CMT model

The proposed MSA may be described by a temporal CMT (TCMT) model [Reference Fan, Suh and Joannopoulos36, Reference Piper and Fan37]. The resonator comprising the MSA supports a single mode in the frequency range of interest, with the resonances being well-separated. The TCMT equation is of the following form [Reference Fan, Suh and Joannopoulos36]:

(1)\begin{equation} \frac{d\alpha(t)}{dt} = j\omega_t\alpha(t) - (\gamma_i + \gamma_e) \alpha(t) + \kappa^+ s^{+}, \end{equation}

with $\alpha(t)$ being the resonator mode amplitude, $|\alpha(t)|^2$ the energy stored in the resonator, ω_t the real part of the resonant frequency with all losses included, $s^+$ the amplitude of incident wave, γ_i the intrinsic decay rate, accounting for dielectric and resistive losses and γ_e is the external decay rate, which in the case of a MSA, corresponds to radiation losses. The term $\kappa^+ = \sqrt{2\gamma_e}$ is the coupling coefficient between the resonator and the incident wave. A second equation, describing the de-excitation of the MSA, is also required:

(2)\begin{equation} s^{-} = c s^{+} + \kappa^{-} \alpha(t), \end{equation}

in which $s^{-}$ is the outgoing wave mode amplitude and $\kappa^{-} = \sqrt{2\gamma_{e}}$ is the coupling coefficient between the resonator and the outgoing wave. The parameter c accounts for the direct scattering process and is evaluated from the reflection imposed by the metal-backed substrate with appropriate referencing.

By combining the above equations and considering $\Gamma = s^{-}/s^{+}$ and $c = -1$, we obtain:

(3)\begin{equation} \Gamma = \frac{-j(\omega- \omega_t) - (\gamma_i-\gamma_e)}{j(\omega- \omega_t) + (\gamma_i+\gamma_e)}. \end{equation}

The absorbance $A = 1 - |\Gamma|^2$ is in turn calculated as [Reference Piper and Fan37]:

(4)\begin{equation} A = \frac{4 \gamma_i \gamma_e}{(\omega-\omega_t)^2+(\gamma_i+\gamma_e)^2}. \end{equation}

Perfect absorption is attained on resonance ($\omega = \omega_t$) and when the critical coupling condition is satisfied, with the intrinsic and external decay rates being equal, $\gamma_i = \gamma_e$. We note that c may assume any value depending on the choice of the reference plane, as long as the time-reversal symmetry condition is satisfied $c\kappa^{-*}=-\kappa^-$. The critical coupling condition is directly related to the impedance-matching condition, used in the absorber-related literature [Reference Costa, Genovesi, Monorchio and Manara11]. The normalized impedance $Z/Z_0$ is related to the coupling coefficient Γ as:

(5)\begin{equation} \frac{Z}{Z_{0}} = \frac{1+\Gamma}{1-\Gamma} = \frac{j(\omega - \omega_{t}) + \gamma_{i}}{j(\omega - \omega_{t}) + \gamma_{e}}. \end{equation}

The impedance matching condition, $Z = Z_0$, is satisfied when $\gamma_i = \gamma_e$, hence verifying the equivalence between the two conditions.

The decay rates, resonance frequencies, and Q-factors are calculated by eigenfrequency full-wave FEM simulations, within the COMSOL Multiphysics environment. As the proposed MSA is uniform, we model a single resonator, and enforce periodicity, by imposing periodic boundary conditions (PBCs) on the side boundaries of the MSA unit cell, i.e., along the x- and y-axes. The complex resonance angular frequency $\omega^c_{e/t}$ can be written in terms of its real part $\omega_{e/t}$ and the decay rate $\gamma_{e/t}$ as follows:

(6)\begin{equation} \omega^c_{e/t} = \omega_{e/t} + j\gamma_{e/t}. \end{equation}

The Q-factors are calculated as:

(7)\begin{equation} Q_{e/t} = \frac{\omega_{e/t}}{2 \gamma_{e/t}}. \end{equation}

