## 1 Introduction

In recent years, terahertz (THz) electromagnetic waves with the frequency range from 0.1 to 10 THz have become the hotspot of research. Benefiting from the newly emerging metamaterials, THz devices have spawned a series of significant applications in many kinds of fields such as imaging spectrum systems communications, and nondestructive sensing [1], [2], [3], [4], [5]. Among these applications, THz absorbers have become the crucial component. Since Landy et al. first proposed the concept of metamaterial absorbers (MMAs) [6], MMAs have attracted great research attention. The mechanism of MMAs to realize the near-perfect absorption is the impedance matching with the surrounding air. Recently, different kinds of MMA designs with matched impedance have been investigated in a wide frequency region [7], [8], [9], [10], [11], [12]. However, these perfect MMAs usually work in a narrow band, typically not more than 20% of the center frequency, which is caused by their resonance features. Nevertheless, broadband absorption is highly desirable in many applications, such as electromagnetic stealth, detectors, and THz imaging [8], [13], [14].

Thus far, a lot of efforts have been made to break the bandwidth limitation of THz absorbers. Typically, the common approaches to extend operation bandwidth are stacking multiple layers consisting of different antenna arrays and constructing composite supercell with different resonance modes. In 2014, Zhu et al. implemented a metal-dielectric multilayer composite to realize a broadband absorber with an absorption above 80% and a full-width at half-maximum of 127% in the range of 0.7–2.3 THz [15]. In 2017, Kenney et al. proposed a planar THz absorber composed of a supercell of fractal crosses to acquire an average absorption of 83% from 2.82 to 5.15 THz [16]. These works demonstrated the ability to broaden the bandwidth of the approaches but at the expense of a complex supercell or an increased fabrication complexity. A strategy to fix the problems is engineering heavily doped silicon. In 2012, Pu et al. initially exploited the heavily doped silicon to realize broadband absorption. Then, they proposed a binary grating structure on heavily doped silicon to acquire an absorption larger than 90% in frequencies from 1 to 4 THz [17]. Then, a lot of works based on doped silicon realized broadband absorption in the THz region [18], [19], [20], [21]. These works have made a great progress in broadband THz absorbers. However, fabrication is complex in multilayer doped silicon and the bandwidth needs to be further improved. Recently, a better strategy, dispersion management of metasurfaces for bandwidth extension, has been proposed [22], [23], [24]. By conjugating the compensation of the dispersion of metasurfaces and the frequency-dependent phase shift of the dielectric spacing layer, ultra-broadband THz devices can be realized. In particular, with the help of coherent control, ultra-broadband absorption with ultrathin film has been demonstrated [25], which breaks the fundamental limitation of bandwidth and thickness.

In this paper, we used the capability of dispersion management to extend the bandwidth of THz absorbers. As a proof-of-concept, a metamirror with the dual-metasurface was designed for a broadband near-unity THz absorber, which was composed of two polarization-independent metasurfaces separated from a metallic ground by dielectric layers with different thickness. Benefiting from the fully released dispersion management ability in adjusting the dimensions of the metasurface, we obtained the ultra-broadband THz absorber in the frequency range from 0.52 to 4.4 THz with an absorbance above 90%. The experimental results verified the ability of dispersion management in designing broadband absorbers and the performance of the designed absorber. The underlying physical mechanism of dispersion management was interpreted in the general equivalent circuit theory and transmission line model. Besides, we employed catenary electric fields to further interpret the physics behind the dual-metasurface. The absorber is polarization insensitive due to the symmetry of dual metasurfaces. Importantly, we investigated the impact of alignment deviation between the dual metasurfaces on the performance of designed absorber and simulated results showed a slight declination, which make it easy to be fabricated. We believe that the ultra-broadband absorber could find important applications in THz devices.

