## Abstract

Metamaterial (MM) absorber is a novel device to provide near-unity absorption to electromagnetic wave, which is especially important in the terahertz (THz) band. However, the principal physics of MM absorber is still far from being understood. In this work, a transmission line (TL) model for MM absorber was proposed, and with this model the S-parameters, energy consumption, and the power loss density of the absorber were calculated. By this TL model, the asymmetric phenomenon of THz absorption in MM absorber is unambiguously demonstrated, and it clarifies that strong absorption of this absorber under studied is mainly related to the LC resonance of the split-ring-resonator structure. The distribution of power loss density in the absorber indicates that the electromagnetic wave is firstly concentrated into some specific locations of the absorber and then be strongly consumed. This feature as electromagnetic wave trapper renders MM absorber a potential energy converter. Based on TL model, some design strategies to widen the absorption band were also proposed for the purposes to extend its application areas.

©2009 Optical Society of America

## 1. Introduction

Metamaterial (MM) is a composite structured materials, formed either from periodic or random arrays of scattering elements [1]. Negative refraction [2,3], perfect lens [4], invisibility cloaking [5,6] and some other interesting phenomenon have been realized using this novel material. Electromagnetic (EM) metamaterials are geometrically scalable which translates into operability over a significant portion of the electromagnetic spectrum. To date, metamaterial has been demonstrated in almost every technologically relevant spectral range, from radio, microwave, mm-Wave, terahertz (THz), infrared, to the near optical [7]. Generally, the effective permittivity (ε) and effective permeability (µ) are used to describe metamaterial in the framework of effective medium theory. However, transmission line model is also a powerful tool for characterization and interpretation of this novel material [8–11].

Metamaterial absorber, as firstly reported in 2008, is a kind of three-layer structure with thickness of each layer significantly smaller than the wavelength [7]. The three layers are an electric split-ring resonator (eSRR) layer, a metal wire or plate layer, and a dielectric layer between them. Theoretical results show that MM absorber can absorb the EM wave almost completely in a narrow frequency band, which makes it an ideal candidate for bolometric pixel elements. Though more and more attention has been paid to this novel material [12–15], the mechanism of the near-unity absorption is still under studying. It has been suggested the matching between the effective permittivity and permeability may be able to interpret the perfect absorption. However, the effective medium theory has flaws in describing MM absorber because the three-layer structured device doesn’t exactly satisfy the homogeneous-effective limit, according to Caloz [11]. A typical case is that the strong asymmetric phenomenon cannot be fully explained by effective medium model, as we have pointed out previously [16].

Quite a few previous works attempted to figure out the time-domain working mechanism of the MM absorber. But it is still not clear that why the absorption is extensively enhanced after the eSRR structure and the wires structure being integrated into an absorber, and it is also unknown why the absorptions is highly sensitive to the thickness of separated layer. In this paper, a transmission line (TL) model for the MM absorber is proposed aiming to answer these questions. Basing on the proposed TL model, the asymmetric absorption was demonstrated by calculating the S-parameters. The average absorption power densities of the absorber were analyzed, which shows that that the absorber traps the EM wave inside a specific space and finally make it be consumed. The trapped EM wave can be converted into thermal energy, electric energy or any kinds of other energy depending on the functions of the spacer materials. Finally, with the TL mode, it is easy to develop some strategies to widen the absorption bands. Therefore, the investigation on TL model of the MM absorber is also a step toward wide-frequency applications.

## 2. The TL model of the metamaterial absorber

Generally, MM absorber is constructed by three layers: eSRR layer, separation layer and wires layer. In the TL model, it is assumed that the transverse electromagnetic (TEM) wave propagates through free space and the substrate with intrinsic impedances *Z _{i}* and

*Z*respectively. There are two assumptions for constructing the TL model. One is that coupling capacitor or coupling inductor between the eSRR layer and wires layer should be ignorable, so that these two layers can be individually modeled, as demonstrated in Fig. 1. Another is that the THz wave normally incidents into the absorber plane with the electrical field parallel to the split gap of the eSRR. The TL model of eSRR proposed by A. K. Azad [17] is used to describe the eSRR layer, in which the LC resonance and dipole resonance each is represented by one group of L, C and R respectively, and the coupling between these two resonances is specified by the parameter M. The wires layer part is mimicked by the TL model developed by L. Fu [18], with the only resonance expressed by one group of L, C and R. The function of isolation layer is modeled by a transmission line which contains all EM related properties of the isolation layer such as ε, µ and thickness. It connects the eSRR part and wire part. All the parameters are needed to be optimized until the S-parameters calculated by the TL model fit the simulation results.