The resonance angular frequency in the absence of ohmic losses ω_e, Q-factor Q_e, and decay rate γ_e are extracted by performing an eigenfrequency full-wave simulation, accounting only for the external losses, which correspond to the resonator’s coupling to the incident wave. Hence, the metal surface is modeled as a perfect electric conductor (PEC), and the dielectric constant assumes a purely real value. The intrinsic resonator parameters cannot be directly calculated by an eigenfrequency full-wave simulation, as the external losses cannot be artificially switched off. To that end, we calculate the total decay rate γ_t, including all loss mechanisms, and retrieve the intrinsic decay rate γ_i using the relation:

(8)\begin{equation} \gamma_i = \gamma_t - \gamma_e. \end{equation}

Dielectric losses are taken into account by the complex dielectric constant of the PET substrate. The MSA patch is treated as a resistive ohmic sheet of zero thickness with a specified surface resistance value, imposed by a surface current density boundary condition. The lower layer of the substrate is modeled as PEC. The computational domain is truncated along the positive z-axis by a $1{\mathrm{st}}$ order absorbing boundary condition.

By using the CMT, we can calculate the reflection coefficient of the MSA. As the patch is a symmetric structure, the response is identical for a normally incident TE- and TM-polarized wave, so we can calculate and illustrate the results for only one polarization. Our goal is to determine a suitable patch size, so that a resonance around 11 GHz will appear. We vary the unit cell filling factor v (Figure 1) in the range $[0.95, 0.99]$ as in Figure 2 and treat the patch as PEC. As expected, as the unit cell filling factor increases, the resonant frequency decreases. A value of v = 0.98 sets the resonance close to 11 GHz.

Figure 2. Refection coefficient vs. frequency, varying the MSA unit cell filling factor v. Results are obtained by the CMT, with the patch considered as PEC.

As a next step, we determine the surface resistance R_s, that maximizes the absorption. We vary R_s in the range $[0.05,2]$ $\Omega/$sq and calculate the external and intrinsic decay rates associated with each value of R_s. The intrinsic and external decay rates are depicted in Figure 3. The external decay rate is not influenced by R_s, as it depends solely on radiation losses. The intrinsic decay rate experiences a monotonic behavior, increasing along with R_s. We observe that the external and intrinsic decay rates are equal to each other, γ_i = γ_e, for surface resistance values around 1 $\Omega/$sq, hence fulfilling the critical coupling condition [Reference Piper and Fan37, Reference Isić, Sinatkas, Zografopoulos, Vasić, Ferraro, Beccherelli, Kriezis and Belić38], and thus leading to maximum absorption. It is noted that the requirement for such low values of the surface resistance is strongly associated to the ultra-low thickness of the substrate, as also concluded in [Reference Costa, Monorchio and Manara10] and confirmed by the above simulations. In addition, such thickness values result in high filling factors, in order to place the resonance within the desired range.

Figure 3. Intrinsic γ_i and external γ_e decay rates vs. the surface resistance.

Semi-analytical approach quantifying mutual coupling

The MSA supercell is conceived as an array of N coupled elements, excited by an incident wave. Near-field coupling, quantified by the mutual impedance matrix, has been used to accurately model various MSs [Reference Olk and Powell41]. The upcoming analysis considers the mutual coupling between closest-neighboring absorbing elements, in terms of the surface currents [Reference Balanis42].

By modeling the MS as a set of N coupled elements, the current flow at an arbitrarily selected reference point $\mathbf{r}'_{0,m}$ on the $m{\rm th}$ element, I_m = $I_m(\mathbf{r}'_{0,m})$, is expressed as [Reference Balanis42]:

(9)\begin{equation} I_m=I_{0,m}+\sum_{\scriptsize{\begin{matrix}n=1 \\ n\neq m\end{matrix}}}^NH_{mn}I_n, \end{equation}

where $I_{0,m}$ is the current on $\mathbf{r}'_{0,m}$ when mutual coupling is neglected, and H_mn are mutual coupling coefficients. We note that $\mathbf{r}'_m$ refers to the position vector of an arbitrary point located on the $m{\rm th}$ array element. Hence, (9) can be written as:

