1. Introduction

Metamaterials are artificially engineered materials that are made of periodic sub-wavelength-sized unit cell structures and these attain exotic and unique properties that are difficult to realize with natural materials such as negative reflection/refraction, the reversal of Doppler shift, and Cherenkov radiations [1–7]. Metamaterials have become a research hotspot because of their potential applications in sensing, stealth technology, imaging, perfect absorption and so forth [8–15]. Within the context, Landy et al. firstly proposed the concept of metamaterial-based absorber [16]. Metamaterial-based absorbers achieve near-unity absorption by exciting both electric and magnetic resonance modes simultaneously. To achieve perfect absorption, several designing techniques have been investigated like square rings, split-ring resonators, trapezoidal arrays, cross resonators, and all- dielectric based metasurfaces in a single band, multiband and wide-band spectrums, ranging from microwave to optical frequency regimes [17], [18–22].

Terahertz band, frequency ranging from 0.1 THz to 10 THz is attractive research topic due to their immense applications in biomedicine, communications and security [23–28]. In the last decade, THz technology came into limelight of several researchers because of paucity of functional materials with suitable performance in this regime. Terahertz wave shows a lack of response with the naturally existing media and it has been ignored for a long time; therefore, it hindered the development of devices in this regime [29,30]. Fortunately, metamaterials have been attracting tremendous attention to solve this problem. Many research groups have proposed and experimentally realized high-performance metamaterial-based absorbers in THz regime [18,19,31][32]. Nevertheless, most of the proposed designs have drawbacks of narrow bandwidth, which limits their usage for a specific range of frequencies and makes them unsuitable for practical applications. Generally, the bandwidth of THz absorbers can be extended by three ways; in the first approach different sized sub-unit cell structures are combined to form a coplanar super unit cell [31,33]; in the second method different metal-dielectric-metal layers are vertically stacked to form a unit cell [34,35], in the third one, some composite structure or graphene or doped silicon is used [36–37]. However, the main drawback of the first approach is, it makes angular dependent structures as larger size unit-cell leads to many local interactions between sub-unit cells. Additionally, the extension of bandwidth in this approach is also limited because a few sub-unit cells exist in the coplanar structure. The second and the third approach are the most suitable options for realizing broadband or ultra-broadband absorption. Nonetheless, cost and fabrication complexity restrict the use of these approaches. Numerous THz metamaterial based absorbers have investigated to achieve wide absorption response with different metal-dielectric-metal combination including single and multi-layers [38–41]. In Ref. [42], Cumming et al. proposed a cross fractal metamaterial absorber. It is comprised of three stages of fractal to form a supercell. It is depicted that absorber exhibits above 80% absorptivity from 3.01 to 4.84 THz. All of these mentioned absorbing devices either composed of multilayer stacking configuration or manifest less absorption bandwidth. Therefore, designing and realization of high-performance, ultra-broadband, miniaturized, cost-effective and single layer THz metamaterial absorber is a highly significant and desirable research topic in the field of optics.

A geometrical structure whose sub-unit cells contain the same statistical characteristics as the basic unit cell is called fractal geometry. Due to unique space-filling and self-similarity, fractal geometry can be used in many fields to attain wideband and miniaturized features. They create multiple resonance response due to their repeated patterns. In the field of metamaterial absorbers, the fractal structure can be used to enhance the bandwidth of the perfect absorbers. Due to their inherent multiple resonance characteristics owing to the self-similar structures, the bandwidth of the microwave and optical devices (antennas, absorbers, and filters) can be extended [43,15,44]. By using such geometry capacitance and inductance of a resonator can be tuned to equalize the metasurface impedance with the free-space impedance. Resultantly fractal geometries provide a new degree of freedom to achieve a variation in resonance frequency without conventional physical dimensions scaling rule [45–47]. Consequently, an efficient and wideband metamaterial absorber can be designed with a more compact size.

