1 Introduction

Metamaterials, composed of the artificially subwavelength structure cells arranged in periodic arrays, have been extensively studied at wavelengths from visible to microwave band recently [1–4], due to versatility of manipulating incident light wave, and possess exotic physical properties that are unattainable with naturally available materials, such as negative index of refraction [4], invisible cloaking [5], and extraordinary optical transmission [6, 7]. Metamaterials have quite wide range of immediate and potential applications, such as super lenses [8], wavefront shaping devices [9], and absorbers [10–12] for different frequency bands. Among multifarious metamaterials, perfect metamaterial absorbers can be employed in many application areas like plasmonic sensors [10, 13], selective thermal emitters [14, 15], imagers [16], and solar cells [17], have attracted great attention. Since the first perfect metamaterial absorber (PMA) was experimentally demonstrated by Landy et al. [18], many kinds of perfect metamaterial absorbers have been demonstrated: wide band [19–21], polarization-independent [22, 23], controllable band [24, 25], and multiband [26, 27]. In principle, the perfect absorber is achieved by minimizing the reflection and eliminating the transmission. Most perfect metamaterial absorbers are consisted of two metallic layers separated by a dielectric layer. The front metallic subwavelength structure layer is for minimization of reflection by impedance matching, and the back continuous metallic layer is for blocking the transmission. In order to achieve, especially, multiband PMA, single-layer structure with two or multiple resonators in a super-unit and multiple vertically stacked metallic layers have been proposed. For example, Huang et al. [28], Bhattacharyya et al. [29, 30], Shen et al. [31], Wang et al. [32], Cao et al. [33], and presented the multi-band absorbers based on multiple different-sized sub-unit elements in a coplanar super-unit structure. Zhu et al. [34], Hu et al. [35], He et al. [36], and Bhattacharyya et al. [37] demonstrated the broadband absorbers by stacking multiple different-sized metallic patterns. However, these designs are mainly achieved by utilizing the combination of the fundamental resonance of the regular single-band absorber. There is little novelty in designing the new types of multi-band absorbers. Moreover, these approaches face two challenging problems: The former has a problem of larger unit size, and the latter has technical difficulties in fabrication, in particular at higher frequencies such as terahertz, infrared, and visible regions. Therefore, in order to design the multi-band (or broadband) PMA, it is urgently necessary to reduce the unit size and the stacked layers as much as possible.

In this paper, we propose a simple and effective design of four-band terahertz PMA utilizing single #-shaped strip pattern. The absorber shows four distinctive absorption peaks at frequencies of 1.09, 2.76, 3.23, and 4.04 THz, with absorption rates of 99.87, 98.46, 99.77, and 99.22 %, respectively. Compared with the previous reported multiband absorbers [33, 38–41], the proposed absorber has two advantages: First, and most importantly, the straight strip is the simplest structure to achieve multiband perfect absorber and thus overcomes the problems of both the large unit size and fabrication. Secondly, the physical mechanism of the proposed absorber stems from the combination of LC resonance and electric dipole response of the single patterned metallic structure, making the resonance absorption mechanism has a new novelty. Therefore, such a simple device may provide a desirable absorption method and operation to select frequency band at THz frequencies and thus lead to most practical applications.

2 Design and simulation

Figure 1 shows the proposed structure consisting of a single-layer #-shaped metallic strip and a metallic ground plane separated by a dielectric layer, which are located on a \(500\hbox {-}\upmu{\hbox{m}}\)-thick silicon substrate with a refractive index constant of 3.4. The optimized geometry dimension parameters of the unit cell are as follows: \(l_{1}=60\,\upmu{\hbox {m}}, l_{2}=28\,\upmu\hbox {m}, l_{3}=20\,\upmu\hbox {m}, w=4\,\upmu\hbox {m}\), and \(p=70\,\upmu\hbox {m}\). The thickness of the dielectric polyimide layer is set to \(t_{2}=11.5\,\upmu\hbox {m}\) with a refractive index constant of \(1.8+i0.06\) in order to optimize the impedance matching to the free space [22]. Both of the two metallic layers are made of gold with a conductivity of \(\sigma =4\times 10^{7}\,\hbox {S/m}\) and a thickness of \(t_{1}=0.5\,\upmu\hbox {m}\) [22]. We perform the simulation using finite-difference time-domain (FDTD) software package [42], where a single unit cell is illuminated by a normally incident plane wave with the electric field polarized along the x-direction as the excitation source. A perfectly matched layer (PML) is used along the z direction, and period boundary conditions are used in the x and y directions. The absorption coefficient A is obtained by using \(A=1-T-R\), where T is transmission coefficient and R is reflection coefficient. The transmission is zero because the thickness of the metal ground plane (\(t_{1}=0.5\,\upmu\hbox {m}\)) is much larger than its skin depth, and then the absorption is calculated by using \(A=1-R\). Perfect absorption may be achieved when R is close to zero [i.e., the impedance (\(z=1\)) matches that of free space].