## 2 Structure and results

The schematic of the array and unit cell of this ultra-broadband THz absorber is shown in Figure 1A and B. The absorber consists of dual metasurfaces separated from the metallic ground plane by SU-8 photoresist layers with different thickness (i.e. metamirror), and an extra SU-8 layer is capped on top of the metasurface. Figure 1C and D gives the top views of the dual metasurfaces. The metasurfaces are composed of arrays of square loops and patches, and metasurface 2 has twice the period of metasurface 1. The metallic parts are both made up of chromium with a conductivity of 2.2×10^{5} S/m, and the thickness of the ground plane and dual metasurface are 300 and 150 nm, respectively. The permittivity of SU-8 photoresist is 2.79+*i*0.31 [15], and the thicknesses of trilayer insulators are *h*_{1}=20 μm, *h*_{2}=16 μm, and *h*_{3}=20 μm. The other detailed geometric parameters of the broadband THz absorber are optimized as *P*=2×*P*_{1}=100 μm, *l*_{1}=35 μm, *l*_{2}=23 μm, *l*_{3}=80 μm, *l*_{4}=60 μm, and *w*=16 μm.

The broadband THz absorber was investigated with a commercial software package (CST Microwave Studio) using the finite-element method. In the simulations, the unit cell boundary conditions were applied in the *x* and *y* directions, whereas the top and bottom boundaries normal to the *z*-axis were set as open. Due to the symmetry of the structure, we applied the normally incident TM-polarized plane wave with the polarization of the electric field along the *x* direction. The absorption of the THz absorber *A*(*ω*) can be calculated by

where *T*(*ω*) and *R*(*ω*) are the transmittance and reflectance of the absorber, respectively. Because the metallic ground plane is thick enough to prevent the THz wave transmission, the absorption can be simplified as *A*(*ω*)=1 – *R*(*ω*). Figure 1E shows the calculated absorption spectra of the ultra-broadband THz absorber in the frequency range from 0 to 5 THz. As depicted in Figure 1E, the incident THz wave ranging from 0.52 to 4.4 THz can be efficiently absorbed with an absorption above 90%, and the corresponding relative absorption bandwidth is nearly 143%. For a physically realizable broadband absorber, the Rozanov limit indicates that, for any metal-backed absorber under a normal incidence, its total thickness *d* must be larger than a theoretical limit for the given frequency response of the absorption [21], i.e.

where *λ* is the free-space wavelength. For the broadband absorption shown in Figure 1E, the corresponding theoretical limit is estimated to be 53.07 μm, which is close to the thickness (56 μm) of the designed absorber.

To prove the simulated results, a sample was fabricated by a standard optical lithography process by employing the same structural and material parameters used in the above simulations (more fabrication details are depicted in the supplementary material). Figure 2A and B shows the optical microscopy image of the fabricated sample. The sample was measured by a commercial THz-time domain spectroscopy (THz-TDS; ADVANTEST TAS7400SP) with a frequency range from 0.1 to 1.5 THz. In the measurement, an opaque gold film was used as the reference reflection mirror. Figure 2C depicts the measured results at an incident angle of 20°, which is the minimum incident angle of THz-TDS for TE and TM polarizations. For a broader study of the THz properties, the sample was also characterized by a Fourier transform infrared (FTIR; Bruker v80) spectroscope equipped with a bolometer detector that offered a valid range from 1.5 to 5 THz. The measured spectral characterization was obtained using an 11° angle reflection accessory. Figure 2D gives the measured result under polarization-independent incidence due to the lack of a linear-polarized plate in this frequency range. From the spectrum, we can obtain an absorbance that exceeds the 90% range from 1.8 to 4 THz. Looking at the measured absorption rate in Figure 2 as a whole, the fabricated THz absorber remains a high absorbance (almost more than 90%) in the designed frequency range. The measured results agree well with the simulated results in all, which demonstrates the ultra-broadband and high absorbance of the absorber. Note that there is also deviation between measured results and simulated results, which may be caused by the fabricated imperfection and accuracy of measured instrument. Besides, an oblique incident angle makes a slight difference of absorption between TE and TM polarizations (as depicted in Figure 2C).

