1. Introduction

Terahertz (THz) radiation encompasses a range of frequencies in the electromagnetic spectrum between 0.1 THz and 10 THz. There has been an increasingly strong driving force to make use of this portion of the spectrum because of the potential benefit of THz technology in a range of application including, but not limited to, medical imaging [1], environmental monitoring of earth [2], remote sensing of explosives [3], and semiconductor electrical property determination [4,5], as terahertz radiation has the unique ability to safely penetrate a wide variety of non-conducting materials including clothing, paper, cardboard, wood, masonry, plastic and ceramics, and to interact with molecules without any ionizing effect [6]. However, this scientifically rich spectrum has been technologically underdeveloped. On the other hand, commonly used devices and techniques in the microwave and optical regime are not applicable to measure and manipulate terahertz radiation. THz detection techniques and devices like Terahertz Time Domain Spectroscopy (THz-TDS) and bolometer suffer from some major drawbacks like larger size, lower sensitivity at room temperature, higher cost or indirect measurement of THz radiation in frequency domain. Therefore, there exist great demands for a compact, low cost and sensitive THz detector capable of directly measuring THz radiation.

Metamaterial which is an artificial subwavelength composite material that can manipulate electromagnetic waves by proper designing its effective permittivity and permeability could be an alternative approach to meet this need [7,8]. Among different devices based on metamaterials, metamaterial terahertz absorbers (MMTA) have been recently attracted attentions due to their potential which promise a novel compact and perfect THz detector. The first MMTA proposed by Hu Tao et al. [9], was based on three major components including a patterned metallic layer as a frequency selective surface (FSS), a dielectric spacer and a metallic backplane, which absorbs THz radiation in a very narrow bandwidth. Soon after, more research was done to enhance the bandwidth of the absorber. Similar structures were reported either to make multiband MMTAs by using different FSS for various resonance frequencies [10–12],or to make broadband absorbers by bringing resonance frequencies closer to one another and by engineering the frequency dispersion of FSS to mimic an ideal absorbing sheet in infrared range [13–16]. Other structures were also designed and fabricated to improve polarization insensitivity of MMTA by using four-fold symmetric FSS [11, 17].A thorough study of the recent advances in MMTAs has been recently reported by C. M. Watts et al [18].

There have been reports of a few different physical mechanisms to understand operational principles of the metamaterial terahertz absorbers; anti-parallel currents [9], out of phase currents inside the absorber [19], standing waves resonances inside spacer [10],and destructive interference between reflected waves of the FSS and metallic back layer [20]. Several articles have reported modeling of split ring resonators (SRR) based on RLC resonator circuits by using either Transmission Line method (TL) or qausistatic approach [21–25]. Lorentz oscillator model has been considered in other works for determining effective permittivity and permeability of metamaterials by quasistatic and nonquasistatic circuit models [16, 22–29]. Inspired by SRR electric models, several researchers have proposed similar models applying for FSS and filters in microwave and to understand their principal function and improve their responses [30–33]. In those models, metallic patterns have been considered as an inductor in series configuration with a resistor and the dielectric gap between metallic surfaces as a capacitor. Similarly, Y. Pang et al. have proposed a simple TL model to accurately describe the operation of their GHz absorber [34]. This model is composed of a series RLC circuit, to account for resonance and loss in the FSS, which is in parallel to a transmission line accounting for the dielectric spacer. A similar work has also been reported in more detail in THz domain [35]. Furthermore, Q. Wen et al modeled the first reported MMTA by considering both dipole and LC resonances in the FSS as well as the coupling interaction between the two [36]. However to the best of our knowledge, a quasistatic RLC circuit model has not been introduced yet for this type of absorbers to explain its functionality by considering the effect of both FSS and the spacer.