Fig. 1
figure 1

a Schematic of the #-shaped strip of the proposed four-band PMA and b the cross section of the complete four-band PMA

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3 Results and discussion

Fig. 2
figure 2

a The simulated reflectance, transmittance, and absorption spectra by the four-band PMA as a function of frequency. b Dependence of the absorption spectra of the proposed four-band PMA on two different loss conditions (lossy and lossfree)

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Figure 2a presents the simulation results of the S-parameters and absorption spectrum. It can be seen from Fig. 2a that there are four distinct absorption peaks located at \(f_{1}=1.09\,\hbox {THz}, f_{2}=2.76\,\hbox {THz}, f_{3}=3.23\,\hbox {THz}\), and \(f_{4}=4.04\,\hbox {THz}\), with absorption rates of 99.87, 98.46, 99.77, and 99.22 %, respectively. The large light absorption of the four resonances is mainly attributed to the hybrid of the LC resonance and electric dipole response of the patterned structure, and understanding of such four-band absorption is illustrated by investigating the surface current, electric field and magnetic field distributions (see below Figs. 6, 7, and 8).

We next investigate the origin of loss to understand the contributions of each part of the proposed structure. Figure 2b shows the absorption spectra of the proposed structure under two different loss conditions (lossy and lossfree) of the dielectric layer. It can be found from the simulated results that the positions of these absorption peaks are unchanged by changing loss conditions of dielectric, which agree well with the previous reports [43]. Furthermore, the simulation results also demonstrate that the majority of the energy is dissipated as dielectric loss in the dielectric layer and only a fraction of the energy is dissipated as ohmic loss in the metal layer. This feature makes it less sensitive to the conducting properties of the metal layer and could have great application prospects in some fields where the electromagnetic energy needs to be collected by the dielectric layer.

Fig. 3
figure 3

a Simulated absorption spectra at different polarization angles for TM wave. b Dependence of the simulated absorption spectra on different incident angles for TM wave

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We also analyze absorption spectra as a function of polarization angle for TM wave, where the angle of incidence is fixed at \(0^{\circ }\). The polarization angle is defined as the angle between the polarized direction and the x axis of the coordinate, which is equivalent to rotating the designed absorber structure while fixing the polarized direction. Figure 3a presents the simulated absorption spectra of the proposed absorber at different polarization angles of the normal incidence. It can be found from Fig. 3a that the first two absorption peaks hardly relay on the polarization angle, while the latter two absorption peaks depend strongly on the polarization angle of the incident light. Thus, with this structure, we are able to tune the absorption intensity of the proposed absorber by changing the polarization angle of the incident light. This feature makes the proposed structure very useful in manipulation polarization of light and detecting the electromagnetic waves with specific polarization [44]. In addition, we investigate the effect of incidence angle on the absorption spectra. The incidence angle is defined as the angle between the wave vector and the z axis of the coordinate. Figure 3b shows the simulated absorption spectra of the proposed absorber at different incidence angles, where the azimuthal angle is fixed at \(0^{\circ }\). From Fig. 3b, we can see that the absorption spectra of the proposed four-band PMA are sensitive to the incidence angles. Also, the additional peaks increase or decrease depending on the incident angles.

Fig. 4
figure 4

The relative impedance \([z(\omega )]\) for the proposed four-band PMA shown in Fig. 2a. The dash straight lines denote the positions of the four absorption peaks

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To understand the physical origin of the proposed four-band perfect absorber, we calculate the relative wave impedance \([{\tilde{z}}(\omega )]\). The relative wave impedance is retrieved by using the simulated scattering parameters. The relative wave impedance equation is given as follows [45]:

$$ {{\tilde{z}}}(\omega ) = \sqrt{\frac{{{{(1 + {S_{11}})}^2} - S_{_{21}}^2}}{{{{(1 - {S_{11}})}^2} - S_{_{21}}^2}}} $$
(1)

where \(S_{11}\) and \(S_{21}\) are the scattering parameter i.e., complex reflection and transmission coefficients. It can be seen from Fig. 4 that the real and imaginary parts of the relative wave impedances close to unity and zero, respectively. It demonstrates the relative wave impedances are nearly matched to free space, so the reflection coefficients of the proposed four-band PMA are very low. Due to the absence of transmission, nearly perfect absorption at these four frequency positions can be realized.

