption at optical frequency when the period of the surface structure is large enough compared with the wavelength [4,5].

In another theoretical regime, transmission line theory (TLT) is used to study the properties of metamaterials from microwave frequency to optical frequency [16–18]. In this regime, the relation between structure parameters and electromagnetic characters become more obvious. Recently, a transmission line model has been proposed to analyze the three-layer MM absorber in the terahertz band [19]. Although the results are in agreement with simulated ones, the magnetic coupling between two metallic layers is not taken into account and the design methodology is still not clear.

In this paper, the proposed absorber consists of a metallic surface structure (metasurface) and a thick metal layer separated by a thin dielectric spacer. The ideal impedance for the metasurface is theoretically deduced to absorb all electromagnetic radiation. Infrared dual band absorption is numerically demonstrated and the conditions of perfect absorption are presented by utilizing equivalent circuit models. The angle-independent absorption is found to be related to the localized plasmon modes, while the SPs induced absorption is eliminated by reducing the period of the surface structure.

## 2. Perfectly impedance-matched sheet

As shown in Fig. 1
, a metasurface is placed in front of a thick metal layer (denoted as ground plane) with a distance *d*. The metasurface can be thought as an impedance sheet characterized by its impedance. By matching the impedance of ground plane to that of vacuum, perfect absorption can be realized and the corresponding sheet is called as perfectly impedance-matched sheet (PIMS).

The impedance of PIMS can be obtained by setting reflection coefficient *S*
_{11} to be zero. Assuming the dielectric spacer is non-magnetic (*μ* = 1), the boundary conditions for the sheet at normal incidence can be obtained (*L* denotes the left side of the impedance sheet and *R* denotes the right side):

where *Y*
_{0}, ${Y}_{1}=\sqrt{{\epsilon}_{1}}{Y}_{0}$ and ${Y}_{m}=\sqrt{{\epsilon}_{m}}{Y}_{0}$ are the intrinsic admittance of vacuum, dielectric spacer and metal respectively, *ε*
_{1} and *ε _{m}* are permittivities of dielectric and metal.

*k*

_{0}and $k=\sqrt{{\epsilon}_{1}}{k}_{0}$ are the wave vector in the vacuum and dielectric spacer. ${r}_{m}=({Y}_{1}-{Y}_{m})/({Y}_{1}+{Y}_{m})$ is the reflectivity of thick metal layer.

*J*is the current flowing in the sheet while

*a*is the amplitude of incident

*E*field. By eliminating

*a*in Eq. (1), the admittance of PIMS can be obtained:

The ideal impedance (*Z _{s}* =

*R*-

*iX*, where

*R*and

*X*are resistance and reactance) for PIMS at normal incidence is independent of polarization. As depicted in Fig. 2 , the impedance should be pure resistive (

*R*= 377Ω,

*X*= 0) corresponding to the traditional Salisbury screen when

*d*is near

*λ*/4

*n*($n=\sqrt{{\epsilon}_{1}}$is the refractive index of the dielectric spacer). In Zone1, the required resistance should be very small and the corresponding reactance should be capacitive (R>0,

*X*<0). In contrast, the required reactance is inductive (R>0,

*X*>0) for Zone2.

## 3. PIMS design

As the thickness of the absorber is much smaller than the working wavelength in Zone1 as shown in Fig. 2, Zone1 is typically the case for most practical applications. To mimic the impedance of PIMS, lossy metal should be deposited in appropriate geometrical patterns such as square patches, crosses as proposed in the recent designs [11,12,14].

Here, we choose circular patches arrayed in hexagonal lattice due to its high azimuthal symmetry. As illustrated in Fig. 3
, the radius of cylindrical patch is *r* and the center distance between adjacent patches is *a*. The thicknesses of patch and dielectric spacer are *t* and *d*, respectively. The thickness of thick metal layer *h* is far larger (100nm in our case) than the penetration depth to suppress transmission. All metallic parts are chosen as gold and experimental value of permittivity is used [20]. Al_{2}O_{3} is used as dielectric spacer and the dielectric constant is 2.92 at wavelengths around 3μm (Al_{2}O_{3} is lossless in this frequency regime [20]).

