The proposed absorber was designed by using a commercial full-wave finite integration technique (FIT) based on high-frequency electromagnetic solver, CST microwave studio. Nowadays, CST makes it possible to use very difficult numerical calculations in the electromagnetic field through a number of software packages. Thus, the MTM characteristics can be determined using a number of numerical calculations. During the simulation phase, very complex and long calculations can be easily performed, and the behaviours of large-scale and very different shapes of MTM structures, under the selected frequency range and selected boundary conditions, can be demonstrated. Scientists have had the opportunity to test electromagnetic materials in laboratory conditions under various boundary conditions. In numerical analysis, various boundary conditions were used to analysis the structures such as PEC/PMC, PEC, free space, periodic and unit cell. In order to obtain the effective dimensions of the proposed structure and to simplify the simulation processes, a unit cell was assigned in the x-/y-directions while an open add space was assigned to the z-direction. The perspective view and layers of the structure are illustrated in Figure 1A. In this design, a flexible substrate made of polyamide was used to be backed by a gold film with electrical conductivity of 4.561 × 10⁷ S/m. The polyimide substrate has a dielectric constant and loss tangent value of 3.5 and 0.0027, respectively. On top of the flexible polyamide, the resonator layer was designed, which was made of vanadium dioxide. The layer parameters are shown in Figure 1C, while the dimensional parameters are given in Table 1. The structure includes a combination of square split rings and a quatrefoil resonator at the center, as shown in Figure 1B. The quatrefoil structure within split ring resonators was specifically designed to give an alternative to the current state of the art metamaterial absorber in the studied frequency band. Polyamide intermediate layer was chosen as the flexible substrate for its dielectric properties and flexibility for future research developments.

Figure 2 shows the approach of presenting a step-by-step design of the proposed absorber. The design is a three-layer quatrefoil and circle-loaded complementary square split ring resonator shape that consists of three types of material (Gold-Polymide-VO₂). This design is especially developed for easy production. The parametric dimensions of the proposed deign were purposely tuned by using parametric study and genetic algorithm to simulate the broad bands of perfect absorption in Terahertz frequency range. In the design 1 and 2, a double split-ring resonator was used and rotated with equivalent scales to observe how the traditional split ring resonator works in the terahertz regime. A fractal circular resonator was added to the center of the split rings, as shown in design 3 and 4, followed by examining its effect on the performance of the unit cell. Finally, the proposed design was selected by combining all the layouts in one single design (see Figure 2).

The reflection responses and absorption capabilities of the studied layouts were comprehensively investigated, while absorptivity of the layouts was calculated using:

A ( ω ) = 1 − T ( ω ) − R ( ω ) (1)

Where T ( ω ) and R ( ω ) defines the transmission and reflection responses, respectively, such that transmission T ( ω ) = | S 12 | 2 and reflection R ( ω ) = | S 11 | 2 . This is where S 11 and S 12 are reflection and transmission coefficient, respectively. To eliminate the transmission response T ( ω ) , the backside of the structure was covered by a metallic layer. Therefore, the absorptivity equation becomes:

A ( ω ) = 1 − R ( ω ) (2)

Supported by Eq. (2) and Figure 3 shows that the absorptivity of all the layouts is inversely proportional to the reflection response. The absorption bandwidth of the layouts was compared at the corresponding absorption of 90%. The greater absorption bandwidth was observed for the design one in comparison to that of the design 2, which is the scaled and rotated version of the design 1. Moreover, the absorption bandwidth of the design 3 and 4, which was obtained by adding the fractal circular resonator, was almost at the same level of that for the design 1 and 2, respectively, as illustrated in Figure 3B. Finally, based on the achieved results, the proposed design was selected and the absorption bandwidth was studied in the frequency range from 0.58 to 1.65 THz, which corresponds to the absorption of 90% at 1.07 THz.

One of the most important parameters is the split gap in a resonator design, which corresponds to a capacitive element in the resonator structure. The split ring resonator corresponds to an LC resonance circuit with resonance frequency as follows:

f 0 = 2 π L C (3)

Where C and L are capacitance and inductance from the current path of the ring resonator. As shown in Equation (3), the resonance frequency can be controlled depending on the values of L and C of the current path of the ring resonator. Figures 4A,B show that the change in the split gap with a step size of 2 µm caused a pronounced variation in the reflection and absorption spectra around 1.15 THz in the center frequency. This can be attributed to the change of capacitance in the LC equivalent circuit. Therefore, considering the optimum bandwidth and reflection magnitude, the optimum split gap was found to be 6 µm for the proposed resonator. Although the results of 6 and 10 µm dimension seemed to be similar, the best absorption response was obtained when g = 6 μm, which will be discussed in the following sections.