In this paper, we aim at investigating Pythagorean-tree fractal geometry metasurface absorber operating in the THz regime. Three different fractal order structures are designed and simulated to achieve the desired goal. Finally, second order fractal geometry is selected to achieve broadband absorption behavior. It is depicted that, the designed metasurface absorber shows near-unity absorption from 8.4 - 9.7 THz and above 80% absorptivity is observed for the frequency span of 7.5 - 10 THz. Further, both the transverse electric (TE) mode and transverse magnetic (TM) mode are excited to explore the absorption properties for different oblique incidence angles of EM waves. To understand the physical mechanism of absorption, surface current density and electric field are taken up. Proposed absorber shows excellent results to oblique incidences up to 30°. By proper scaling and carefully selecting the geometrical dimensions of the proposed metasurface absorber, the absorption characteristics can be tuned to any required frequencies.

2. Designing and simulation

Figure 1 shows the schematic of unit cell of the proposed metamaterial absorber operating in the THz regime. The unit-cell of our proposed absorber is comprised of three layers with top metasurface made of assembly of gold (Au) square patches, called Pythagorean-tree fractal structure. A lossy material of Polyimide with refractive index (RI) as n = 1.68 + 0.06i [42] is used as a dielectric spacer backed by a bottom ground metallic layer of Au, which serves as back reflector. The conductivity of Au is 4×10⁷ S/m. The thickness of top metasurface is kept as ${t_t}\; = \; 1.2\; \mathrm{\mu} m$, the thickness of the dielectric spacer as ${t_d}\; = \; 19\; \mathrm{\mu} m$, and the thickness of the back reflector is kept as t_g = 2 µm.

 

Fig. 1. Schematic of the unit cell. (a) Top surface view (b) 3-D view and (c) Configuration of the proposed PTFMA.

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Figure 1(c) illustrates the proposed PTFMA configuration. The TE- and TM-mode are shined from + z-axis. All simulations are performed for a single unit cell by using commercially available CST Microwave Studio software. The unit cell boundary conditions were applied in x and y-axis whereas open add space for z-axis. The unit cell periodicity is kept as P = 69 µm while length and width of main square patch is taken as L = 20 µm. Dimensions of higher order fractals are taken with respect to L (labelled in Fig. 1(b)). Thickness of dielectric layer is taken as t_d=19 µm.

3. Results and discussion

Now the absorption through PTFMA is taken up. Firstly, we analyzed the absorption features of zeroth-order to the second-order fractal structure to achieve wideband characteristics as shown in Fig. 2. Absorption characteristics of both TE- and TM-modes are taken into account. We started with the zeroth-order fractal stage which consists of a single square-shaped block of size L× L. The absorption feature of the zeroth-order fractal is illustrated in Fig. 2(a). It is examined that the absorber manifests above 80% absorption for the frequency span of 7.5 THz to 10 THz for both the TE- and TM-mode, and also, two near-unity absorption peaks are noticed at 8.45 THz and 9.2 THz. Due to the four-fold symmetry in the zeroth-order structure of the unit cell, it shows the polarization-insensitive response to incident wave; therefore, it attains the same absorption results for both TE- and TM-modes (as obvious in Fig. 2(a)). Secondly, to improve the absorption bandwidth, the first-order fractal stage is designed from the previous zeroth-order fractal through connecting the two equidimensional squares of size 4L/5 × 4L/5 at the top edges of the main square. Figure 2(b) exhibits the absorption response of the first order fractal structure and it shows above 80% absorptivity from 7.4 THz to 10 THz for TE-mode excitation and more than 76% absorption is depicted for TM-mode ranging from 6.8 THz to 10 THz. The four-fold symmetry is lost in this case, hence both TE- and TM-modes exhibit different results under the normal incident (θ = 0⁰) and it behaves as a wideband band absorber with the improved absorption as compared to the zeroth-order fractal structure. Finally, the second-order fractal geometry is introduced by connecting two equal-sized smaller square patches with a size of 2L/5 × 2L/5 at the two different top edges of all the squares that were present in the first-order fractal stage. The absorption performance of the final stage is depicted in the Fig. 2(c) and it displays above 98% absorption between 8.4 THz to 9.7 THz for TE-mode, similar behavior is observed for TM-mode, nevertheless, a small dip is observed at 8.7 THz with absorption is about 90%. In broadband terms, this structure attains above 80% absorption from 7.5 THz to 10 THz for both the TE- and TM-mode as shown in Fig. 2(c). Overall, the proposed second-order fractal structure manifests better absorption as compared to zeroth-order and first-order fractal.

