## Abstract

The resonant absorption in a planar metamaterial is studied theoretically. We present a simple physical model describing this phenomenon in terms of equivalent resonant circuit. We discuss the role of radiative and dissipative damping of resonant mode supported by a metamaterial in the formation of absorption spectra. We show that the results of rigorous calculations of Maxwell equations can be fully retrieved with simple model describing the system in terms of equivalent resonant circuit. This simple model allows us to explain the total absorption effect observed in the system on a common physical ground by referring it to the impedance matching condition at the resonance.

© 2013 Optical Society of America

## 1. Introduction

A large variety of metallic and semiconductor metamaterials have been recently found to exhibit total absorption (TA) of electromagnetic radiation. This effect has been reported in broad frequency range including microwave [1–4], THz [5–8], NIR [9–12], MIR [13–16] and visible [17–19] regions. A lot of diverse applications have been already proposed. Among them radar [20–23], electromagnetic compatibility [24], photovoltaics [25], biosensing [11], thermal source emission [13, 26], and thermal bolometer [15].

Although the structures that exhibit total absorption might differ considerably, they possess some common characteristics: the energy transmission through entire structure is forbidden, the structure possess an intrinsic resonance, and total absorption is achieved at a specific condition imposed on the geometry of the structure or on the material properties.

One of the metamaterial structures widely discussed in literature in respect to the highly efficient absorber is a planar metamaterial with lattice of metallic patches separated from the ground plane by absorptive layer. Such structure with subwavelength thickness can exhibit the effect of total absorption by trapping the electromagnetic radiation under the patch. Several geometries of the patch have been studied. Among them square [10, 15, 27, 28] and circular patches [11,16,29], cut wires [8,30], closed-ring [4], split-ring [1,6,12], and cross-shaped resonators [14]. Even the leaf-shaped [31] and flower-shaped patch configurations [32, 33] have been considered. In each of this studies an exhaustive information about the resonant total absorption effect and even the formation of the absorption bands are presented although limited to a particular geometry. Nevertheless, they fail to offer the basic physical principles for parameter choice of a perfect metamaterial absorber. To date, in spite of extensive engineering search for the patch geometry the lack of general strategy is still observed.

In this paper along with rigorous electromagnetic calculations we present a simple physical model describing the properties of planar metallic metamaterial in terms of equivalent oscillating-current resonant circuit. In this analytical approach we describe the absorptive layer squeezed between the periodic arrangement of metallic patches and ground plane by an effective load impedance. This allows us to go beyond the geometry and to find the universal ruler for TA effect to be achieved.

The paper is organized as follows. We start from the simplest geometry of a square patch in order to tackle the problem and understand the physics behind. In Sec. II we show the results of the rigorous electromagnetic calculations performed with use of the commercial software COMSOL Multiphysics [34] based on the finite element method. In Sec. III we present a simple physical model that describes the resonant absorption in terms of equivalent resonant circuit. In Sec. IV we develop the condition for the perfect absorber and discuss the role of radiative and dissipative damping of resonant mode supported by a metamaterial in the formation of absorption spectra. Finally, we tested our model with the planar metamaterials with arbitrary patch geometry.

## 2. Resonant absorption by a planar metamaterial

In Fig. 1 we present the structure under consideration (metasurface): it is a periodic arrangement of square metallic patches of length *l* located above a ground plane at a distance *s* ≪ *λ* (*λ* is the wavelength). Between the patch and the ground plane we include an absorptive layer with a weak loss tangent (tan*δ* = *ε″/ε′* ≪ 1, where *ε′* and *ε″* are the real and imaginary part of dielectric function). We assume at the beginning that the ground plane as well as a patch are perfect metals.

In general, a structure presented in Fig. 1 poorly absorbs the electromagnetic energy. However, if one could match the free-space impedance (purely resistive), *Z*_{0} = 120*π* Ω, to the impedance of the metasurface, the incident energy would be completely transformed into Ohmic losses inside the absorptive layer and the total absorption effect would be achieved. If we consider that the electric current induced in absorptive layer by an incident plane wave is oscillating along the patch side, the electronic resistance *R* of the absorptive layer is inversely proportional to the layer thickness *s* and, therefore, extremely large compared to the free-space impedance *Z*_{0}. The problem of the impedance matching may be solved by using an intrinsic resonance. At the resonance the real part of the effective impedance (i.e. the effective electric resistance) can be drastically decreased, while the inductive reactance inherent to the resonance may be canceled by an capacitive one, thus, leaving the effective impedance purely resistive. The absorptive layer squeezed between the patches and ground plane supports a transverse magnetic (TM) mode with a dispersion that can be easily obtained by imposing the zero electric field conditions above and below the absorptive layer:

*ω*is a frequency,

*ε*=

*ε′*(1 +

*i*tan

*δ*) is the complex dielectric function of absorptive layer, and

*c*is the speed of light. The resonance frequencies of the structure in Fig. 1 can be obtained by quantization of the TM mode (Eq. (1)) by the cavity formed between the patch and ground plane. The metallic patches play the role of a Fabry-Perot resonator for the cavity TM standing wave squeezed between the patch and the ground plane, propagating along the absorptive layer and reflected at the terminations of the patch. Assuming that the modes are strongly localized under the patch and the field does not undergo the fringing at the patch edges, we can obtain the mode dispersion by considering rectangular cavity (

*l*×

*l*×

*s*) with zero tangential electric field at the metallic walls and zero magnetic field at the other four walls of the cavity. Since

*s*≪

*l*and analysis presented in this paper concerns only the lower frequency modes for 2D case we have where ${f}_{mn}={\omega}_{mn}/2\pi =c\sqrt{{n}^{2}+{m}^{2}}/2l\sqrt{\epsilon}$ are actually the resonant frequencies of classical square patch antenna [35],

*n*,

*m*= 0, 1, 2,... are integer and

*n*

^{2}+

*m*

^{2}≠ 0.

In Fig. 2(a) we present the absorption spectra of metasurface for different patch length *l* (12.2, 13.2 and 15.2 mm). The period of the structure is *l* + Δ*l* with patch-to-patch separation Δ*l* = 2 mm and absorptive layer thickness *s* = 0.3 mm. For the absorptive layer we chose *ε′* = 4.4 and tan*δ* = 0.02 (*ε″* = 0.088) that are characteristic of FR-4 glass epoxy widely used as insulator for printed circuit boards. We have done it by purpose in order to demonstrate that a subwavelength layer of good isolator can be converted into the perfect absorber once it is squeezed between the ground plane and judiciously structured metal surface. The calculations have been performed for normal incidence with use of the commercial software COMSOL Multiphysics [34] based on the finite element method. One can observe a series of resonances associated with the excitations of the first (*n* = 1, *m* = 0) and the next (*n* = 3, *m* = 0) patch antenna’s modes. One can notice nearly total absorption of electromagnetic radiation at the resonance of the first TM mode. Note, that the mode (*n* = 2, *m* = 0) is the dark mode and can not be exited since it does not correspond to the symmetry of the incident field.

The near field distributions *E _{z}* in absorptive layer (see Fig. 2(b)) calculated along the line

*z*= 0 and

*y*= 0 at frequencies A and B of the spectra shows the field oscillations that are typical for classical patch antenna [35]. Note that the field variations along the height of the absorptive layer are nearly constant since for a given resonance wavelength the phase varies only slightly with |

*z*| <

*s*/2 ≪

*λ*. We have checked the

*z*-dependence numerically (results not shown here). One can also notice the effect of field fringing at the edges of the patch. It is rather weak due to the small layer thickness. Therefore, the modes are strongly localized under the patch and the coupling between patches are extremely weak. That is why the resonant frequencies in spectra are approximately defined by Eq. (2) and independent from patch to patch distance within the studied range of Δ

*l*as one can perceive from Fig. 2(c).

## 3. Equivalent resonant circuit model

In order to provide a general view on the effect of total absorption regardless the particular geometry of the patch we apply RLC circuit model. In RLC model a structured surface is considered as a load impedance at the end of the transmission line modeling the free space. Similar model has been successfully applied to explain the total light absorption effect exhibited by the nanostructured metal inherent the plasmon resonance in visible [36].

Let us first introduce the effective surface impedance *Z*_{eff} of the metasurface. In this case the reflection from metasurface can be described on a common basis independently of type of structuring. We describe the resonance of TM mode excited by a normally incident plane wave and squeezed between the patch and ground plane in terms of its equivalent resonant RLC circuit. In Fig. 3(b) we present the equivalent RLC circuit of the TM mode for the metasurface of Fig. 3(a). This circuit describes the main physical features of the resonant structures and contains (i) the total resistance *R* that determines the amount of power absorbed due to Ohmic losses, (ii) the total inductance *L* created by the finite electric currents oscillating in metallic patch and ground plane, and (iii) the total capacitance *C* that describes the charge accumulation induced by the external field. Note that inline with discussion above and results of Fig. 2(b) and 2(c) we did not take into account the patch-to-patch mode coupling. Elsewhere, an addition capacitance responsible for these coupling should be connected in series to the circuit presented in Fig. 3 (see, for example, Ref. [29]).