We simulated the reflection coefficient *S*
_{11} using frequency domain Finite Element method (FEM). The unit cell is shown in Fig. 3 (a) with dashed line and periodic boundary condition is used to mimic infinite cell array. The absorption is calculated as *A* = 1-*S*
_{11}
^{2} since there is no transmittance. For TE (TM) polarization, the *E* (*H*) field direction is along y-axis.

As an example, the geometrical parameters are adjusted to optimize the absorption at frequency of 100THz (*λ* = 3μm). Following the same strategy as Eq. (1) and Eq. (2), the effective admittance of the surface structure can be retrieved from the simulated reflection coefficient *S*
_{11}:

In fact, the ideal admittance given in Eq. (2) is obtained on a more specific condition (*S*
_{11} = 0). When *r* = 277nm, *a* = 693nm, *t* = 32nm, *d* = 30nm, the retrieved impedance for TM polarization overlap with the ideal one at *f* = 100THz and near-perfect absorption is realized as shown in Fig. 4 (a)
. Interestingly, another absorption peak occurs at the right side of the resonant frequency *f _{0}* = 245THz. The simulated absorption coefficients are 99.96% and 98.89% for 100THz and 280THz while the bandwidths are very narrow for both peaks.

It is worth noting that the two absorption peaks both occur at the intersection points of the retrieved and ideal impedances. In fact, the retrieved impedance is similar with that of a typical parallel RLC circuit with resonance frequency at *f* = 245THz. Although the resonant frequency of the retrieved impedance does not correspond to any absorption peak, the resonant characteristic is the key of impedance matching for the two peaks. To obtain an accurate circuit model, the field distributions in the structure should be explored at different frequencies. As shown in Figs. 5 (a, c)
, two different resonances in the voids between the patches and ground plane are found at 100THz and 280THz.

Using the classical equivalent circuit theory [16,17] to analyze the electromagnetic properties of the two resonant modes, the effective impedance in the whole simulated frequency region can be written as

where *f*
_{c}≈200THz is the frequency where mode begins to change. The corresponding equivalent circuit models are shown in the insets of Figs. 5 (a) and (c). At frequency below *f*
_{c}, only the lowest resonance mode is responsible for the circuit model. However, the applicability of the single resonance model breaks down and higher order modes must be taken into account at higher frequencies.

Although the circuit elements in Eq. (4) are frequency dependent in general [18], they can be assumed to be constant in a narrow frequency band. In general, the resistors (*R*, *R*
_{1}, *R*
_{2}) come from the intrinsic loss in metal (the dielectric spacer is lossless here) and become larger for smaller patch thickness *t*. The capacitors (*C*, *C*
_{0}, *C*
_{1}, *C*
_{2}, *C*
_{3}) result from the electric field distribution in dielectric between two metallic elements and tend to be larger for smaller distance between them. The inductors (*L*, *L*
_{1}, *L*
_{2}) are related with the current distribution in the metallic patches. These relations between the circuit elements and geometrical parameters are used in the optimization of absorptance.

To obtain the circuit parameters, one must fit the impedance calculated using Eq. (4) to the retrieved one. Because there are so many parameters to be determined, the relations between these parameters and impedances must be taken advantage to solve the problem. As an example, the resonance frequency and bandwidth are determined by parallel connected *R* (*R*
_{1}, *R*
_{2}), *L* (*L*
_{1}, *L*
_{2}) and *C* (*C*
_{1}, *C*
_{2}) while the overall capacitance can be tuned by the series connected capacitor (*C*
_{0}, *C*
_{3}). The fitted parameters for the dual-band band absorber are: *R* = 1.7Ω, *L* = 20fH, *C* = 20aF, *C*
_{0} = 27aF, *R*
_{1} = 4.1Ω, *L*
_{1} = 75fH, *C*
_{1} = 5.7aF, *R*
_{2} = 4Ω, *L*
_{2} = 80fH, *C*
_{2} = 2.3aF, *C*
_{3} = 12aF. The corresponding impedances and absorption are depicted in Fig. 4 (c), in good agreement with that in Fig. 4 (a). Thus, the impact of cavity modes on effective impedance is successfully modeled through parallel connection of RLC circuits.