Fig. 5
figure 5

Comparison of extracted constitutive electromagnetic parameters: a real parts of \(\varepsilon _{\text{eff}}\) and \(\mu _{\text{eff}}\) and b imaginary parts of \(\varepsilon _{\text{eff}}\) and \(\mu _{\text{eff}}\)

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In addition, The constitutive electromagnetic parameters for the effective permittivity and effective permeability of the absorber are retrieved according to the references [46, 47], as shown in Fig. 5. It can be found that they are very close with each other at the four absorption peaks, as shown in Table 1. The results demonstrate that the input wave impedance of the medium is perfectly matched with the wave impedance of free space, and thus lead to minimum reflection from the surface of the absorber and consequently occurrence of maximum absorption.

Table 1 Comparison of retrieved constitutive electromagnetic parameters of the proposed four-band PMA
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Fig. 6
figure 6

The surface current distributions on the top metal surface and bottom ground metal plane of the proposed four-band PMA at the frequencies of absorption, respectively

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To understand the underlying physical mechanism of the observed four-band absorption behavior, we investigate the surface current distributions on #-shaped strip resonator and the bottom ground metal plane at the four absorption frequencies, as shown in Fig. 6. The indicative red arrows are added to mark the foremost direction of the surface current density in Fig. 6. For resonance at 1.09 THz, both the LC and electric dipole resonances contribute to the resonance absorption peak. Surface currents flowing around the upper and lower gaps of the #-shaped strip resonator are originated from LC resonances, as the arms serve like an inductance L and the gap between the two arms provides the capacitance C. the straight red arrows indicates the strong electric dipole resonances are excited, which arise from charges accumulation at the left and right corners (see Fig. 6a). In addition, there also exist other electric dipole resonances arising from charges assembly at the left and right arms of the middle rectangular loop in the #-shaped strip resonator, as indicated by the crooked red arrows in Fig. 6a. For the resonance at 2.76 THz, the currents concentrated on the upper and lower gaps of the #-shaped strip resonator are also originated from the LC resonances. The weak currents flowing around the top and bottom arms of the middle rectangular loop in the #-shaped strip resonator on account of the electric dipole resonances, as shown by the crooked red arrows in Fig. 6b. For the resonance at 3.23 THz, it is caused by only the electric dipole resonances (see Fig. 6c). The fourth absorption peak is caused by the mixture of LC and electric dipole resonances of the patterned (see Fig. 6d). For the four resonance absorption peaks, the antiparallel currents occur at the #-shaped strip resonator and the ground metal plane (see Fig. 6). As a result, the magnetic polaritons can be formed, and thus induce strong magnetic resonances.

Fig. 7
figure 7

Distributions of the electric field [abs (E)] in the interface between the top metal layer and the dielectric layer for the proposed four-band PMA at resonance frequencies of 1.09 (a), 2.76 (b), 3.23 (c), and 4.04 THz (d), respectively

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Fig. 8
figure 8

Magnetic field [abs (H)] distributions in the \(x\hbox {-}z\) plane at \(y=35\upmu\hbox {m}\) at resonance frequencies of 1.09 (a), 2.76 (b), 3.23 (c), and 4.04 THz (d), respectively

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To further understand the underlying mechanism of the four absorption peaks, we also investigate the simulated electric field distributions in the interface between the top metal layer and the dielectric layer corresponding to four absorption peaks in Fig. 7. At 1.09 and 2.76 THz, the electric field is mainly distributed in the ends of the #-shaped strip shown in Fig. 7a, b, and it is mainly concentrated in the ends of the #-shaped strip as well as the left and right arms of the middle rectangular loop in the #-shaped strip resonator at 3.23 THz in Fig. 7c. In addition, in Fig. 7d, the electric field is distributed in the left and right ends of the horizontal bars and the middle arms of the vertical bars at 4.04 THz. According to the electric field distribution at different resonance absorption frequencies, it is demonstrated here that the resonance absorption of the PMA is induced by the #-shaped strip resonator. The magnetic field distribution in the \(x\hbox {-}z\) plane at \(y=35\,\upmu\hbox {m}\) is shown in Fig. 8a–d at four resonance absorption frequencies. Thus, it is easy to understand that the resonance absorption peaks are generated by magnetic resonance between two metal layers so that the magnetic field distributions between two metallic layers at four resonance frequencies can be observed. Thus, in regard to the discussion above, we can propose that all resonance absorption frequencies are attributed to the LC resonance and electric dipole response of the #-shaped strip resonator, and although the field distribution characteristics from the #-shaped strip resonator are different, their mechanisms are mainly the combination of the LC resonance and electric dipole resonance.