(10)\begin{equation} \mathbf{I}=\mathbf{H}^{-1}\mathbf{I_{0}}, \end{equation}

where $$\mathbf{I}=\begin{bmatrix}I_1 & I_2 & \cdots & I_N\end{bmatrix}^T$$ and $$\mathbf{I_0} = \begin{bmatrix} I_{0,1} & I_{0,2} & \cdots & I_{0,N} \end{bmatrix} ^T$$, with T denoting matrix transpose. The coefficient matrix H, that quantifies near-field coupling, is expressed in the following form:

(11)\begin{equation} \mathbf{H}=\begin{bmatrix} 1 & -H_{12} & \cdots & -H_{1N} \\ -H_{21} & 1 & \cdots & -H_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ -H_{N1} & -H_{N2} & \cdots & 1 \\ \end{bmatrix}. \end{equation}

The coefficients H_mn are considered to be non-zero only if resonators m and n are nearest neighbors. We also assume that the current distribution profile $i_n(\mathbf{r}'_n)$ is not affected by mutual coupling, i.e.,

(12)\begin{equation} \frac{I_n(\mathbf{r}'_n)}{I_n(\mathbf{r}'_{0,n})}=\frac{I_{0,n}(\mathbf{r}'_n)}{I_{0,n}(\mathbf{r}'_{0,n})}=i_n(\mathbf{r}'_n). \end{equation}

Hence, (10) may be used to calculate matrix elements H_mn and H_nm, provided that the currents ${I_{0,m}}$, ${I_{0,n}}$, which are assumed to be independent of mutual coupling, as well as the currents ${I_{n}}$ and ${I_{m}}$, that account for mutual coupling, are already determined.

To determine each current $I_{0,n}$ and the current distribution profile i_n, we perform full-wave simulations under plane wave excitation, involving only the $n{\rm th}$ element of the array. Similarly, H_mn and H_nm are obtained, by a set of pairwise simulations considering elements m and n. All simulations are performed in a $3\times3$ supercell configuration, with the structure being excited in all cases by a normally incident plane wave, the computational domain is truncated by PBCs, and losses are included. We note that I_n may also represent a current density or a surface current density instead of a current, without essentially affecting the overall formulation. We will use the surface current density on the patches, as it obtained by COMSOL Multiphysics, with $\mathbf{r}'_{0,n}$ selected at the patch midpoint.

Validation of the semi-analytical approach on the 1 x 3 MSA supercell

We extend the uniform MSA design to form a $1 \times 3$ supercell, synthesized by patches of different size lying on the metal-backed PET substrate. Figure 7 illustrates the non-uniform MSA $1\times 3$ supercell, with three unit cells placed along the x-axis and a single unit cell along the y-axis.

Figure 7. Configuration of the $1\times 3$ non-uniform metasurface supercell.

Based on the results of the parametric analysis of Figure 6, we choose the filling factor values $v_{1}=0.85$, $v_{2}=0.9$, and $v_{3}=0.95$ for the supercell design, aiming at absorption in the frequency range $[10,11.5]$ GHz. The Ohmic loss of the $n{\rm th}$ element is calculated as [Reference Balanis42]:

(13)\begin{equation} L_n=\frac{1}{2}\iint_S R_s \rvert I_n(\mathbf{r}'_n) \lvert^2dS. \end{equation}

The R_s value used in (13) to determine the Ohmic losses is R_s = 1 Ω/sq, as in Figure 6. By combining (12) and (13), we derive the following expression

(14)\begin{equation} L_n=\frac{1}{2}\frac{\iint_SR_s\rvert I_{0,n}(\mathbf{r}'_n) \lvert^2dS}{\rvert I_{0,n}\lvert^2} \rvert I_n\lvert^2 =\frac{L_{0,n}}{\rvert I_{0,n}\lvert^2} \rvert I_n\lvert^2, \end{equation}

where $L_{0,n}$ is the Ohmic loss of the $n{\rm th}$ element in the absence of mutual coupling.