In this work we present a simple and straightforward quasistatic dynamic RLC model that can fit well with absorption response of MMTAs. Based on Lorentzian shape of the absorption spectrum, a simple RLC band-pass circuit is first employed to model the perfect absorption (100%) from the absorber. Then, in order to understand the effect of dielectric spacer on absorber function, a parallel RLC loop is inserted into circuit. In addition to simplicity, our model describes well the physics governing the operation of MMTAs based on destructive interference theory [20].

2. RLC circuit model of MMTA

The circuit model of MMTA was originated from our observation that absorption spectrum of MMTA is a Lorentzian function that can be described as,

 

Fig. 1 (a) Quasistatic RLC model for a perfect MMTA with 100% absorption. (b) Schematic of the FSS of absorber which gray band is copper deposited on polyimide (c) three dimensional picture of the absorber. The space between patterned plane and metal backplane is polyimide.

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If the MMTA absorption is not perfect, such a simple circuit model will not suffice, as it will always give 100% absorption. To make the circuit model realistic and accommodate the possibility for non-perfect absorption, we should add a component to account for the spacer layer to the model. In order to decrease the peak of absorption for non-perfect cases, another resistor between LC and R should be considered. The peak of absorption changes based on the value of this resistor, and perfect absorption will be obtained when it is equal to zero. In actual MMTA, spacer thickness and permittivity of dielectric tend to change the absorption frequency slightly, while a resistance of the spacer does not change the resonance frequency of the model. Therefore, variation in resistance of the spacer cannot be sufficient to account for the spacer. In order to realize the spacer in the model another series LC component which is parallel to the resistor should be added. Figure 2(a) depicts the complete circuit model that is proposed in this work. R_p and L_pC_p account for the spacer so that L_pC_p can bypass the resistor R_p in the perfect case compared to the non-perfect cases, where L_pC_p can vary both absorption amplitude and frequency slightly. The magnitude square of the transfer function for this circuit shown in Fig. 2 can be written as,

 

Fig. 2 (a) Complete quasistatic RLC model for a MMTA with current loops corresponding to first reflected beam (I₁) and collective reflection from spacer cavity (I₂). (b) Schematic of reflection beams in a MMTA. I₁ is the first reflected beam and I₂ is collective reflection spacer cavity. In perfect absorption case, I₁ and I₂ are out of phase and equal-magnitude.

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In this model, the absorption frequency is dominantly determined by FSS as a given LC. L_pC_p and R_p play the role of resonance inside spacer, between metal backplane and FSS which its resonance frequency is determined by L_pC_p. In more details, R_p becomes parallel with impedance Z_LpCp and represents any loss in the absorber energy due to reflection. Whenever LC and L_pC_p resonance frequencies match, R_p will get short-circuited and the structure will operate as a perfect absorber with 100% absorption. In this case, the currents I₁ and I₂, have the same magnitude but are out of phase so that they will cancel out each other in R_p. If we view I₂ as the reflected wave of the resonance inside spacer toward the source and I₁ as the first reflected beam on FSS interface and air, this model works well to explain interference theory as shown in Fig. 2(b).

In the case of a non-perfect absorber, the LC and L_pC_p resonance frequencies cannot be equal anymore, thus impedance Z_|| = R_p || (Z_LpCp) will lead to a drop in the output voltage V_O. It means that the reflected wave I₂ will no longer be out of phase with I₁ and consequently the total reflected power will not be zero (absorption not 100%). Non-perfect absorption can come from two sources: one is the dielectric spacer layer where the thickness or permittivity varies, and the other is variation in metal conductivity. The former case is explicitly explained by changes in L_pC_p and R_p directly. Unfortunately, effects of changes in metal conductivity are more subtle.