## 3 Theoretical analysis

Subsequently, we resort to the transmission line model and general equivalent circuit theory to interpret the underlying physical mechanism of the broadband absorber. Due to the deep-subwavelength thickness, such dual-metasurface layer can be regarded as thin impedance sheets. We present an analytical circuit model to interpret the sheet impedance. In this case, arm and gap behave as inductors and capacitors, respectively. Figure 3A and B shows the general equivalent circuit theory for the dual metasurface, and according to the classic circuit, the frequency-dependent impedance can be expressed as [26], [27]

First, we consider the equivalent impedance of metasurface 2. In Equation (4), *Z_{g}*

_{1}and

*Z*

_{g}_{2}are the equivalent frequency-dependent impedance of two resonant circuits in metasurface 2. Besides,

*Z*

_{g}_{1}and

*Z*

_{g}_{2}can be expressed as

The equations for calculating the resistance, capacitance, and inductance in the equivalent circuit can be expressed as [28], [29]

*R*_{0}=1/(*σ*_{Cr}*t*) is the resistance of Cr film (*σ*_{Cr} and *t* are the conductivity and thickness of chromium, respectively). *ε*_{eff} is the permittivity of SU-8 photoresist. In Equation (9), *C*′ and *C*″ are induced by the outer and inner gaps, which can be expressed as

In addition, *F* can be expressed as

*θ* is the angle of incidence, *λ* is the wavelength, and *G* is the correction term [29]. Interestingly, the first term of Equation (12) is characterized by the catenary of equal strength, which was called catenary dispersion [30]. In the same way, we can obtain the resistance, inductance, and capacitance in the general equivalent circuit theory of metasurface 1 as follows:

Assuming that a plane wave normally impinges on the metamirror, due to reflections occurring at the metasurface and background plane, both the forward and backward scattering waves exist in the dielectric spacer and surrounding space, as depicted in Figure 3C. According to the transmission line model, we can obtain the general equivalent circuit theory of the THz absorber, as shown in Figure 3D. *Z_{s}*

_{1}and

*Z*

_{s}_{2}are the equivalent impedance of the dual metasurfaces.

*Z*

_{d}_{1},

*Z*

_{d}_{2}, and

*Z*

_{d}_{3}are the characteristic impedance of trilayer SU-8 colloids, respectively.

*β*

_{0}and

*β*are the propagation constant of the free space and medium, respectively.

_{d}*Z*

_{0}and

*Z*

_{in}are the characteristic impedance of free space and input impedance of the whole circuit model. The input impedance of each layer can be described as [31]

In Equations (14) and (15), *β_{d}* and

*Z*(

_{di}*i*=1, 2) can be expressed as

*η*_{0}≈377Ω is the free-space impedance. *ε_{d}* and tan

*δ*are the relative permittivity and dielectric dissipation factor of SU-8 colloids, respectively. Using the iterative algorithm, we can recursively derive the input impedance of the entire general equivalent circuit theory from the bottom layer. The reflection coefficient can be calculated by the following equation:

Then, we can calculate the absorbance of the proposed broadband absorber by the simplified form of Equation (1). The theoretical results of the transmission line model are shown in Figure 3E, which shows good agreement with the numerical results by full-wave simulation. Based on this approach, we can further expand the bandwidth of the THz absorber by stacking more metasurfaces.