We perform both single-element and pairwise simulations, to determine the uncoupled $I_{0,n}$ and $L_{0,n}$ and coupled I_m and I_n currents, respectively. To obtain $I_{0,n}$ and $L_{0,n}$, we simulate the $n{\rm th}$ element, embedded within a $1\times3$ supercell environment, by a periodic full-wave solver. The center of each patch is set as the element’s reference point $\mathbf{r}'_{0,n}$. The magnitude of the uncoupled surface current density $I_{0,n}$ is shown in Figure 8(a). Both an x- and y-polarized incident wave are considered for each filling factor value v_n = $\{0.85, 0.9,0.95\}$. The absorbance of each element is also obtained from the corresponding reflection coefficient calculated by the periodic full-wave solver, as shown in Figure 8(b) for both polarizations. The absorbance of each element is different in the two polarizations, due to the asymmetric $1\times3$ supercell arrangement. The coupled I_m and I_n currents are obtained by pairwise simulation of the $m{\rm th}$ and $n{\rm th}$ elements. The coupling coefficient between elements #1 and #3 is non-zero, as they are first-nearest neighbors as well, considering an infinite periodic layout. Having determined $I_{0,n}$, $I_{0,m}$, I_n, and I_m, we calculate the H_mn coefficients using (10). Figure 9(a) and (b) presents the magnitude of the calculated coefficients H_mn for both polarizations. All possible pairwise combinations are taken into account.

Figure 8. (a) Uncoupled surface current density $I_{0,n}$ magnitude and (b) absorbance vs. frequency, considering both x- and y-polarized incident waves for filling factor values v_n = $\{0.85, 0.9,0.95\}$.

Figure 9. Magnitude of the calculated H_mn coefficients for an (a) x- and (b) y-polarized incident wave.

Once all H_mn and H_nm coefficients and the uncoupled currents $I_{0,n}$ are determined, we solve the system of equations (10) and calculate the currents I_n of the MS. The total Ohmic loss (i.e., the total absorbance) of the non-uniform MS is calculated by

(15)\begin{equation} L_{\rm tot}=\sum_{n=1}^3L_{0,n}\frac{\rvert I_n\lvert^2}{\rvert I_{0,n}\lvert^2}. \end{equation}

The overall semi-analytical approach described by equations (9)–(15) is, therefore, applied as follows: we restrict the possible resonator sizes to a small number of values, covering a reasonably wide range of resonant frequencies and we conduct full-wave simulations of both the individual resonators and all possible pairs of resonators in proximity, which form a very limited set of low-complexity simulations. The actual synthesis of the full MS will be, subsequently, performed via an exhaustive but fast search over all possible combinations among the limited preselected resonator sizes, based on the total absorbance by (15). We note that a full FEM model of a large sized MS, especially when employing all possible combinations of resonator placement, would result in an excessive number of simulations, which is computationally prohibitive.

For a thorough validation of the semi-analytical method, we compare the results against spectral full-wave simulations of an indicative $1\times3$ non-uniform MS shown in Figure 10, which will exhibit a very good agreement. As this MSA configuration is asymmetric, a different spectral response is observed according to polarization of the incident wave. It is evident, that a broader response is obtained, in case the electric field is oriented along the x-axis of supercell. The reflection coefficient is lower than roughly −5 dB in the range $[10.05,11.5]$ GHz, corresponding to a bandwidth of BW = 1.41 GHz, for the case of an x-polarized incident wave. A narrower bandwidth of about 1.1 GHz is attained for a y-polarized incident wave.

Figure 10. Reflection coefficient (dB) vs frequency, for the $1\times3$ non-uniform MSA supercell, considering both a x- and y-polarized normally incident plane wave. Semi-analytical results are compared against spectral full-wave results of the entire MSA and very good agreement is observed.

Synthesis of the non-uniform symmetric 3 x 3 MSA supercell

We extend the $1 \times 3$ MSA design to form a $3\times3$ non-uniform MSA configuration, with a polarization-insensitive response. The supercell dimensions are $3D\times3D$, with D = 8.26 mm. The patch resonators are symmetrically placed with respect to the supercell main diagonal, to achieve polarization-insensitivity. Hence, six distinct patches of different size are used, each one having a different resonant frequency, as in Figure 11 (inset). We will employ a similar approximate analysis to identify the best possible resonators’ layout for wideband absorption.

Figure 11. Reflection coefficient (dB) vs frequency, for the $3\times3$ non-uniform MSA supercell, obtained by the semi-analytical and full-wave simulation results of the entire MSA. The $3\times3$ supercell with all filling factors are shown as an inset.