_{o}Once all of the parameters in Fig. 1 are determined, the S-parameters of the absorber can be derived as follows. The ABCD matrix of eSRR structure layer, isolation layer and wires structure are

Where ${X}_{1}=\genfrac{}{}{0.1ex}{}{1}{j\omega {C}_{1}}+{R}_{1}+j\omega \left({L}_{1}-M\right),{X}_{2}=\genfrac{}{}{0.1ex}{}{1}{j\omega {C}_{2}}+{R}_{2}+j\omega \left({L}_{2}-M\right),{X}_{3}=\genfrac{}{}{0.1ex}{}{1}{j\omega {C}_{3}}+{R}_{3}+j\omega {L}_{3},$
*k* is the wave vector of the TEM wave, *l* and *Z _{c}* is the thickness and characteristic impedance of the isolation layer.

So the ABCD matrix is

$$=\left[\begin{array}{cc}\multicolumn{1}{c}{\mathrm{cos}\left(\mathrm{kl}\right)+\genfrac{}{}{0.1ex}{}{{\mathrm{iZ}}_{c}\mathrm{sin}\left(\mathrm{kl}\right)}{{X}_{3}}}& \multicolumn{1}{c}{{\mathrm{jZ}}_{c}\mathrm{sin}\left(\mathrm{kl}\right)}\\ \multicolumn{1}{c}{\left[\genfrac{}{}{0.1ex}{}{1}{\genfrac{}{}{0.1ex}{}{{X}_{1}{X}_{2}}{{X}_{1}+{X}_{2}}+M}+\genfrac{}{}{0.1ex}{}{1}{{X}_{3}}\right]\mathrm{cos}\left(\mathrm{kl}\right)+\genfrac{}{}{0.1ex}{}{j\mathrm{sin}\left(\mathrm{kl}\right)}{{X}_{3}{Z}_{c}}\left[{X}_{3}+\genfrac{}{}{0.1ex}{}{{Z}_{c}^{2}}{\genfrac{}{}{0.1ex}{}{{X}_{1}{X}_{2}}{{X}_{1}+{X}_{2}}+M}\right]}& \multicolumn{1}{c}{\mathrm{cos}\left(\mathrm{kl}\right)+\genfrac{}{}{0.1ex}{}{{\mathrm{jZ}}_{c}\mathrm{sin}\left(\mathrm{kl}\right)}{\genfrac{}{}{0.1ex}{}{{X}_{1}{X}_{2}}{{X}_{1}+{X}_{2}}+M}}\end{array}\right]$$

Then the S matrix thus can be calculated as

## 3. The simulation details

The first proposed MM absorber, following by some other designs, aimed to obtain near-unity, polarization insensitive, flexible or wide-angle absorption. All these devices include two metallic elements: a eSRR layer and a wire layer. Actually, each metallic layer with its underlayer composes a metamaterial. Therefore, the MM absorber can be regarded as a composite of two different metamaterials separated by a functional material layer. In this work, we take the most familiar and basic structure as an example to investigate the validity of the TL model, as shown in Fig. 2(a). The dimension details of the cell of metal layer are shown in Figs. 2(b) and 2(c) with unit of µm.

The metal used in the calculation is gold with conductance of 4.09×10^{7} S and thickness of 800nm. The distance between eSRR structure layer and wires structure layer is 7.8µm and this space filled with polyimide with *ε*=3.5+0.0105i, *µ*=1. The substrate material is a slice of GaAs with *ε*=12.9+0.0774i, *µ*=1. In this work, two propagation direction of the EM wave is defined and comparatively studied. A positive direction is defined as from eSRR surface to wires and then GaAs substrate, and the reverse sequence (substrate – wires - eSRR) is defined as negative propagation.