Resonator line width is one of the other useful parameters which contribute in designing a split ring resonator. The split ring can be considered as a microstrip line, which has corresponding impedance given in Eq. 4 [37]:

Z 0 = 60 ε e f f ln ( 8 ( H W ) + 0.25 ( W H ) ) (4)

Where the microstrip line width of W and height H satisfy the condition of W/H<<1. Also, the resonator corresponds to an LC circuit as mentioned before, and hence the width of the resonator changes the inductive characteristics of the LC resonance. As a result, a variation in the resonator leads to the change in both impedance and inductance of the microstrip resonator. Figure 5 shows the effect of the variation of width W , with the step size of 1 µm, on the reflection coefficient magnitude and resonance bandwidth. In addition, the absorption characteristics were changed by the resonator width variations. In this scenario, the optimum bandwidth and its magnitude was estimated to be at 12 µm.

The substrate thickness (H) of the resonator, as shown in Eq. (4), presents a significant effect on the microstrip line impedance of the split ring resonator. In addition, the thickness variation changes the capacitive stabilization between the microstrip split ring line and back metallic ground. According to the mentioned criteria, the substrate thickness plays a significant role in the resonance characteristics. As shown in Figure 6, the change in the thickness from 28.8 to 32.8 µm has caused a clear change in the reflection and absorption characteristics. Consequently, the optimum thickness for the substrate was assigned to be 30.8 µm, leading to an improved bandwidth and absorption magnitude compared to those of the other thicknesses. It was seen that the simulation results at the substrate thickness of 29.8 and 30.8 µm are almost similar. Since the maximum bandwidth was achieved at 30.8 µm, we have chosen this dimension for the proposed design.

The effect of the radius of the central resonator on the absorption and reflection coefficient was also investigated. Technically, the change in the resonator radius corresponds to the change of the operating wavelength. This in turn causes the change in the resonance characteristics, as shown in Figures 7A,B for both reflection and absorption characteristics. According to the obtained results, the optimum radius of the inner resonator of the split ring is 14 µm for wideband applications at around one THz center frequency. Due to the parametric optimization results, the authors suggested to use 14 µm for the R dimension.

The incident angle and polarization dependency of the proposed hybrid metamaterial were examined, as shown in Figure 8. When the step size of change in the incident angle was 15°, no significant change was observed in the absorption characteristics, as shown in Figure 8A. Due to symmetric and unique design of the proposed structure, a polarization and limited incident angle independency was achieved. The stable absorption characteristics confirmed the incident angle independence for the proposed structure under both TE and TM wave modes. Furthermore, another important phenomenon was investigated, which is the polarization dependency of the proposed design. According to the absorption characteristics given in Figure 8B, the proposed hybrid metamaterial has polarization independence under both TE and TM wave incidence around one THz. Both studied factors are specifically important in various application fields such as medical, military, stealth technology, and communication.

In this section, taking into account different material properties of the proposed structure, the absorption mechanism of the proposed MTM absorber was investigated. First, the effect of the electrical conductivity and loss tangent of the metallic layer on the absorption response of the structure was simulated, as shown in Figure 9. This section was completely prepared by creating new material techniques in the CST. As such, the effect of loss tangent and conductivity on the absorption relation was ruled out. Figure 9A shows that the absorption response is more than 90% with a broad bandwidth of one THz over the frequency range of 0.7–1.7 THz, which is called a conducting (metallic) state with the conductivity of 3 × 10⁵ S/m. However, in the insulating state, when the conductivity is reduced to 3 × 10² S/m, the absorption is less than 5% in the frequency range of interest. As a result, the proposed absorber can be considered as a reconfigurable device with a dynamic absorption range of 3–90% in the frequency range of 0.7–1.7 THz. Moreover, Figure 9B shows the variation of the loss tangent, tanδ = 0.027, 0.0027, and 0.00027, for the proposed design. One can see that loss tangent has a trivial effect on the absorption response.

In addition to the material properties, two different configurations were utilized aiming at investigating the absorption response of the structure. In the first configuration, the VO₂ resonators were placed above PI substrate, which is backed by a gold ground plate (VO₂-PI-Gold). However, in the second configuration, the resonators were made from gold, and the ground plate was formed from VO₂ (Gold -PI- VO₂). The two configurations were investigated in both insulating and conducting (metallic) states of the VO₂ layer and the obtained results are illustrated in Figures 9C,D, respectively. Figure 9C shows the absorption spectrum of both configurations when VO₂ is in the isolating state. The absorption spectrum in the first configuration, i.e.VO₂-PI-Gold, was found to be more than 45% in comparison to the second configuration (Gold -PI- VO₂), which was less than 20% in the frequency range of interest.