Similarly to the $1\times3$ structure, we calculate the H coefficients of all permitted pairwise combinations, regarding size and position. We restrict the design only to filling-factor values v = $\{0.85$, 0.9, $0.95\}$. The value R_s = 1 Ω/sq is used for the synthesis process. Due to the symmetric placement, only six values of v are sought, within the above set. We conduct pairwise simulations in horizontal, vertical, and diagonal directions. Based on these simulations, we assign the H_mn coefficients, where m and n are the indices that map the patch to the $3\times3$ placement, hence m, n = 1⋯9. By considering all permitted combinations, of the position and filling factors of elements m and n, we calculate the absorption by (15) and select the best combination. We note that the required pairwise simulations are drastically reduced, due to the symmetric placement of the patches, the restriction to only three filling-factor values and by considering only first-nearest neighbors. The aforementioned systematic design process is also very time- and memory-efficient, compared against parametric spectral full-wave simulations of the entire supercell. The parameter set resulting from the semi-analytical synthesis approach is v ₁ = 0.95, v ₂ = 0.9, v ₃ = 0.85, v ₄ = 0.95, v ₅ = 0.85, and v ₆ = 0.9. Semi-analytical and full-wave simulation results of the MSA reflection coefficient are presented in Figure 11. As the $3\times3$ MSA is a symmetric structure, the response is identical for a normally incident TE- and TM-polarized wave, so we can calculate and illustrate the results for only one polarization. A very good agreement is attained between the two methods.

The adopted parameter set, obtained by the semi-analytical approach, serves as a starting point for the fine-tuning process. We use a genetic-algorithm optimization in conjunction with full-wave simulations. The average reflection coefficient in the range 10–12 GHz is chosen as the fitness function, which we try to minimize, aiming at lower reflection coefficient values across an even broader frequency range. Apart from the filling factors we also allow for small variations in the unit cell size and the surface resistance, thus providing additional degrees of freedom, which may lead to an even more optimized configuration. By enforcing lower and upper bounds on the parameters, we help the algorithm converge toward a more optimized parameter set faster, by searching in a limited parameter space. The following finalized design parameters are: D = 8.275 mm, R_s = 0.34 Ω/sq, v ₁ = 0.957, v ₂ = 0.893, v ₃ = 0.934, v ₄ = 0.854, v ₅ = 0.861, and v ₆ = 0.889.

We also provide a physical interpretation of the ultra-low substrate thickness on R_s values, tuned for high absorption in the case of the $3 \times 3$ non-uniform MSA. Hence, we determine the value of R_s that fulfill the critical coupling condition for each one of the supported modes of the supercell, as follows. A periodic eigenfrequency analysis of the $3\times3$ supercell is carried out, identifying all supported modes of the configuration, in the frequency range of interest $[10,12]$ GHz. Eighteen modes are supported in this range, pairwise degenerate. Nine distinct modes are plotted in Figure 12. It is evident that different modes correspond to different combinations of excited patches. The resonance frequencies of the supercell modes differ from those of individual resonators due to the coupling between neighboring elements. The real part of the eigenfrequency of each supported supercell mode is reported in Table 1, considering R_s = 1 $\Omega/$sq, while the resonance frequencies of individual resonators are f ₁=10.423 GHz, f ₂=11.069 GHz, f ₃=10.63 GHz, f ₄=11.479 GHz, f ₅= 11.413 GHz, f ₆=11.185 GHz. By direct inspection of Table 1, one can correlate the supercell resonant frequencies with those of individual resonators. Clearly, there are observable shifts due to the inter-resonator coupling. By varying the surface resistance R_s, the imaginary part of the eigenfrequency, which is related to the MS losses, also changes. Meanwhile, the real part of the eigenfrequency experiences a slight shift.

Figure 12. Normalized electric field norm at mid-substrate of the supported modes of the $3\times3$ non-uniform MS supercell. Intrinsic (γ_i) and external (γ_e) decay rates are shown for modes #1 and #4, varying R_s.