## 4. Localized and delocalized plasmons

The angle dependences of the absorption of the dual-band absorber are shown in Fig. 6 . The absorption for TM (TE) polarization at 100THz is larger than 90% at incidence angle of 70° (55°), while the absorption is still larger than 80% at incidence angle of 30° (45°) at 280THz. This kind of wide-angle absorption has been found at other nano-structured metal surfaces and is attributed to the localized void plasmons [4,21]. The field distribution in Fig. 5 demonstrates that most energy is localized in the cavity formed by the patches and ground plane. As shown at the top right corner of Fig. 6 (b), there is an angle dependent absorption curve, which is related to SPs. The SPs cannot be excited when the period of the structure is in sub-wavelength scale. However, this effect will become much more obvious when the period is comparable with the working wavelength, as discussed later. At large incidence angles, other absorption peaks also occur at frequencies around 180THz for TM polarization due to excitation of horizontal resonance modes.

To further exploit the influence of localized modes and delocalized SPs on the absorption properties, another absorber at 100THz with increased geometrical parameters (Sample2) is designed. Compared with the absorption of the first design (Fig. 4), two additional absorption peaks around 270THz are observed (Fig. 7 ). The circuit parameters are not calculated here because the retrieved impedances are sufficient for the optimization. As can be seen from the retrieved impedances, there are three resonances at frequencies around 155THz, 255THz and 260THz, corresponding to the drastic variations of the curves.

The angle dependences of the absorption for Sample2 are shown in Fig. 8 . The absorption for TM (TE) polarization at 80° (55°) is still 90% at 100THz, which is less angle dependent compared with the first absorber. Absorptions at frequencies above 150THz are characterized by very bad angular stability. Obviously, the absorption peak at 100THz is mainly related with the localized plasmon resonance, while absorption peaks at higher frequencies are related with the interplay of localized and delocalized plasmon resonances. As wide-angle perfect absorber is needed in most applications, the influence of SPs should be eliminated in the design.

In general, the coupling of propagating wave to delocalized SPs in the nanostructured periodic surfaces is realized by a reciprocal vector added to the wave vector of the incident wave and this mechanism can be explained using Bragg scattering theory [21]:

where ${n}_{eff}\approx 1/\sqrt{{\epsilon}_{m}^{-1}+{\epsilon}_{d}^{-1}}$ is an approximation of the effective refractive index of SPs mode and $\left|{\overrightarrow{q}}_{\overline{1}\overline{1}(11)}\right|=4\pi /a$ is the lowest reciprocal vector along z-axis. The theoretical result is shown by red dashed lines in Fig. 8. According to this theory, reciprocal vector will decreases and the coupling frequency will shifts to lower frequency as the patch separation *a* increases. For the first absorber, the patch separation *a* is 693nm and there is only a SPs tail at the top right corner in Fig. 6(b). For sample2 (*a* = 1385nm), however, the SPs coupling is obvious and the angle-stability of the high frequency absorption is dramatically decreased. In the design process, the patch separation *a* should be chosen to be less than 2*λ*/(*n _{eff}* + 1) in order to remove the impact of SPs. This sub-wavelength scale is just one of the conditions for the homogenization of MMs.

## 5. Conclusion

In conclusion, we have proposed a methodology based on plasmonic circuit theory to design metamaterials absorber at infrared frequency. Based on this methodology, a dual-band polarization-insensitive absorber at 100THz and 280THz with absorption of 99.96% and 98.89% is numerically demonstrated. Circuit models are constructed to describe the frequency-dependent impedance property. The influence of localized and delocalized plasmonic modes on the angle dependence of absorption is discussed. It is shown that wide-angle absorption can be realized by utilizing the properties of localized plasmonic modes and eliminating the impact of surface plasmons. Although the design procedure is devoted to perfect absorber at infrared frequencies, our conclusions can be extended to other frequency bands.

## Acknowledgments

This work was supported by 973 Program of China (No. 2011CB301800).

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