If the metal conductivity remains close to a perfect conductor, the resonance frequency associated with the FSS will be most likely fixed since it comes from L and C of the FSS layer which are independent of metal conductivity. Therefore the dominant resonance frequency of the absorber will not be directly changed by metal conductivity as we observed through simulation of the FSS resonance frequency by using different metal conductivities. However, instead, it is important to note that any change in metal conductivity will eventually change the resonance condition inside the cavity of spacer. Since the permittivity of a metal is related to its conductivity through ${\epsilon }_r={\epsilon }^{\text{'}}+j\sigma /\omega$ and its refractive index is related to the permittivity through $n=\sqrt{{\epsilon }_r}$, the conductivity of a metal changes its refractive index. This change in metal refractive index will then alter the phase and amplitude of the reflected and transmitted waves at both facets inside the spacer cavity resulting in a change in the resonance condition of the cavity and hence L_pC_p resonance frequency. The L_pC_p resonance frequency will shift further from LC (the major and dominant resonance frequency associated with the FSS) and the impedance (Z_LpCp) of L_pC_p will no longer be zero. According to the change in cavity condition, Z_LpCp will be a positive or negative imaginary number resulting in a phase shift in I₂. Therefore reflected waves of I₁ and I₂ cannot be completely out of phase anymore. In this case, R_p will get involved in the circuit and will decrease absorption and L_pC_p will shift the structure absorption resonance frequency.

3. Results and discussion

Table 1 summarizes parameters used to simulate two different MMTAs, A and B which are studied. In this table, the parameters are used to simulate MMTAs for perfect absorber and they are indicated in Fig. 1. Both structures are simulated to obtain absorption magnitude and resonance frequencies and we applied our model to fit the absorption spectra.

Table 1. Dimensions of the simulated and fabricated absorbers

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Figure 3 shows the results of fitting Lorentzian Eq. (1) to the simulated data for the absorber A with 7 µm spacer thickness. Polyimide with 7 µm thickness gives nearly perfect absorption (99.8%) at 0.87 THz. We obtained x₀ = 0.8713 THz, $\gamma =$0.01967 THz, and I = 1 from the fitting results respectively, which confirms our assumption for perfect absorber discussed in previous section. Additional simulations confirmed that reasonable changes in spacer thickness from 5.5 up to 15 µm, spacer permittivity from 3 up to 3.5 or spacer conductivity do not change the resonance frequency more than half width at half maximum bandwidth of the absorber. Thus, we used this limit on L_pC_p and R_p to avoid undesired values for parameters. If the small change in L_pC_p and R_p results in large deviation in resonance frequency, those set of values are not acceptable so that fitting will get continued to find acceptable values.

 

Fig. 3 Lorentzian fitting of the simulation data of absorber A with 7 µm polyimide thickness.

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To examine non perfect case in circuit model, we first started with fitting absorber A (perfect case with 7 µm spacer thickness) with Eq. (3). Results of the obtained fitting parameters are summarized in Table 2. Then we varied the spacer thickness to both 6 µm and 9 µm in the same absorber A structure to achieve non-perfect cases. Since everything is same except the polyimide thickness, we can use the values of L, C and R obtained for the 7 µm case in the Table 2 as a reference. Once L, C and R are fixed, the L_p, C_p and R_p can be determined by fitting Eq. (3) again for absorber A with 6 and 9 µm. Z_LpCp is almost zero for 7 µm but it is positive and negative imaginary numbers for 6 and 9 µm (non-perfect case) respectively, confirming that I₂ is not out of phase with I₁ anymore, hence the absorber loses some energy by non-zero reflection.

Table 2. Parameters extracted from fitting for simulation data of the absorber A with three different polyimide thicknesses

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Figure 4 illustrates fitting results from the absorber A and compares those three absorbers with different polyimide thicknesses. The dashed curve is the fitting result for the 7 µm or perfect case while the solid curves belong to the fitting results for non-perfect absorber with 6 and 9 µm spacer thickness in Figs. 4(a) and 4(b), respectively. Circular dots represent data obtained from FEM simulation for 7µm case and square dots are the data from simulation for non-perfect MMTAs. We also simulated the absorber with finer frequency step sizes around the peak to verify the matching peak position between simulation data and fitting curves. We confirmed resonance frequency and absorption value at peak matched very well for both fitting and finer simulation. From the Fig. 4, we found f_res is 0.867 THz with 97.00% absorption for 6 µm MMTA, and f_res is 0.875 THz with 97.5% absorption for 9 µm MMTA.