Table 1. Real part of the eigenfrequency (GHz) of each supported mode of the $3\times 3$ non-uniform MS supercell

We calculate the intrinsic and external decay rates, γ_i and γ_e, respectively, related to each of the nine supported modes of the $3\times3$ non-uniform MS supercell, varying the surface resistance value R_s in the range $[0.05,1]$ $\Omega/$sq. As in the case of the uniform MS, the external decay rate of each mode remains constant, as it only depends on leakage or radiation losses, whereas the intrinsic decay rate experiences a monotonic behavior, increasing along with R_s. In contrast to the uniform MS, which experiences critical coupling, γ_i = γ_e, when R_s = 1 $\Omega/$sq, the $3\times3$ non-uniform MS requires lower values of R_s, i.e., below 0.2 $\Omega/$sq, to achieve critical coupling, as shown in Figure 12 for modes #1 and #4.

We confirm the findings of the eigenfrequency analysis of the MS supercell, by calculating the reflection coefficient of the $3\times3$ non-uniform MS, by employing the full-wave periodic solver and considering three different R_s values, 0.05, 0.34, and 1 $\Omega/$sq. The results corresponding to a normally incident plane wave are presented in Figure 13, for the filling factors and unit cell size obtained by the optimization process. We observe that lower values of R_s lead to sharper and more pronounced resonances. Specifically, in case of R_s = 0.05 $\Omega/$sq four resonances are visible, in case of R_s = 0.34 $\Omega/$sq we observe three resonances and R_s = 1 $\Omega/$sq results in a smoother response. Based on the preceding discussion, the resonances are in all cases nine. However, the lower quality factors associated with higher R_s values, results in resonances being merged and a broadband response. The first two cases have a very similar performance. This claim is verified by the average reflection coefficient, equal to −4.4 dB and −4.6 dB, for R_s = 0.34 $\Omega/$sq and 0.05 $\Omega/$sq, respectively. We choose R_s = 0.34 $\Omega/$sq for the non-uniform MS configuration, as it provides an acceptably large bandwidth with reasonably low reflection coefficient values. The reflection coefficient is retained lower than −5 dB in the frequency range of [10.2,11.4] GHz, corresponding to a bandwidth of BW = 1.2 GHz. The $3\times3$ non-uniform MS is associated with a higher bandwidth compared to the uniform configuration, with a 50% relative improvement, similar to that of the $1\times3$ non-uniform MS, while offering a polarization-insensitive design.

Figure 13. Reflection coefficient (dB) vs frequency, for the $3\times3$ supercell, under normal incidence, for R_s = 0.05, 0.34, and 1 $\Omega/$sq.

Performance evaluation based on the Rozanov limit

The performance of the MSA may be further assessed by calculating the Rozanov limit [Reference Rozanov43], which provides an estimation of the largest possible bandwidth that may be attained, considering a predetermined thickness and reflectance of the absorber. Specifically, the maximum reflection coefficient ρ ₀ that may be attained within $\Delta\lambda$ = $\lambda_{\rm max}-\lambda_{\rm min}$ is related to the relative magnetic permeability µ_i and thickness d_i of each layer i of the absorber through the following inequality [Reference Rozanov43]:

(16)\begin{equation} |\ln{\rho_0}|\Delta\lambda \leq 2\pi^2\sum_i{\mu_i d_i}. \end{equation}

As the proposed $3\times3$ non-uniform MSA consists of a single nonmagnetic layer, inequality (16) is transformed into

(17)\begin{equation} |\ln{\rho_0}|\Delta\lambda \leq 2\pi^2 d. \end{equation}

Hence, the minimum theoretical value of the absorber thickness d _min is calculated as

(18)\begin{equation} d_{\rm min}=\frac{|\ln{\rho_0}|\Delta\lambda} {2\pi^2}. \end{equation}

According to the results of the non-uniform $3\times3$ MS (Figure 13), the reflection coefficient Γ₀ is less than −5 dB within the frequency range $[10.2,11.4]$ GHz. The minimum substrate thickness d _min is thus calculated equal to 0.09 mm, i.e., about 0.1 mm.