The equivalent impedance of the circuit can be written as

Equivalently, Eq. (3) can be cast in the form where the dissipative mode damping*ν*=

*R*/2

*L*and ${\omega}_{0}=1/\sqrt{LC}$. The resonant frequency can be derived by Im(Z) = 0 as ${\omega}_{\mathit{res}}^{2}={\omega}^{2}-{(R/L)}^{2}=\omega -4{\nu}^{2}\simeq {\omega}_{0}^{2}$. In the vicinity of the resonance

*ω*≃

*ω*

_{0}assuming that 2

*ν*≪

*ω*

_{0}(i.e. high quality resonance) we obtain The total frequency-dependent equivalent impedance of the metasurface can be expressed as where we introduce the coefficient |

*β*|

^{2}< 1 responsible for the coupling of the incident field with the resonant mode. The coupling coefficient |

*β*|

^{2}depends both: on the mode number and geometry of the structure (namely patch-to-patch distance).

Now let us calculate the absorption by a thin layer characterized by an effective impedance *Z*_{eff}. Since there is no transmission through the structure we apply the impedance boundary condition [37] and obtain a complex-valued reflection coefficient for the normal incidence in the form

*C*on the geometry of the structure and dielectric function of absorptive layer. Note, that the absorption resonance described by Eq. (9) has Lorentzian lineshape with full width at half maximum

*FWHM*= 2Γ = 2(

*ν*+

*γ*).

## 4. General condition for the perfect absorber

Having introduced the radiative and dissipative damping of the resonant TM modes, we now discuss their contribution to the shape of absorption spectra. From Eqs. (8) and (9) one finds at resonance (*ω* = *ω*_{0})

*𝒜*

_{res}= 1) occurs when or Obviously, for

*𝒜*

_{res}= 1 reflectance drops to zero (

*ℛ*

_{res}= 0).

Note that if we ignores ohmic losses in the absorbing layer (*R* = 0, hence *ν* = 0), the resonance with entire linewidth given by 2*γ* disappears since in this case reflectance reaches unity and absorbance drops to zero. Therefore, the resonance mode sustains the radiative and dissipative damping simultaneously. In order to distinguish the radiative and dissipative contribution to the resonance linewidth we calculate numerically the absorbtion spectra at the first resonant mode for different ohmic losses in absorptive layer (see color curves in Fig. 4(a)). For this purpose we multiplied the originally used imaginary part of dielectric function *ε″* = 0.088 (green curve in Fig. 2) by factor 0.125 and 2. This results in the narrowing and broadening of the resonant curve, respectively. The absorption is reduced in both cases indicating that *ε″* = 0.088 is the optimal value. In the inset we plot Γ = *ν* + *γ* extracted from the spectra of Fig. 4(a) as a function of *ε″*. As we expect we observe the linear dependence of the resonant width Γ on *ε″*. Thus, by extrapolating the curve in the inset of Fig. 4(a) to *ε″* = 0 (i.e. *ν* = 0) we can extract the radiative damping of the mode *γ* = 2.83 × 10^{8}s^{−1}. Now with this result we find that the condition for total absorption *ν* = *γ* is satisfied at *ε″* = 0.088. In this case the half width of the resonance is Γ = 2*γ* = 5.66×10^{8}s^{−1}. In Fig. 4(a) the absorption spectra calculated numerically for *γ* = *ν* at *ε″* = 0.088 (blue line) exhibits the total absorption of electromagnetic radiation at resonant frequency.

Above we have considered in detail the properties of the first resonant mode. The resonant circuit model can be equally applied to achieve the TA effect when higher order TM mode is excited. For this purpose, we repeated the procedure discussed above. In Fig. 4(b) we plot the absorption spectra calculated for different ohmic losses in absorptive layer for the third TM mode (*n* = 3, *m* = 0). We observe the narrowing and broadening of the absorption resonance, while none of them exhibits the TA effect. In the inset of Fig. 4(b) we plot the width of the resonance extracted from resonant absorption curves as a function of *ε″*. The extrapolation of the curve gives us the radiative damping *γ* = 3.14 × 10^{8}s^{−1}. Now the TA condition *ν* = *γ* is satisfied at *ε″* = 0.0347. In Fig. 4(b) we present the absorption spectra for optimal *ε″* by black line.