In an effort to further illustrate the relationship between the effective impedance and the geometry of the metasurface, the amplitude of electric field for the dual metasurfaces in the *x*-*z* plane at resonant frequencies (0.6 and 2.5 THz) is plotted in Figure 4A, C, D, and F. In addition, we illustrate the instantaneous electric field distributions under normal incidence at these resonant frequencies, which is depicted in Figure 4B and E. Apparently, the strong resonance occurred in the gap of the adjacent unit cell are responsible for the absorption at low frequency (0.6 THz). For higher frequency (2.5 THz), the resonance occurred in the gap of a single unit cell; additionally, there appears a hybrid mode between the two metasurfaces. To take a closer look at this issue, we employed the extracted E-field amplitude at these resonances. Interestingly, all profiles in the gaps can be well described by the well-known catenary curves in architecture (the fitting curves are shown in blue solid lines and the generalized catenary model is given in the supplementary material). Due to the THz frequency range, metal can be treated as the perfect electric conductor (as mentioned above) and surface plasmon polaritons do not exist. Thus, the amplitude of the E-field along the center of the gap can be fitted with a curve using the catenary function, and excellent agreement is found [32], [33]. This catenary optical field is also called as microscopic meta-surface-wave (M-wave) [34]. When changing the geometry of the metasurface, the catenary field distributions will be altered as well, leading to the modulation of the effective impedance at the corresponding metasurface. By properly optimizing the profile of the catenary field and coupling the different catenary field in structure, the desired effective impedance of metasurfaces can be obtained, thus achieving broadband absorption. The discrepancy between the extracted field and the fitted catenary in Figure 4D is caused by the coupling of closely adjacent catenary field in metasurface 1, which implies a hybrid mode. Akin to the dispersion management (general equivalent circuit theory and transmission line model), we believe that this catenary E-field model offers a new interpretation to understand the physics behind resonance-based metasurfaces. Based on the above analysis, engineering the dispersion of metasurfaces has the potential to realize broadband absorption.

Then, we investigated the impact of the alignment deviation between the dual metasurfaces on the performance of the THz absorber. The parameters *x*- and *y*-shifts (as depicted in Figure 5A) represent the deviation distance between the dual metasurfaces along the *x* and *y* directions. The range of *x*- and *y*-shifts is from −25 to 25 μm, including the strict alignment to the maximum deviation. The simulated results of the parameter sweep of *x*- and *y*-shifts with an increment of 1 μm are plotted as a band in Figure 5B. The results reveal that the performance of the broadband high-efficiency THz absorber is sustained in all cases. Thus, the designed absorber exhibits high manufacturing tolerance, which makes the absorber easy to be fabricated. For practical applications, the incident THz wave is not always impinging normally into the THz devices. Therefore, we have to investigate the performance of the absorber under oblique incidence. The simulated absorption at different incidence angles for TE and TM polarizations are shown in Figure 5C and D. For TE polarization (Figure 5C), the absorption intensity drops when the incident angles are larger than 50°, but the bandwidth seems to experience little changes. For TM polarization (Figure 5D), the absorption intensity has weak dependence on the incident angle even up to 70°. The results indicate that the absorber remains at high performance in the case of wide-angle incidence.

## 4 Conclusion

In summary, we apply the dispersion management for metamirror with the dual-metasurface to extend the operating bandwidth of the THz absorber. As a proof-of-concept, an ultra-broadband THz absorber with an operating bandwidth range from 0.52 to 4.4 THz is designed and experimentally characterized. As the general equivalent circuit theory and transmission line model confirm that the broadband absorption is induced by the coupling of multiple resonances in dual metasurface, broader bandwidth and better performance could be obtained by stacking more metasurfaces or adopting elaborate metasurface. With the help of the catenary optical model, we can better understand the E-field distribution of the metasurface at resonant frequency and design the structures more efficiently. Moreover, ignoring the deviation of the alignment makes the absorber closer to the applications and potential for further bandwidth extension. The strategy of dispersion management to extend the working bandwidth can be also applied for other designs of broadband metamaterial-based devices.

See the supplementary material for the supporting content.

The authors declare no competing financial interest. We thank Ms. Yuan for her help with the fabrication. This work was supported by the National Natural Science Funds of China (grant 61575032).

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## Footnotes

## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2018-0110).