Fig. 9
figure 9

Dependence of the absorption spectra on the thickness change of the dielectric polyimide layer for the proposed four-band PMA

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It is worth noting that the off-resonance absorption of the proposed four-band PMA is quite large between 2.5 and 3.5 THz. That means the second (2.76 THz) and the third (3.23 THz) absorption peaks are not clearly distinguished. In fact, the off-resonance absorption can be depressed by decreasing the thickness of the dielectric polyimide layer, as shown in Fig. 9. It can be seen that the off-resonance absorption gradually decrease with the decrease in the thickness (\(t_{2}\)) of the dielectric layer. Particularly, the off-resonance absorption is only about 33 % when \(t_{2}\) is equal to \(8.5\,\upmu\hbox {m}\), and the four separated absorption bands whose peaks with high absorption (average over 97 %). In addition, it can be also seen that all the absorption peaks shift to low frequency when increasing the thickness of the dielectric polyimide layer. According to LC circuit model: \(f\propto \frac{1}{\sqrt{LC}}\), when \(t_{2}\) is increased, the coupling capacitance between the metal #-shaped strip and the ground plane decreases and the metal #-shaped strip inductance increases, the total result that is the resonance frequency decreases [48]. Therefore, when \(t_{2}\) is increased, the resonance absorption peaks show red shift. From the point of interference theory, it can verify our results. As the thickness of dielectric layer (\(t_{2}\)) increases, the surface destructive interference induced the absorption peaks gradually red shift [49, 50].

Fig. 10
figure 10

a Dependence of the absorption spectra on the thickness change of the over-layer. b Dependence of the absorption spectra on the change of the surrounding index

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In fact, the proposed four-band PMA can be used for sensor by adding a dielectric layer on the top of the absorber. Figure 10a shows the simulated results for different thickness of the over-layer. It can be seen from Fig. 10a that all the four resonance absorption peaks shift to the low frequency with the increase in the over-layer. The shift of the absorption peak can be explained by the variation of the capacitance of the overall structure. Based on the equation \(f\propto \frac{1}{C}\), the increment of the thickness of over-layer leads to increase capacitance; therefore, the resonance absorption peak shows a red shift with the increase in the over-layer [48, 51].

We next investigate the refraction index (RI) sensing with the proposed absorber, as shown in Fig. 10b. It can be found from Fig. 10b that the frequencies of the modes \(f_{1}, f_{2}, f_{3}\), and \(f_{4}\) are all red shift gradually when increasing the RI. It is clear that when the RI of the added dielectric layer increases, the capacitance of the overall structure also increases. Consequently, the resonance absorption peaks gradually shift to low frequency with the increase in the RI. Interestingly, the frequency decrease speed of modes \(f_{1}\) and \(f_{3}\) are smaller than that of modes \(f_{2}\) and \(f_{4}\). Specially, when the RI is increased from 1.0 to 1.4, the frequency shifts of the modes \(f_{1}, f_{2}, f_{3}\), and \(f_{4}\) are 0.061, 0.164, 0.124, and 0.174 THz, respectively. In other words, the proposed four-band spectral sensor, in particular the modes \(f_{2}\) and \(f_{4}\) of the structure could provide a high bulk RI sensing if a high-resolution spectrometer is used.

4 Conclusion

In conclusion, we present a simpler and effective design of four-band terahertz PMA composed of a metallic #-shaped strip resonator and metallic ground plane separated by a dielectric layer, which locates on a \(500\,\hbox {-}\upmu\hbox {m}\)-thick silicon substrate. The structure not only provides a simple unit cell but also apparently decreases the stacked layers, which are required for the four-band application. Four distinct absorption peaks are seen at 1.09, 2.76, 3.23, and 4.04 THz, with the average absorption over 99 %. The mechanism of four-band absorption is mainly originated from the LC resonance and electric dipole response of the patterned structure, and understanding of the multi-band absorber is illustrated by investigating the current density, electric field, and magnetic field distributions at four resonance absorption frequencies. Moreover, the roles of the parameters are investigated to explore the sensing performance of the absorber.