Noteworthy, a promising result was achieved when the VO₂ was in the conducting state for the first configuration (Gold -PI- VO₂), as shown in Figure 9D. Herein, the absorption spectrum was calculated to be more than 90% in the frequency range of 0.7–1.7 THz. In the second configuration of VO₂-PI-Gold, only four absorption peaks were observed to be more than 90% at the frequencies of 0.67, 0.88, 1.5, and 1.8 THz. These results can be attributed to the VO₂ resonators, which have an active role in producing multiple electric and magnetic resonances close to each other, leading to a broad absorption spectrum. The VO₂ resonators are sensitive to the change in the constituent materials. Therefore, the resonators were no further responsible for producing the same resonances after changing VO₂ properties and materials.

Figure 10 shows the performance of the proposed structure at different temperatures from 300 to 380 K. The temperature-dependent absorption response was investigated by importing the data of the change in the conductivity of VO₂ into the numerical simulation. This was validated by considering the experimental effect of temperature on the conductivity of VO₂ [38]. Results showed that the VO₂ based resonator behaves in its insulating phase up to the temperature of 330 K, while beyond 330 K the VO₂ becomes metallic due to the phase transition. It is worth mentioning that the absorption peak in the metallic state is at a lower frequency (red shifted) compared to that of the insulating state. This can be attributed to the plasmon resonance effect due to the interaction of incident photons with the free electrons of the metallic state. It is concluded that by the control of VO₂ temperature, it is possible to manipulate the absorption band of the proposed MTM structure for desired applications.

The use of flexible polyamide was motivated by its importance for the non-planar and conformal geometry applications, extending extra capabilities of the metamaterial for stealth technologies application. We carried out a simulation to show the effect of the structure curvature in the insulating state and conducting state, as shown in Figures 11A,B (b). It was found that when the proposed structure is bent, the absorption response between the operation channels is almost similar to that of the normal condition, as shown in Figure 9C. This is because of the resonator and VO₂ properties. However, by looking at the VO₂-Pl-Gold configuration, one can see a considerably lower absorption value. It was observed from Figure 11B that for the conducting sate when the structure is conformal, the absorption characteristics of the VO₂ (Gold-PI-VO2) presented similar absorption peaks to that of the normal state (VO₂-PI-Gold). In the metallic state, when the resonator is made of VO₂ the absorption response is above 93% at higher frequency.

To further verify the formation mechanism of the absorption and phase in degree response, we have numerically investigated the phase in degree corresponding frequency of interest for both insulating and metallic state as shown in Figure 12. It was seen from Figure 12A that a change in sign has occurred for the transmission phase at the first and second resonance dip. Also, peaks and dips at the insulating states were observed at the second resonance point during the transmission and reflection phases. In the metallic state, the transmission phase was suddenly changed at the resonance points, as can be seen in Figure 12B.

The effect of waveguide ports on the performance of the proposed MTM device was investigated by deploying waveguide ports on the different axes of the structure. First, a pair of waveguide port was used at either side of the proposed structure along the X-axis with the existence of the primary ports along the Z-axis, as shown in Figure 13A. As can be seen from Figure 13B, the S₂₁ curve is below −10 dB from the starting frequency to about 1.2 THz with two resonance frequencies at 0.8, and 1.2 THz having the reflection values of −25 dB and −21dB, respectively. However, the S₁₁ curve is below −10 dB from the starting frequency to 2 THz, except at the center frequencies of the band, for which it was above 10 dB from 0.7 to 1.2 THz, with a relatively similar fluctuation in the S₂₁ response.

In the second case, the waveguide ports were utilized along the Y-axis in the same manner, as in the first case shown in Figure 14A. The results of Figure 14B showed that S₂₁ value is below 10 dB from the starting frequency to 1.2 THz, with observing two resonances at 0.8 THz, and 1.2 THz having values of −25 dB and—21 dB, respectively. Noticeably, the S₁₁ curve is below −10 dB from the starting frequency up to 2 THz, while in the range from 0.7 to 1.2 THz, the S-parameters are above 10 dB. Similar results were observed in deploying the waveguide ports along the x or y-axis, as shown in Figure 13B, implying the independence of the axes on the wave propagation.

In this section, the proposed broadband metamaterial absorber was analyzed by utilizing different unit cell arrangements. Generally, a number of absorber unit cells and arrangements are required for different applications. For example, the 1 × 2 array is mostly utilized as a decoupling structure between antenna elements of a large antenna array. However, in military radar applications, large numbers of unit cell arrangements are used to cover a large area. Therefore, it is crucial to investigate different absorber arrangements before being applied in any desired applications.