 

Fig. 2. Simulated absorption results for different order fractal structures. (a) zeroth-order (b) first-order (c) second-order.

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Next, the effect of the obliquity incidence is analyzed considering the second-order proposed PTFMA. Figures 3(a) and (b) illustrate the absorption for incidence angles from 10˚ to 40˚ for proposed PTFMA for both TE- and TM-mode, respectively. We observe almost perfect absorption for frequency range 8.4 THz to 9.7 THz at normal incidence excitation (θ = 0⁰) (see Fig. 2(c)). It is observed in Fig. 3, that absorption is little lower for TM-mode as compared to TE-mode. It is noticed that the proposed PTFMA has above 90% absorptivity for the wide frequency span of 8.4 THz to 10 THz for θ = 10⁰ to 30⁰ for TE-mode excitation. For θ = 40⁰, absorptivity largely reduces at lower frequencies; nevertheless, more than 90% absorption is still observed from 9.2 THz to 10 THz. Figure 3(b) corresponds to TM-polarization case, it is noticed for θ = 10⁰, it shows above 98% absorptivity for 8.4 THz to 9.6 THz. As the incidence angle increases absorptivity considerably gets reduced. For θ = 20⁰, more than 90% absorptivity is observed for the broad spectral range from 7.8 Hz to 10 THz. For θ = 30⁰, the absorptivity is again decreased, however, it still shows a broadband absorption phenomenon from 8.4 to 10 THz. Finally, for θ = 40⁰, the absorptivity significantly reduces at smaller frequencies but shows large absorption at higher frequencies.

 

Fig. 3. Frequency-dependent absorption of PTFMA for different excitations angles (a) TE polarized wave and (b) TM-polarized wave.

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3.1 Performance analysis of the proposed PTFMA

To evaluate the performance of the proposed PTFMA, we analyzed the figure of merit (FOM) and operational bandwidth (OBW). The optimized results of the proposed absorber can easily be attained by FOM and OBW. The FOM and OBW can be defined as [48].