Inequality (16) provides the ultimate minimum thickness that may be attained. A strict estimation of the minimum was calculated above, considering the reflection coefficient Γ₀ less than −5 dB within the frequency range $[10.2,11.4]$ GHz, and 0 dB elsewhere. A more realistic calculation of the minimum thickness, may be calculated, based on the following inequality [Reference Rozanov43]:

(19)\begin{equation} \left|\int_{0}^{\infty} \ln{|\rho(\lambda)|}d\lambda\right| \leq 2\pi^2\sum_i{\mu_i d_i}, \end{equation}

keeping in mind that

(20)\begin{equation} |\ln{\rho_0}|\Delta\lambda \leq\left|\int_{0}^{\infty} \ln{|\rho(\lambda)|}d\lambda\right|. \end{equation}

By limiting the integral within λ _min = 0.025 m and λ _max = 0.033 m, corresponding to a frequency range of $[9,12]$ GHz, we estimate the minimum thickness required to obtain the reflection coefficient of the R_s = 0.34 $\Omega/$sq case of Figure 13, found to be equal to d _min = 0.19 mm. The realized substrate thickness of the $3 \times 3$ MSA is h = 0.27 mm, which is reasonably close to the above estimated theoretical minimum. It is also noted that higher absorption levels, in the order of 90%, may appear in the literature but this most commonly refers to significantly thicker substrates. For ultrathin MSAs, especially for aviation applications, with a thickness in the order of $\lambda/100$, lower absorption levels are naturally anticipated for reasonably wideband operation. This is dictated by Rozanov’s limit, demonstrating the trade-off between substrate thickness, minimum absorbance, and operating bandwidth. Therefore, for the particular choice of −5 dB reflection, a chief advantage of the proposed design is that the realized thickness of $0.27\,\rm{mm}$ is reasonably close to the theoretical minimum of $0.19\,\rm{mm}$, which is nevertheless very hard to approach. Obviously, the level of absorption is a matter of an application-dependent choice and higher levels could be attained either by thicker substrates or within narrower bandwidths. We have confirmed via extensive simulations that the proposed MSA design framework can achieve a 90% absorption level (corresponding to −10 dB reflection reduction), naturally over a narrower bandwidth. We present in Figure 14 such a design with design parameters D = 8.275 mm, R_s = 0.4 Ω/sq, v ₁ = 0.930, v ₂ = 0.900, v ₃ = 0.880, v ₄ = 0.885, v ₅ = 0.875, and v ₆ = 0.850, which achieves a 90% absorption level over 0.59 GHz (5.5% fractional bandwidth). We have to note that the −10 dB design delivers a narrower −5 dB reflection bandwidth compared to the original one, i.e., 0.9 GHz instead of 1.2 GHz. Therefore, we have specifically opted for an increased −5 dB bandwidth, accepting absorption levels of 68% (corresponding to −5 dB maximum reflection), since these were deemed sufficient for our application. It is emphasized that via the proposed approach we can always trade off bandwidth for absorption, depending on the application.

Figure 14. Reflection coefficient (dB) vs frequency, for the $3\times3$ supercell, under normal incidence, for the non-uniform MSA that achieves 90% absorption.

Uniform MSA variation

We provide a comparison between experimental and scattered-field full-wave simulation results of the uniform MSA sample, comprised by a 10×10 repetition of the MSA unit cell. The surface resistance is set to R_s = 0.34 $\Omega/$sq in the simulations as well. Figure 16(a) presents the normalized monostatic RCS, along with a photograph of the fabricated MSA sample, shown as an inset. Good agreement between measurements and simulations is observed, with the resonance downshifted by about 0.4 GHz in the measurements. An RCS reduction of at least 5 dB is achieved within a 0.85 GHz band, according to the experimental results.

Figure 16. Normalized RCS (dB) vs. frequency (GHz) for the (a) uniform 10×10 and (b) non-uniform 9×9 MSA sample, considering a normally incident TE-polarized wave. The fabricated MSA sample is shown as an inset.