In order to determine the *L _{i}, C_{i}* and

*R*(

_{i}*i*=1,2,3) in the TL model, each metallic layer, as a metamaterial, was simulated by CST. They were modeled as eSRR-MM and wire-MM, respectively, as demonstrated in Fig. 3. From positive direction along z axis, the eSRR-MM is constructed with the sequenced of port 1- vacuum- eSRR - polyimide- GaAs slice- port 2, and wire-MM is constructed as port 1- vacuum- polyimide- wire structure-GaAs slice- port 2. In all of the simulations, the TEM waves are radiated from port 1 or port 2 with wave vector being perpendicular to the absorber plane and the electric field paralleling the x axis and the magnetic field parallels the y axis.

## 4. Results and discussions

Figures 4(a) and 4(b) show the S-parameters of eSRR and wire metamaterials (see Fig. 3) calculated by using the TL model proposed by A. K. Azad [17] and L. Fu [18], respectively (dotted lines). The results from CST simulation (solid lines) were also depicted for a comparison. It can be seen that for the eSRR, the S-curves calculated from optimized TL model match the CST simulation results very well in the whole frequency region under studied. This indicates that the TL model of the eSRR is reasonable and effective. However, as to the wire-MM, it seems that one group of RLC is not perfect to describe it since in the high frequency band there is a small but clear discrepancy between the calculated results and the simulation ones. For instance, the simulated transmission curves (S_{21} and S_{12}) increase quickly in the high frequency region and reach 0.64 at 1.7 THz, while the calculation from TL model only yield the value of 0.55 at 1.7THz. Nevertheless, in the frequency range near and lower than the resonance point, all S curves have a very good match between the simulation and calculation results. Since our researches mainly focus on the frequency region near the resonance point, we think that the TL model is good enough to characterize the responses of the wire-MM. By fitting to the CST results, *L _{i}, C_{i}* and

*R*(

_{i}*i*=1,2,3) were determined, which could be used in the TL model of the absorber for further investigation.

Figure 5 shows two series of S-parameters of the absorber. One series of curves (solid line) are the simulation results with CST, and another series of curves (dotted lines) are calculated from the TL model shown in Fig. 1 with the parameters derived from TL model of eSRR and wire metamaterials. It can be seen that the basic shape of all S curves agrees well with the experimental results [7,12]. Furthermore, the two series of S-curves are coincident with each other in the frequency range of 0 ~1150GHz. But the discrepancies become obvious in the high frequency band. Transmission curves (S_{12} and S_{21}) are exactly the same, which decrease with the frequency and then increase with a minimum of 0.05 at about 1100GHz. However, as to the reflection curves it is quite different for the positive and negative cases. The reflection curve of S_{22} increases with frequency to a maximum of 0.9 at about 1130GHz and then drops. However, for S_{11} curve, there is a resonance peak locates at about 1130GHz with a minimum reflection of 0.12. The off-resonance regions are similar to S_{22} in curve shape. For the positive and negative incidence of the EM wave, a remarkable difference in the absorbance is unambiguously indicated with identical transmission (S_{12} and S_{21}) but different reflection (S_{11} and S_{22}). That is, the phenomenon of asymmetric absorption in MM absorber we reported previously was well reproduced by the TL model.

So it can be summarized that the proposed TL model is able to describe the electromagnetic property of MM absorber, especially at low frequency or near the resonance frequency. As mentioned before, the two components in the TL model are copied from the TL model of eSRR and wire metamaterials. When these two layers are put together, the distance between them is small (a few micrometers) thus the coupling capacitor cannot be ignored in the high frequency, and that is thought to be the reason for the observed discrepancy between the calculated and simulated results.