Figure 4 exhibits the figure of merit ${\eta _{OBW}}$ (average value of absorption in the given operational bandwidth) and operational bandwidth OBW of PTFMA for TE- and TM-mode excitation corresponding to different operating conditions. Throughout the forthcoming discussion, ${f_{min}}$ is chosen as 7.5 THz, whereas, the value of ${f_{max}}$ depends on the threshold value of the absorption, and the threshold value depends on the operating conditions. Figures 4(a) and (b) illustrate ${\eta _{OBW{\; }}}({f,{\; }\theta } ){\; }$ and OBW as a function of incidence angle, keeping the other geometric parameters as t_d = 19 µm, L = 20 µm, P = 69 µm and t_t = 1.2 µm. In this case, different threshold values of the absorption have been selected. Initially, for θ = 0-20⁰, the threshold value of the absorption is selected as 60% for both the TE- and TM-mode. It is noticed that figure of merit gets lowered with the increase of the incidence angle. The highest value of ${\eta _{OBW}}({{\; f},{\; }\theta } )$ is 88% for normal incidence (θ = 0⁰) of TM-mode with OBW as 2.5 THz (as shown in Fig. 4(b)). This corresponds to the situation of Fig. 2(c) for TM-mode, it is obvious that threshold absorption is above 85% (average absorption$\; {\eta _{OBW}}(f,\theta {\; }$) = 88%) from 7.5 to 10.0 THz. Considering the situation θ = 20-30⁰, under TE- and TM-wave polarization, the absorption threshold is chosen to be 40%. For this case, the maximum value of ${\eta _{OBW}}({{\; }f,\; \theta } )$ is 60% for TE mode at θ = 20⁰ and 53% for TM mode at θ = 22⁰ is observed whereas the corresponding OBW as 2.5 THz. For the higher incidence angles (θ = 30-40⁰), the threshold absorption condition is picked as 35%, and the highest value of figure of merit is 51% at θ = 31⁰ for both the TE and TM mode with an OBW value of 2.5 THz. Next, the effect of the dielectric spacer thickness on the figure of merit ${\eta _{OBW}}$ and operational bandwidth OBW of PTFMA under TE- and TM-mode are studied, Figs. 4(c) and (d) corresponds to this situation. In this situation, the other operating conditions are kept as θ = 0⁰, L = 20 µm, P = 69 µm and t_t = 1.2 µm. It is noticed that for the smaller thickness of the dielectric spacer, PTFMA manifests the poor figure of merit. When dielectric spacer thickness t_d = 16 um, the threshold value of absorption was kept as 30% with f_min= 7.5 THz whereas the f_max= 10 THz, the ${\eta _{OBW}}(f,{t_d}$) = 34% with OBW = 2.5 THz. By increasing the thickness of the dielectric spacer from 18 um to 22 um, and the absorption threshold condition is selected as 70% for both TE- and TM-mode, ${\eta _{OBW}}$ $(f,{t_d}$) increases with OBW. The best figure of merit is noticed for the dielectric spacer thickness from 18 µm to 22 µm, nevertheless, ${\eta _{OBW}}$ and OBW reduce for the larger values of the thickness (t_d > 22 um) of the dielectric spacer, when the absorption threshold value is 60% for both TE- and TM-mode. Further, the figure of merit ${\eta _{OBW}}$ and operational bandwidth (OBW) is analyzed by considering the different values of the thickness of the top metasurface, keeping the other geometric parameters are as θ = 0⁰, L = 20 µm, P = 69 µm and t_d= 19 µm, as shown in Fig. 4(e) and 4(f). It is noticed that for t_t = 0.8 um to 2.4 µm, the average figure of merit and OBW have been observed as ${\eta _{OBW{\; }}}({f,\; {t_t}} )= $85% and OBW = 2.5 THz when the threshold value of the absorption is 80%. As the metasurface thickness increases, the operational bandwidth decreases sharply, however, ${\eta _{OBW}}$ approached towards the maximum value. Further, TM-mode shows higher ${\eta _{OBW}}$ as compared to TE-mode. Through this performance analysis of the metasurface absorber, the absorption characteristics of the proposed PTFMA can be improved by carefully selecting the optimized geometric parameters of the unit cell.

 

Fig. 4. Performance analysis of the proposed PTFMA (a) figure of merit ${{\boldsymbol{\eta}}_{{{\boldsymbol{OBW}}}\; }}({{\boldsymbol f},\; {\boldsymbol{\theta} }} )\; $ and operational bandwidth OBW against incidence angle for TE-mode, (b) figure of merit ${{\boldsymbol{\eta} }_{{{\boldsymbol{OBW}}}\; }}({{\boldsymbol f},\; {\boldsymbol{\theta} }} )\; $and operational bandwidth OBW against incidence angle for TM-mode, (c) figure of merit ${{\boldsymbol{\eta} }_{{{\boldsymbol{OBW}}}}}({\boldsymbol f},{{\boldsymbol t}_{\boldsymbol d}}$)and operational bandwidth OBW against dielectric thickness under normally incident wave for TE-mode, (d) figure of merit ${{\boldsymbol{\eta} }_{{{\boldsymbol{OBW}}}}}({\boldsymbol f},{{\boldsymbol t}_{\boldsymbol d}}$) and operational bandwidth OBW against dielectric thickness under normally incident wave for TM-mode, (e) figure of merit ${{\boldsymbol{\eta} }_{{{\boldsymbol{OBW}}}}}({\boldsymbol f},{{\boldsymbol t}_{\boldsymbol t}}$) and operational bandwidth OBW against top metasurface thickness under normally incident wave for TE-mode, (f) Figure of merit ${{\boldsymbol{\eta} }_{{{\boldsymbol{OBW}}}}}({\boldsymbol f},{{\boldsymbol t}_{\boldsymbol t}}$) and operational bandwidth OBW against top metasurface thickness under normally incident wave for TM-mode.