Non-uniform MSA variation

A comparison between experimental and scattered-field full-wave simulation results is also carried out for the 9×9 non-uniform MSA sample, i.e., a 3×3 repetition of the 3×3 supercell. We set R_s = 0.34 $\Omega/$sq in this case as well. Figure 16(b) shows the normalized monostatic RCS for both cases, along with the corresponding photograph of the fabricated sample, as an inset. Experimental results are in very good agreement with the simulations. An RCS reduction of at least 5 dB is attained, with respect to a metal plate of identical dimensions, for a 1.2 GHz and 1.3 GHz, according to simulations and measurements, respectively. Even though experimental results point to a slightly greater bandwidth, both verify the broader spectral response of the non-uniform MSA design.

We also present measurements of normalized RCS for oblique incidence, where absorption properties will be shown to remain at significant levels. This is despite that the MSA has been designed for maximum absorption at normal incidence, since in applications such as coating of aerial vehicles, the electromagnetic signature is mostly pronounced at normal incidence and considerably lower at oblique angles. Therefore, we provide simulations and measurements of the non-uniform MSA under oblique incidence up to 60^∘, for both TE and TM polarization, in Figure 17. The measurements now refer to bistatic RCS toward the specular direction, for various angles of incidence, which has been normalized with respect to the RCS of an equally sized metal plate for the same directions of incident and scattered waves. It is observed that for the TE case the performance gradually degrades as the angle of incidence increases, still achieving at 20^∘ a normalized RCS of less than −5 dB within a 1.16 GHz bandwidth. For the TM polarization, a good performance is essentially retained for angles up to 60^∘, with normalized RCS of less than −5 dB being achieved within a 1.07 GHz bandwidth at the highest angle. Poorer angular stability under TE polarization is anticipated for ultrathin MSAs, as the free-space impedance will increase along with the incident angle, whereas the MSA’s input impedance will remain almost equal to that of normal incidence, leading to impedance mismatch, and consequently a degraded performance. Angular stability is expected under TM polarization, as both the input and free-space impedance gradually decrease, and hence they remain well-matched [Reference Costa, Genovesi, Monorchio and Manara11].

Figure 17. Normalized RCS (dB) vs. frequency (GHz) for the non-uniform 9×9 MSA sample, considering an obliquely incident (a) TE- and (b) TM-polarized wave. Incident angles of up to 60^∘ are considered.

It is noted that the measured RCS values cannot directly translate into absorbance, though they are highly correlated. This is due to the fact that the sample dimensions are relatively small, and thus, the bistatic RCS does not coincide with the reflection coefficient of the infinite MSA. In the latter case, the absorbance is directly calculated as $A = 1-|R|^2$.

Curved MSAs

As our MSAs are intended for achieving RCS reduction in aviation applications, we also assess our MSA’s performance when mounted on a cylindrical curved surface. Specifically, we place both the uniform and non-uniform MSAs on a cylindrical surface of a 90^∘ arc, as depicted in the insets of Figure 18, which show the curvature and a photo of the mounted MSA sample. We calculate and measure the normalized RCS of the curved MSA samples, with respect to a PEC sample of identical dimensions and curvature, for both a TE- and TM-polarized normally incident plane wave in the range 9–12 GHz. The measured and simulated RCS reduction is shown in Figure 18 for the uniform and non-uniform MSAs, and a satisfactory agreement is obtained. The 68% absorption bandwidth is 1 GHz and 0.63 GHz for the uniform MSA under TE and TM polarization, according to simulations. Similarly, for the non-uniform MSA, the 68% absorption bandwidth is 1 GHz and 1.3 GHz for TE and TM polarization. We conclude that even for the considered relatively large curvature, which is considerably greater than most typical curvatures of UAV constituent surfaces, i.e., UAV wings, our MSAs achieve a performance very close to that of a flat sample. Specifically, the non-uniform MSA achieves a slightly increased 68% absorption bandwidth of 1.3 GHz instead of 1.2 GHz for the TM polarization, while it accomplishes a somewhat narrower bandwidth of 1 GHz for the TE polarization. Therefore, the curved MSA essentially retains its competitive performance, regarding absorption and bandwidth properties, both for TE and TM polarization, even in the case of a considerable curvature.

Figure 18. Normalized RCS (dB) vs. frequency (GHz) for the (a) uniform and (b) non-uniform curved MSA sample for a normally incident TE- and TM-polarized wave. The considered curvature, corresponding to a 90^∘ arc, as well as a photograph of the curved sample are included as insets.