WEN Qi-ye,ZHANG Huai-wu,XIE Yun-song,YANGQing-hui,L I Yuan-xun

(State Key Laboratory of Electronic Fi lm s and Integrated Devices,University of Electronic Science and Technology of China,Chengdu610054,China)

1 Introduction

A Metamaterial Absorber(MA)is a kind of threelayer metamaterial with a thickness significantly s maller than the wavelength,which can absorb an electromagnetic wave completely overa narrow frequency band[1-4].This unique property makes it an ideal candidate for bolometric pixel elements.However,the mechanism of the unit absorption is still being studied.N.I.Landy has suggested using the matching between the effective permittivity and permeability to interpret this phenomenon[5].But the strong non-reciprocity of the MA indicates that this interpretation may be inappropriate,because the MA thickness is too s mall compared to the wavelength.Thismeans that the effective medium theory can not be applied to it and the effective permittivity and per meability approach is not able to describe the MA[6].

Earlier studies have proved that the absorption derives from the ERR structures and is enhanced by the coupling between the ERR and wire structures.We have also tried to understand the time-domain working mechanis m of the MA,which leads to the non-reciprocity.However,the nature of the coupling is still not understood norwhy the parametersof each layer in the MAs,especially the thickness of the dielectric layer,must be carefully chosen. In this paper,basing on the reported TL model for ERR structure and wire structures[7],a TL model for the MA is proposed.Thismodel provides ans wers to all these questions,can be confirmed by the field analysis,and significantly displays the same non-reciprocity as the simulation.Based on the investigation to the TL model and the distribution of average absorption power densities,it is pointed out that the MA does not simply convert the electromagnetic energy into thermal energy,but concentrates the electromagnetic wave into a small space,and then it is absorbed completely.This working mechanis m indicates that the MA is actually an ideal light concentrator and has greatpotential in the application to the solar cells.

2 TL model formetamaterial absorber

Generally,theMA is constructed from three layers:the ERR layer;the isolation layer;and the wire layer.In the TL model of the MA,it is assumed that the TEM wave propagation through free space and the substrate has intrinsic impedancesZiandZo,respectively,and there is no coupling capacitor or coupling inductor between the ERR layer and wire layer.These two layers are individually modelled as shown in Fig.1.The TL model of the ERR proposed byAbul K.Azad is used in the ERR layer part of theMA,because it is simple and able to described the behaviour of theS-parameters of the ERR structure.In this TL model,theLCresonance and dipole resonance are represented by one set of values ofL,CandR,respectively,and the transformer coupling parameterMis used to specify the coupling between the two resonances.The wire layer part is modelled by the TL model reported by L.Fu,and the single resonance of the wire structure is expressed by second set of values ofL,CandR.The effect of isolation layer in the MA is modelled by a transmission line which connects the ERR and wire structure.All of the parameters of the components need to be optimized so that theS-parameters calculated by the TL model fit the simulation results.

Fig.1 Trans mission line model for the metamaterial absorber.In the ERR structure part,the parametersR1,L1,C1andR2,L2andC2describe the LCand dipole resonance,respectively,parametersM describe the coupling between the two resonances.ComponentsR3,L3andC3specify the resonance of the wire structure and TL refers to the trans mission line and represents the isolation layer of theMA.

Once all of the parameters in Fig.1 are determined,theS-parameter values of theMA can be derived as followed.

TheABCDmatrix of the ERR structure layer,isolation layer and wire structure are,respectively

Therefore theABCDmatrix of theMA is

and theSmatrix can be calculated as

3 Simulation ofmetamaterials

The first proposed MA was composed of ERR and wire structures,then laterMAswith other structures were reported to display the near-unit absorption.Here we take a most familiar and commonMA as an example to investigate the validity of the TL model based description of theMA.