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The comparison of the reported THz metamaterial absorbers with the designed PTFMA is given in Table 1. It discusses the features of the previously published THz absorbers with respect to our proposed absorber. The last row of the discussed table corresponds to the PTFMA that designed in present communication. Our developed absorber has a single layer and high absorption bandwidth of 2.5 THz, whereas the other mentioned absorbers consist of multiple layers and less bandwidth as compared to PTFMA.

Table 1. Review of different wideband absorbers operating in terahertz regime.

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For the physical understanding of the absorption mechanism, the electric field patterns of all three fractal-orders structures are plotted for two different operating frequency points, i.e., 8 THz and 9.5 THz. Firstly, considering the zeroth-order fractal structure (for TE-mode excitation), it is noticed that the electric field is maximally accumulated at the four edges of the square-shaped patch for the lower operating frequency, i.e., 8 THz (as shown in Fig. 5(a)). Also, quadrupole electric field profile is observed in Fig. 5(a). Considering the operating frequency of 9.5 THz, it is observed from Fig. 5(b) that most of the electric field is concentrated on the top and the bottom portion of the square-shaped patch. This is dipole like pattern of the surface electric field. The confinement of the surface electric field is due to localized surface plasmon resonance (LSPR) at the interface of free-space and metasurface. Figures 5(c) and 5(d) present the surface electric field profile of the first-order fractal for two independent absorption peaks, i.e., 8 THz and 9.5 THz for TE-mode. Figure 5(c) corresponds to the e-field pattern for lower frequency, i.e. 8 THz. It is obvious that maximum intensity is concentrated at the outer edges of the smaller squares and the top and the bottom parts of the main square. Figure 5(d) presents the electric field distribution for the operating frequency of 9.5 THz. The concentrated electric field corresponds to the LSPR mode. Similar trend to Fig. 5(c) has been observed, nonetheless the intensity of e-field is maximum at the higher frequency because at this point PTFMA exhibits high absorption (ref Fig. 2(b)). Next, Figs. 5(e) and (f) illustrate the electric field distribution (for TE-mode) corresponding to 8 THz and 9.5 THz, respectively. It is noticed that most of the electric field is confined at the bottom edges of the assembly of squares for both the operating frequencies. The electric field is concentrated due to LSPR that cause the absorption through the proposed structure.

 

Fig. 5. Surface electric field patterns (a) for zeroth-order fractal structure at 8 THz and (b) for zeroth-order fractal structure 9.5 THz, (c) for first-order fractal structure at 8 THz and (d) for first-order fractal structure 9.5 THz, (e) for second order fractal structure at 8 THz and (f) for second order fractal structure at 9.5 THz.

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Within the context, surface current distributions remain an interesting topic to study the physical mechanism of the absorption through the proposed PTFMA. Therefore, we try to evaluate the magnitudes of current distribution pattern at the top surface and bottom ground plane by considering three absorption peaks, namely 7.5 THz, 8.4 THz and 9.4 THz for TE-mode excitation. First and second column in Fig. 6 depicts the surface current distribution at the top and ground-sheet of the unit cell of the proposed PTFMA, respectively. It is noticed that for lower operating frequencies, i.e. 7.5 THz, 8.4 THz, the current distribution pattern at the top and bottom metallic sheet are parallel to each other, thereby, the parallel arrangement of currents causing electric resonance in the dielectric spacer. Considering the higher frequency point, i.e., 9.4 THz, it is noticed that current directions are anti-parallel to each other creating the magnetic resonance mechanism between the top and bottom ground plane (dielectric spacer). These electric and magnetic resonances generate current loops in the middle dielectric section. The combined effect of these electric and magnetic resonances ultimately provides the wideband absorption features for this fractal structure.

 

Fig. 6. Current distribution patterns (a) front side at 7.5 THz (b) back side at 7.5 THz, (c) front side at 8.4 THz (d) back side at 8.4 THz, (e) front side at 9 THz (f) back side at 9 THz.

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To better understand absorption mechanism of PTFMA, interference theory is employed. In the Interference theory, PTFMA was taken to be a Fabry-Pérot like resonance cavity shown in Fig. 7.