This section aims to give a useful idea to the reader on how to present the periodic and unit cell boundary conditions. Unit cell and periodic boundary conditions are similar to each other. With Unit Cell boundary conditions and floquet port, we can get the amplitude and phase of the transmitted and reflected waves. So Unit Cell boundary conditions are mostly used for designing elements of reflects array and metasurfaces. With periodic boundary condition, we can consider phase shift along the axes and it is mostly used for getting dispersion diagram of a unit cell. For this reason, different array structures were presented and simulated.

Figure 15 shows the configuration of a 1 × 2 array structure along with the obtained results. For the precise calculation, similar environment and conditions of the proposed structure were set for the 1 × 2 array structure. The reflection and absorption coefficient spectra were compared with the result of the proposed unit cell structure, as shown in Figures 15B,C. It can be observed from the figures that the proposed structure presents a relatively similar result. This is where the reflection of less than 0.3 was realized, corresponding to the absorption of more than 90% over the frequency range of 0.7–1.7 GHz. It is worth to mention that the absorption of the 1 × 2 array structure was lower than that of the proposed design by about 2%.

Using similar procedure to that of the 2 × 1 array structure, the 2 × 2 and 3 × 3 array structures were also designed with their boundary conditions, as shown in Figures 16A, 17A. The obtained reflection and absorption coefficient from both arrangements (2 × 2 and 3 × 3 array) were also compared with the results of the proposed absorber, which is presented in Figures 16B,C, 17(b), and Figure 17C, respectively. The obtained results indicated that the absorption spectrum of both arrangements is almost similar to that of the proposed structure, in which a broad bandwidth of THz with the absorption of more than 90% was achieved over the frequency range of 0.7–1.7 GHz. This result showed that the stability of the absorber is acceptable for the practical applications.

Further investigations were finally made on a 4 × 4 array structure, as shown in Figure 18A. The simulated results of the reflection and absorption responses for the 4 × 4 array absorber were also monitored and presented in Figure 18. It was noted from Figure 18B,C that the reflection and absorption results of the 4 × 4 array structure are similar to that of the proposed results, except for the presence of a small absorption reduction in the lower frequency range from 0.7 to one THz, which is below 90%.

Theoretically, for a perfect absorber to be achieved the reflection and transmission spectra should be zero [39]. In this situation, the absorbance can reach maximum at a resonance frequency of interest, as can be seen from Eq. (2). The reflectance becomes zero if the impedance of the structure (Z) is equal to the impedance of the free space. To maximize the absorbance at the resonant frequency, transmission (T) needs to be at minimum. For an ideal absorber, the real part of Z should be one and the imaginary part should approach zero at a resonance frequency.

It is seen from Figure 19A that the operating band is between 0.7 and 1.7 THz. Moreover, the real part of the impedance is one throughout the operating band and the imaginary part is 0, which makes the proposed structure a good absorber. In the equivalent circuit diagram, the resistive and inductive components (R and L) are mainly attributed to the VO₂ structure in the resonator layer, as shown in Figure 19D. Figure 19C shows the equivalent circuit of the designed unit cell, where R ₁ and L ₁ denote the resistance and inductance of the inner circular shape, while R ₂ and R ₃ define the resistance of the inner and outer rectangular strips, respectively. The equivalent capacitance C _eq1 and C _eq(2) represent the total capacitance of the inner and outer rectangular strips, respectively. Hence, a double mutual inductance can be presented between the L ₁ , L _2, and L ₃ , where L ₂ and L ₃ are the inductance of the inner and outer rectangular strips, respectively. The relative impedance contains both real and imaginary parts as a function of frequency is shown in Figure 19D. During this study, due to the parametric optimization and author experiences, the CST software was chosen.

The surface current distribution, E-field, and H-fields of the proposed absorber are illustrated in Figures 20–22, respectively. The relation between these three can be understood by Maxwell’s equation which relates the magnetic field with the electric field and current distribution as below:

∇ x H = J + ∈ ∂ E ∂ t (5)

Also, the relation between the electric field and current density is: J = σ E (6)

It can be observed from Figure 17 that the currents are flowing dominantly along the y-axis. It is also seen that the flow of current is weak at 0.88 THz compared to that at 1.42 THz.

As a stable and disperse E-field, the proposed structure showed a well-disseminated H-field for two frequencies. However, Figure 21 shows that E-field is very intensive with a little degradation at the split-ring resonators. The H-field is located on the fractal circular resonator at 0.88 THz compared to 1.42 THz and it is widely confined by the dielectric substance of the structure, as can be seen in Figure 22. The current density that is closely connected with the H-field increases the artificial magnetic dipolar moment [40]. The electromagnetic field distribution in Figures 21, 22 is along z-axis and does not represent its absolute value. The arrangement then excites the H-field and hints at a very powerful magnetic resonant dipole and produces an excellent absorption in the entire optical region.