The structure of each metal layer in the MA simulated by CST is shown in Fig.2 with units of μm. The metal in thesimulation is gold with conductance of 4.09×107S/m and thickness of 800 nm.The space between the ERR structure layer and wires structure layer has thickness 7.8μm and is filled with polyimide withε=3.5+0.010 5i,μ=1.The material of the slice of theMA1is GaAswith ε=12.9+0.077 4i,μ=1. From the positive direction to the negative direction,the system is constructed along thezaxis as port 1-vacuum-ERR structure- polyimide-wires structure-GaAs sliceport 2.

Fig.2 Dimension details of theMA1.

In order to deter mine theLi,CiandRi(i=1,2,3)in the TL model,each layer in theMA is simulated.The metamaterial with the same ERR structure(MERR)and the same wire structures(MW)as MA1is illustrated in Fig.3.From the positive direction to the negative direction along thezaxis,the MERR is constructed as port 1-vacuum-ERR structure-polyimide-GaAs slice-port 2,and the MW is constructed as port 1-vacuum-polyimide-GaAs sliceport 2.In all of the simulations,the TEM waves are radiated by port 1 orport 2 and the electric fields are parallel to thexaxis.

Fig.3 Schematic drawing of theMERR(a)andMW(b).

4 Discussion of results

Fig.4(a)and(b)qunatify theS-parameters of the MERR andMW simulated by the CST and calculated using the TL model in the frequency band 0-1 700 GHz.It can be seen that the TL model of the MERR with optimized parameters shows the sameS-parameters as the corresponding simulation results across the whole studied frequency region.However,it seems thatone group of RLC values is not able to fully describe the MW,because there is a small disagreement between the s imulation results and the TL model calculation results of theMW.They share the sameS-parameters in the frequency close to and below the resonance point.But the trans mittance(S21andS12)of the s imulation results rise faster in the higher frequency region up to 0.64 at 1 700 GHz.By contrast,the transmittance calculation results only yield the value 0.55 at 1 700 GHz.There are also some s mall differences inS11andS22between the simulation and calculation results.Since the discussion is largely focussed on the frequency region near the resonance point,it is reasonable to assume that the TL model can characterize the responses of theMW accurately enough.

Fig.4 Simulated and calculatedS-parameters ofMERR(b)andMW(a).

The simulation results of MA1shown in Fig.5 demonstrate the same behaviour as the experiments do.Utilizing the same optimized parameters in the TL model of both MERR and MW,the TL model calculation results ofMA1were obtained and shown by the dotted curve in Fig.5.The calculated results show a strong absorption peak at 1 100 GHz,where most of the input power is absorbed by the componentR1,which is used to represent the loss ofLCresonance of the ERR.TheS-parameters derived by differentmethods almost show the same shapes in the frequency band 0-1 150 GHz,whereS12andS21decrease until the frequency point reaches 1 100 GHz,S11locates at 1 100 GHz with value of 0.15,andS22keeps rising in the low frequency band and yields the maximum about 0.95 at 1 100 GHz.But the variance between the these results become obvious in the frequency band higher than 1 150 GHz.The calculated reflectanceS11andS22end up with value 0.9 in the frequency spectrum,and the simulated reflectance displaying the much higher decreasing speed,yields the value about0.65 at 1 700 GHz.The transmissionS21andS12,naturally,shows the opposite variation tendency comparing to the reflectance.

Fig.5 Simulated and calculatedS-parameters ofMA1.

Therefore it can be deduced that the TL model is able to describe the electromagnetic property exactly at frequencies below the absorption peak,but is less accurate in the higher frequency band.As mentioned before,the components in the TL model ofMA1 are copied from the TL model of the MERR andMW.When these two structures are put together,thedistancebetween is so smallthatthe coupling capacitance can not be ignored at high frequency.This is thought to be the reason for the discrepancy between the calculated and simulated results.

5 Conclusions

The TL model can describe the MA1exactly at frequencies below the absorption peak,but the error becomes larger at higher frequencies. The reason why the thickness isolation layer affects the absorption so much is that the isolation layer actually acts as an impedance transformer in theMA.Both the TL model and the field distribution show that the electromagnetic wave is concentrated in the space near the outside framework of ERR structure and finally absorbed by the isolation layer.

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