In the TL model of the absorber, the *R _{1}, R_{2}* and

*R*represent the components for energy consumption. Figure 6 shows the energy consumed by

_{3}*R*

_{1},

*R*

_{2},

*R*

_{3}in two exciting conditions. When the EM wave is emitted from port 1 (the positive case), nearly 90% of the incident energy is consumed by

*R*

_{1}, and about 10% consumed by R

_{3}. These results mean that the LC resonance of eSRR is predominant in the energy absorption to the EM wave, while the contribution from dipole resonance is very small and ignorable. In the negative case, the total energy consumption of the absorber is very small. It is the wire structure which provides about only 6% energy loss to the incidence wave, and

*R*and

_{1}*R*contributes very little. These results confirmed that, for the absorber discussed here, it is mainly the LC resonance of the eSRR structure contributes to the strong absorption when EM wave propagate along the positive direction. This conclusion is consistent with the previous reports [13, 19]

_{2}Abovementioned results give a insight into the basic function of each components of MM absorber: (1) As hints by the TL model, the LC resonance of eSRR strongly affects the absorption characteristics of the absorber. It is also known that for eSRR, the inductance L is provided by its metallic loops and the capacitance C is induced by the splits (cut) of the ring [20,21]. Thus the absorption curve of the absorber mainly depends on the framework of the eSRR. Of course, the effects of other components such as interlayer coupling and other resonance from L_{i} and C_{i} (i=2, 3) are also not negligible. (2) More importantly, the function of the isolation layer is to adjust the impedance of the metamaterial and enable the EM wave to enter into the device as much as possible. Therefore, the absorption is highly sensitive to the properties of the isolation layer, such as its thickness, permeability and permittivity. (3) The role of wires structure is to enhance the reflection of EM wave thus benefits the trapping and absorbing of wave in the space between the two metallic layers. So, it is not a surprise that a replace of the original wire structure by gold plane can produce the perfect absorption over 99.9% [13]. In addition, the further study shows that the application of Gold plane will bring much convenient in the device design and fabrication, as we will discuss later.

Since the calculation from the TL model shows that it is *R _{1}* consumes most of the EM energy at the absorption frequency, it is worthy to find out what kinds of materials in the absorber response for the strong absorption. Firstly, the conductance of Gold is 4.09×10

^{7}S, which is high enough to be viewed as a superconductor even in THz region thus the ohmic loss of the metal is small and ignorable. Secondly, the

*µ*of all materials (vacuum, Gold, GaAs and polyimide) equal to 1 with zero imaginary parts, thus yields no consumption to the H field of the EM wave directly. Therefore, the dielectric loss of the polyimide spacer and substrate are the only sources for EM consumption since their

*ε*is with non-zero imaginary parts (for the positive case). This result once again addresses the important role of the dielectric spacer in the absorber.

Figure 7(a) demonstrates the distribution of average power loss density in the absorber plane (xy plane), which is calculated by: *P*(*x, y*)=∫*P*(*x, y,z*)*dz*/∫*dz*. Where *P*(*x,y,z*) is the power loss density at the absorption frequency, which indicates how much input power is absorbed in unit volume of the absorber. Apparently, the absorption does not homogeneously occur in the absorber. From Fig. 7(a), it is found that the strong absorption divides into three parts. The first part locates near the two ends of the metal wire and the second one distributes around the outer edges of the eSRR metal framework. The third part takes place in the vicinity of the split gaps of the eSRR, which is far more extensive than the other two portions. As a illustration, the distribution of power loss of 3×10^{11}w/m^{3} is calculated and shown in Fig. 7(b) and (c) from front (3D) and side view. The distribution situation confirms that the absorption mainly occurs in three parts as mentioned above. From a side view (Fig. 7(c)), the power loss only concentrate and compact in a small space near the metal structure, and the most strong power loss is taken place in the vicinity of the split gap. We should note that, as we discussed above, the absorption arise from not the ohmic loss of the Gold metal but the dielectric loss of the isolation material or substrate. This results, on the other hand, indicates that the absorber can concentrate or trap the EM wave in some specific locations of the spacer or the substrate, thus in these spots (for example, the space neighboring the split gap of the eSRR) the energy is significantly reinforced. This unique feature makes it possess many potential applications. For example, MM absorber may be an ideal candidate for applying in solar cell if its dimension size scales to the optical band. By replacing the current anti-reflection coating with MM absorber in the solar cell, the interfacial reflection will be reduced thus more light will be trapped. More importantly, by precise designing, the light can be concentrated and reinforced around the P-N junction thus the photoelectric conversion efficiency will be significantly increase. With the same principle, the absorber can also be used as a powerful thermal emitter.