The radiative damping *γ* also depends on the particular geometry of the structure. To illustrate this effect we study in Fig. 5 the dependence of the radiative damping on the geometrical parameters of our metasurface. We extracted the value of *γ* from the absorption spectra calculations performed for different patch length *l* and absorptive layer thickness *s*. In Fig. 5 additionally to radiative damping *γ* we plot the half width of the resonance Γ for the sake of comparison. Evidently, that the dissipative damping *ν* can be obtained from the difference between the total decay rate Γ and radiative decay rate *γ*. One can observe that the radiative damping *γ* proportional to the absorptive layer thickness *s* and deviates only slightly from the inverse proportionality to *l*^{2} for large *l*. This dependance results from the inverse dependence of *γ* from the effective total capacitance *C* (see Eq. 10). Analyzing the near field distribution of the resonant mode under the patch in Fig. 2(b) one can perceive that it is characteristic of the series of parallel-plate capacitors with total capacitance *C* ∼ *ε*_{0}*εl*^{2}/*s*. The slight deviation from dependence 1/*l*^{2} for large *l* is explained by the fact that even for *l* ≫ Δ*l* a small opening between patches ensures the residual coupling *ξ* of incident wave to TM resonant mode trapped under the patches. We can estimate this residual coupling by fitting *γ*. We obtained that *γ* ≃ *B/l*^{2} +*ξ* (*B* = 4.5 × 10^{10}, *ξ* = 0.13 × 10^{10}). Analyzing the evolution of total mode decay rate Γ and radiative mode damping *γ* one can observe that there is a particular regime for *s* and *l* when Γ ≃ 2*γ* (i.e. *γ* = *ν*). This regime marked in the plot by grey/rose domains characterized by absorption higher than 95%. A weak sensitivity of absorption intensity to the patch length *l* within rose domain (cf. also the absorption spectra of the first resonant mode in Fig. 2(a)) is explained by inverse dependence of dissipative damping *ν* from the effective inductance *L*, that, in turn, is proportional to the length of a conducting layer *l*. Therefore, in this regime, the increase of radiative damping for larger *l* is accompanied by an increase of dissipative damping *ν* ensuring the condition of Eq. (13) being nearly accomplished within rather broad region of parameter *l*.

We can estimate the free parameter |*β*|^{2} by fitting Eq. (9) to the resonant frequency and to the width of resonant absorption curve. In Fig. 6(a) and 6(b) we present the absorption spectra calculated numerically (dashed curves) and that with use of Eq. (9) (thin solid line) for the first and the next higher order resonance of TM mode. In the insets we also plot the effective impedance of the metasurface *Z*_{eff} for each of the resonant modes. We observe that the resistive part of *Z*_{eff} equals to *Z*_{0} = 120*π* Ω at resonance. Therefore, owing to the surface structuring the effective resistance of the metasurface is drastically decreased compared to the resistance of the absorptive layer (approximately by a factor 370 for the first mode and 340 for the higher order resonance). At resonance the imaginary part of *Z*_{eff} passes through the zero since the inductive reactance is canceled by a capacitive one and leaves the impedance purely resistive. We used the following parameters in our model calculations: *R* = 1/*σs*, where the conductivity *σ* = *ωε*_{0}*ε″* (*ε*_{0} is the vacuum permittivity), *L* = *R*/2*ν*,
$C=1/{\omega}_{0}^{2}L$, *ν* = *γ* = 2.83×10^{8}s^{−1} (for the first TM mode) and 2.83 × 10^{8}s^{−1} (for the next higher order mode). We estimated at the resonance the value of the coupling parameter |*β*|^{2} = 2*γCZ*_{0} that equals 1.03 × 10^{−6} and 1.6 × 10^{−7} for the first and the next higher order mode, respectively.

Finally, the TA effect is not the exceptional property of the square patch metasurface. One can obtain TA for other geometry by applying the procedure discussed above. In Fig. 7 we plot the absorption spectra for some examples of planar metamaterials. We have checked that regardless the different geometry of the metasurface the TA absorption effect can be achieved once the radiative and dissipative damping of the mode squeezed between metal parts of the structure are equal. We also performed the exact calculations taking into account the Ohmic losses of cooper instead of model of the perfect metal (not shown here). We found that that neither resonant frequency nor absorption intensity changes. Only the distribution of absorbed energy is influenced. For example, the amount of energy absorbed by the metal patch in the absorption spectra presented by black curve in Fig. 7 is about 2% while less then 0.1% is absorbed by the ground plan.

## 5. Conclusion

In conclusion, we have investigated the resonant absorption phenomenon in planar metamaterials. Along with rigorous electromagnetic calculations we present a simple resonant circuit model that describes the resonant absorption in terms of impedance matching and radiative/dissipative damping of the resonant mode. We show that in spite of evident simplicity of the model it fully reproduces the rigorous calculations. We distinguished the radiative and dissipative contributions to the resonance linewidth and discussed the role of radiative and dissipative trapped mode decay rates in formation of the absorption spectra. In particular, we found that the square patch based metasurface can exhibits the effect of total absorption once the radiative and dissipative damping of modes are equal. Finally, we have shown that this condition of total absorption is universal and can be satisfied for metasurface of arbitrary geometry providing the effect of total absorption in a large frequency range.

## Acknowledgments

The authors are grateful to the Direction Gnrale de lArmement (DGA) for the financial support of the REI project Novel Metamaterial-based Radar Absorbing Structures under Contract No. 2009-34-0030. The authors also thank Sylvain Gransart, Prof. Vyacheslav V. Popov and Prof. Andrey G. Borisov for fruitful discussions.

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