 

Fig. 4 Fitting results for absorber A by using RLC model. Circular dots are simulation data for 7 µm MMTA and square dots are simulation data for 6 and 9 µm in figures a and b respectively. Solid line is the fittings for non-perfect MMTA and dashed line is the fitting for perfect case. Insets are fitting of simulation with finer frequency at peaks by using the electric model.

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Finally the circuit model was applied to the experimental results. Absorber C in Table 1 describes the dimensions of an absorber structure fabricated using standard photolithography. The Cu backplane and FSS Cu elements were deposited by electron beam evaporation with the FSS elements patterned by UV exposure of a positive photoresist through a dark-field photomask. After metal deposition of the FSS elements, excess Cu was removed by lift-off. The polyimide dielectric layer between the ground plane and FSS layer was deposited by spin coating consecutive layers to achieve the desired thickness. Once applied to the ground plane, the polyimide was baked in nitrogen at 350°C for one hour to ensure complete curing. It should be noted that variation in the polyimide curing process can lead to slight differences in the optical properties of the polyimide film. For this work, the polyimide curing procedure was strictly maintained constant to avoid this issue. THz time domain spectroscopy in reflection mode with normal incident beam was used for the experimental absorption measurement. The details of the measurement were reported in previous work [39].

Figure 5(a) shows the absorption spectrum and fitting results to Eq. (3) for both experimental and simulation data (absorbers B and C). The circular dots show the data from simulation for the perfect absorber case (absorber B) as the reference and the square dots correspond to experimentally measured data for the same absorber with 10.7 µm polyimide thickness (absorber C, supposed to be non-perfect). It is clearly observed that the resonance frequency was shifted to the left and the absorption bandwidth was significantly broadened compared to the resonance frequency of 0.899 THz with 99.99% absorption of the perfect absorber with polyimide thickness of 7.5 µm. The fitting parameters which show large variations in R, R_p and L_pC_p are summarized in Table 3. Similarly to what was done for the fitting procedure for absorber A, we first fitted absorption spectrum of the perfect case (absorber B) and determined the values of R, L and C. Then we used the same values of R, L and C for absorber C to fit absorption spectrum of absorber C using Eq. (3) finding R_p,L_p and C_p.

 

Fig. 5 (a) Fitting results for absorber B by using RLC model. Circular dots are simulation data for 7.5 µm polyimide thickness and square dots are measured data for 10.7 µm polyimide thickness in the sample. Solid and dashed lines are the fitting results. (b) Effect of the changing in conductivity of copper on FSS reflection. Solid line is for $\sigma =6\times {10}^{7}S/m$and dashed line is for $\sigma =6\times {10}^{6}S/m$

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Table 3. Parameters extracted from fitting for simulation and experiment data of the absorber B and C.

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However, we found the fitting in the case of absorber C was not straightforward and did not converge at first. The reason was because the values of L, C and R were fixed and changing L_p,C_p and R_p meant reasonable deviation in polyimide characteristics and thickness that could not shift the resonance frequency from 0.9 THz to 0.85 THz. In fact, based on our simulations, we showed that increasing the polyimide thickness (from the 7.5 µm of absorber B to the10.7 µm of absorber C) actually leads to a blue shift of the resonance from 0.9 THz to 0.91 THz, rather than the observed red shift. In order to proper fit the experimentally measured data of absorber C, we found that R should be higher than the value used in FEM simulation, and the resulting fit parameters are shown in Table 3. Because R is related to the resistivity of the copper, the obtained higher R means a lower conductivity of copper, which is the main reason for bandwidth broadening and confirms the similar result reported by other researcher [36]. A reduction in the conductivity of copper might be due to the thin copper layer thickness which makes carrier scattering by lattice defects much larger than scattering by phonons [40].To confirm the impact of a lower copper conductivity, and to show that FEM simulation using a lower conductivity leads to a more reasonable match to experiment, we measured the conductivity of the copper in our structure and obtained a value of 3 × 10⁷ (S/m), which is half the one originally used from the Comsol standard material library. FEM simulation using this reduced conductivity in the geometry of absorber C then led to a red shift in the resonance frequency to 0.87 THz (from 0.91 THz). This was accompanied by a broadening of the absorption peak to close to that in Fig. 5(a).Furthermore, as discussed in the previous section, the conductivity of copper should not affect the resonance frequency of the FSS itself but will change the amplitude and the phase of the reflected and transmitted waves of the FSS. To verify this, we simulated the reflection spectrum from the FSS alone without ground backplane for two different copper conductivities as shown in Fig. 5(b). It can be seen that reducing copper conductivity by 10 times does not change FSS resonance frequency at all. From the above reasoning, we conclude that the differences between the absorption spectra of perfect absorber B and experimentally fabricated absorber C are majorly due to the change in copper conductivity, which results in a resonance frequency red-shift, a peak broadening and leads to large changes in L_pC_p and R_p values in the fitting results.