Narrow-band absorption is desired for some applications such as bolometric pixel elements. However, for many other applications such as thermal emitter, invisible cloaking, or solar cell mentioned above, wide-band absorption is also required in order to enhance the device efficiency. We find that with our proposed TL model, it is very easy to find out some design strategies to widen the absorption band. For the absorber under studied, one method is to increase the value of *R _{1}* since our TL results indicate that

*R*is responsible for most of the absorption. The second one is to introduce several of absorption bands and combine them into a wide one. Following we will give an illustration to show how we can solve these problems with the TL model.

_{1}In the absorber structure applied for following investigation, the wire structure of the device is replaced by a Gold plane and all other structures and parameters keep unchanged. By using a metal plane, the transmission of THz wave (*S _{12}*) through the absorber is zero. Therefore, the absorption can be simply calculated by 1-

*S*

^{2}

_{11}. Figure 8 displays the

*S*parameters with different

_{11}*R*. The black (solid) line is for the original

_{1}*R*, we marked it as

_{1}*R*, and the red (dash), blue (dash dot) and pink (dot) lines are

_{1O}*S*curves of the absorber with 2

_{11}*R*, 4

_{1O}*R*, and 8

_{1O}*R*. It can be seen that the absorption peak is largely widened by increasing the value of

_{1O}*R*. The full width at half maximum of the reflection

_{1}*S*increases from 40GHz to 300GHz as

_{11}*R*increases from

_{1}*R*to 8

_{1O}*R*. As for the device design and fabrication, according to the investigation above, there are at least two ways to increase

_{1O}*R*. One is to lower the conductance of the material of eSRR, similar to the Frequency Selective Surfaces used in some microwave absorber. Another way is to apply an isolation material with large imaginary part of the permittivity, and the enhancement of the leakage capacitance will induce a increase of

_{1}*R*. According to the TL model, the absorption enhancement by the former way is derived from the increase of ohmic loss of the eSRR metal, while with the later method the absorption results from the dielectric loss of the spacer.

_{1}Another method to widen the absorption peak is to combine several absorption peaks together. As indicated by the TL model, the absorption peak of the absorber under studied is mainly decided by the LC resonance of the eSRR structure. Therefore, a specially designed eSRR structure with several overlapped LC resonances is hopeful to realize the wide-band absorption. Furthermore, a preliminary investigation by TL model indicates other resonance (such as the dipole resonance) can also induce strong absorption if the device is properly designed. Further investigation on this topic is still under working.

As compared to the effective medium theory, our proposed TL model is more effective to describe and analyze MM absorber. However, we should emphasize again that there are two limitations for our TL model. Firstly, the coupling between the eSRR and wires layer should be weak enough, otherwise the absorber cannot be model by simply combining the two TL models of the eSRR layer and wire layer. Secondly, the propagation direction of the EM wave has to be normal to the absorber plane, so that the transmission line can describe the isolation layer accurately. We believe this novel device can be better described by further improving our TL model or by some other method. For example, a recent work by Han et al. has studied a tunable semiconductor eSRR pattern in the same shape studied here at THz band, where even the anisotropy can be modeled [22].

## 5. Conclusion

In this work, we have proposed a transmission line model for metamaterial absorber, and the investigations on transmission, reflection and absorption characteristics give a deep insight into the basic physics of this device. By the TL model, the asymmetric phenomenon of THz absorption is unambiguously demonstrated and, for the MM absorber under studied, the strong absorption is found to be mainly related to the LC resonance of the split-ring-resonator structure. The isolation layer in the absorber, however, is actually an impedance transformer and plays key role in producing the perfect absorption. The studies by TL model also show that the electromagnetic wave is concentrated on some specific space in the absorber. The trapped electromagnetic wave can be converted into thermal energy, electric energy or any kinds of other energy depending on the functions of the spacer materials. This feature as electromagnetic wave trapper has many potential applications such as solar cell and thermal emitter. With the TL model, it is easy to develop some strategies to widen the absorption bands. Therefore, the investigation on TL model of the absorber is also a step toward broadband applications.

## Acknowledgements:

This work is support by National Basic Research Program of China (973) (No. 2007CB310407), NSFC (No.60801023, No.60721001), and International S&T Cooperation Program of China (No. 2007DFR10250).

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