 

Fig. 7. Fabry-Pérot like resonance cavity model of proposed PTFMA

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Notably, the top periodic arrays of Pythagorean-tree fractal structures act as an impedance-tuning surface, and the ground sheet behaves as a perfect reflector to the transmitted field through glass. This interference model can be treated as a decoupled system due to minor near-field interaction between top Pythagorean-tree fractal metasurface and bottom ground sheet. The top resonators and ground plane were taken as zero-thickness surfaces [54]. As shown in Fig. 7, when an incident EM wave falls on air-spacer boundary, it partially reflects and transmits with the respective reflection and transmission coefficients $\mathop {{r_{12}}}\limits^{\prime} = {r_{12}}{e^{i{{\varphi }_{\textrm{r}12}}}}$ and $\mathop {{t_{12}}}\limits^{\prime} = {t_{12}}{e^{i{{\varphi }_{\textrm{r}12}}}}$, respectively. The transmitted EM wave continues to propagate with complex propagation phase$\; \beta = n{k_o}{t_d}$, where ${k_o}$ the free-space wavenumber, encounters with the bottom sheet (considered as perfect reflector), and reflects into the dielectric spacer with reflection coefficient as -1. Similarly, another partial reflection and transmission happens again with reflection and transmission coefficients of $\mathop {{r_{21}}}\limits^{\prime} = {r_{21}}{e^{i{{\varphi }_{\textrm{r}21}}}}$ and $\mathop {{t_{21}}}\limits^{\prime} = {t_{21}}{e^{i{{\varphi }_{\textrm{r}21}}}}$, respectively. These multiple reflections result in phase shift $\beta $ that produces the destructive interference and traps the EM waves inside the dielectric spacer, resulting in the maximum absorption. The overall reflection can be calculated from the given equation [54]

Thus, the absorption can be computed through, $A = 1 - {|r |^2}$. Figure 8(a) depicts the amplitudes of the reflection and transmission of the two media (air & substrate) of the proposed PTFMA that were calculated through the Fabry-Perot like cavity model called interference theory. Similarly, Fig. 8(b) corresponds to the phases of the reflected and transmitted energies of the designed PTFMA. Figure 8(c) shows the comparison of the absorption results that were obtained directly through the simulation of the proposed unit cell of the PTFMA and the other was calculated though the interference theory. It is observed in Fig. 8(c) that there is a close agreement between simulated and calculated results.

 

Fig. 8. Plots of (a) amplitude, (b) phase and, (c) absorption of the proposed PTFMA

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By employing the impedance matching theory for normally incident wave on a metasurface, the relative impedance and absorption can be defined as [55,56], [57].

Figure 9 illustrates the frequency-dependent normalized impedance of proposed PTFMA for TE-mode. It is normalized with the free space impedance and its absolute value should be 1 for the perfect matching condition for 100% absorption. It is noticed that real part of the normalized impedance is approaching 1 and the imaginary component is close to zero from 7.5 THz to 10 THz and its absolute value is approaching unity at a few points (indicating unity absorption). It can be concluded that the impedance of metasurface has a good matching with the free-space impedance for the wide frequency range (7.5-10 THz). Notably, due to LSPR free-space impedance match with the metasurface impedance, and metamaterial allows to penetrate the light. Therefore, the metamaterial shows broadband absorption characteristics for the afore-mentioned operating frequency span (7.5-10 THz).

 

Fig. 9. Frequency-dependent impedance of the proposed PTFMA for TE-mode.

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4. Conclusion

In the aforesaid discussion, we have investigated wideband terahertz absorber by exploiting Pythagorean-tree fractal geometry. Observations reveal that proposed PTFMA exhibits excellent absorption for a wideband spectrum ranging from 7.5 THz to 10 THz for both TM- and TE-mode. Moreover, it is noticed that obliquity incidence has profound effects on the absorptivity. Furthermore, interference theory is employed to verify the simulated results. Different physical dimensions are tuned to explore the performance analyzing the FOM and OBW. To understand the absorption mechanism, surface current distributions and electric field patterns are studied. Due to the simplicity in the geometry of the proposed device, this is scalable to other operating frequencies from microwave to infrared. Such a type of broadband absorption characteristics of the proposed PTFMA has potential applications in communication and THz detection.