To better understand the influence of metamaterial geometries on the RLC values, we can gain some insight on the physical origin of L and C by first observing in Fig. 6(a) the electric field strength and current density distribution on the front layer of perfect absorber B. For clarity, we have added white arrows to show the direction of current. It clearly shows that resonance in the FSS arises not only in the gaps between the two vertical legs but also as a dipole on the square ring. By contrast, the horizontal legs do not contribute to the resonance. From this observation, we can infer that the capacitance C originates primarily from the electric charges in upper and lower parts of the FSS pattern where the electric field looks most intense. On the other hand, the origin of inductance L should be straightforward to understand, since we know the path of the resonant current. It comes from the self-inductance of metal bars in which the current flows. According to the current path on the FSS, the simplified equivalent circuit model for the FSS should be like Fig. 6(b) which will result in the equivalent inductance:

 

Fig. 6 (a) Electric field strength and current density distribution on FSS. White arrows indicate to the path of the resonance current. (b) Equivalent circuit model of the FSS with red arrows indicating to the resonance current path. (c) Current density and electric field profile on the backplane with black arrows showing the current path.

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The self-inductance of a metal wire is proportional to its length, therefore by increasing the length of the wire its inductance will increase. This is why the inductance L of absorber A is larger than B, because, in absorber A, w_p is bigger than that of B and the trapezoidal part of vertical legs contributes to the resonance current, therefore the value of L₁ and hence L increases in absorber A. Since the capacitance C is almost the same between absorbers A and B, the resulting resonance frequency decreases.

Figure 6(c) shows the current density and electric field profile on the backplane. As seen in the figure, the resonance current on backplane is in opposite direction to that in the FSS. These opposite currents make a mutual inductance between the FSS metal pattern and the backplane which is at the origin of L_p in our model. We can use the method of image charges to understand this mutual inductance such that we can remove metal backplane and place the image of the FSS pattern with respect to the backplane. Whenever the distance between the FSS and its image increases, the mutual induction between them, i.e. L_p, decreases, which is consistent with the fitting results for L_p for absorber A when the spacer thickness was changed.

As for the resistance R in the model, it is the sum of the resistances from the FSS metal bars and the metal backplane, which can be calculated approximately through R = ρl/A on the FSS bars by considering skin depth at terahertz range. Because the current on the backplane is distributed on a large area, the backplane resistance should be much smaller and probably negligible compared to the resistance from the FSS.

4. Conclusion

In summary, we showed that MMTA can be fitted well to Lorentzian function and hence we can model it as a quaistatic RLC circuit model. Our RLC model is based on a band-pass circuit with an additional loop of L_pC_p || R_p as the polyimide positioned between LC and R as the FSS and loss in metal respectively. This model can explain well the physics of the MMTA, similar to the previously reported interference theory. We analyzed our designed and fabricated absorbers by this model and we investigated intentional changes in fabricated sample based on the electric elements values of the circuit model.