1 Introduction

During World War II, scientific research focused on designing electromagnetic (EM) wave absorbers to minimize the signature of air craft and submarines’ periscopes. Since the first conception of absorbers, researchers are working rigorously to provide significant solutions for designing high quality absorbers to meet the advanced requirements of modern wireless communication applications. As an effective solution, with the advent of metamaterials, remarkable changes have been witnessed in design principle and electromagnetic characteristics of the absorbers. The metamaterial absorbers (MMAs) offer several advantages such as reduced mass and low thickness and can be made from standard materials widely used in the industry unlike the conventional absorbers such as magnetic absorbers (Michielssen et al. 1993), analog circuit-based absorbers (Gau et al. 1997), and structural absorbers (Zadeh and Karlsson 2009). In modern times, MMAs find applications in various fields such as Communication (Hannan et al. 2022b), Biomedical (Abdulkarim et al. 2022), Sensing (EL-Wasif et al. 2023), Solar energy (Wu et al. 2022), and shielding (Zhang et al. 2020).

Generally, when an MMA structure is exposed to an EM wave, it exhibits negative permeability or negative permittivity, or both (Hrabar 2006; Ziolkowski 2003). Therefore, it absorbs the maximum of EM waves if there is proper matching between its characteristic impedance and the impedance of the free space, hence a low coefficient of transmission and reflection. Three layers typically make up an MMA structure: a metallic material forms the top (patch) and bottom (ground plane) layers, with a dielectric substrate situated in the between. Regarding the choice of substrate, various studies have utilized commercial substrates such as Rogers RO or RT and FR-4 (Hannan et al. 2020). Additionally, MMAs can be designed using double and multi-layers (Saadeldin et al. 2023; Ren et al. 2020; Zhang et al. 2020).These materials find applications across different frequency ranges, including optics, infrared, terahertz, and microwave (Mahmud et al. 2020; Shuvo et al. 2022; Lv et al.2022; Abouelez and Abdulaleem 2023).

In the microwave field, MMAs can be used in many applications, including absorption of radar waves, improvement of antenna performance, or reduction of electromagnetic interference (Moniruzzaman et al. 2020; Hakim et al. 2023; Ali et al. 2022b; Munaga et al. 2015). In Landy et al. (2009). For the first time, a double-negative MMA with an absorbance of 88% is reported that operates in the microwave range and specifically in the X application band, this MMA used a split ring resonator (SRR) as a base. After that, scientific researches have been expanded which brings different varieties of MMAs that work in several bands of applications like, S, C, X, and Ku (Hakim et al. 2022). The recommended MMA for Ku band application is based on a modified square SRR. The dimension of this absorber unit cell is \(0.44{\uplambda } \times 0.44\lambda \times 0.088{\uplambda }\), where \({\uplambda }\) is the wave length at the lower resonance frequency, it can reach two absorption peaks of 99.99% in the Ku-band with an average bandwidth absorption of 420 MHz and 480 MHz, respectively. In Hossain et al. (2022), a double E-shaped symmetric SRR MMA is proposed. The electrical size of this unit cell is \(0.179{\uplambda } \times 0.179\lambda\) and it is able to absorb in all three regions in the S and C-bands with an average absorption greater than 99%. Moreover, this structure shows an almost perfect absorption profile across a broad angle of incidence up to 60° in both modes (TM and TE). A double elliptical resonator is proposed in Hossain et al. (2023). The quad-band absorption achieved a wide range of polarization angle up to 60° for TE mode. A penta band MMA is presented in Hannan et al. (2021) which is made of a swastika-shaped patch with a size of \(0.169{{\uplambda }} \times 0.169\lambda\), and can absorb EM waves belonging to the bands X, C, and Ku. Further, this absorber presents an absorption spectrum that does not depend on the co-polarized and cross-polarized EM waves, and the refractive indices are near zero at resonance frequencies. An MMA is presented in Ali et al. (2022a) which is based on the combination of three Closed Circular Ring Resonators (CCRR). The size of this absorber is \(22\,{\text{mm}} \times 22\,{\text{mm}}\) and it shows polarization-independent characteristics for TE and TM modes. This absorber provides six resonance frequencies for C and X application bands. The MMA presented in Dewangan and Mishra (2023) requires a total dimension of \(0.32{\uplambda } \times 0.32\lambda \times 0.098\lambda\) based on a combination of L-shaped resonators. This structure provides three peaks of absorption in X, Ku-bands. The absorption profile is shown over a wide polarization angle and incidence up to 50° for TE and 60° for TM modes. This brief review of the scientific literature reveals that the current focus is on developing multiband MMAs with desirable properties like a broad bandwidth FHWM, high absorption, high stability depending on the polarization state of the EM wave and the angle of incidence. But achieving all these characteristics for a monolayer absorber based on an FR4 substrate remains a challenge.

In this article, a compact MMA using the classical low cost FR4 substrate is presented. This structure is based on the capacitive coupling between three resonators that offers seven peaks of absorption in the microwave S, C, X, and Ku-bands. The unit cell of this MMA achieves insensitive polarization, and a wide angle of incidence for both TE and TM modes which reaches up to 50°. Various characteristics parameters such as permittivity of dielectric and magnetic permeability are extracted and analyzed in order to verify the metamaterial behavior of this absorber. The analysis of the field (E and H) and surface current distributions are carried out and discussed to offer an intuitive discussion of the absorption mechanism. Furthermore, numerous parametric analyses are executed to study the geometric effect of the patch on the absorption spectrum. The proposed MMA offers several advantages as outlined follows:

  • The MMA provides seven absorption bands, characterized by broad polarization independence.

  • A wide total bandwidth FHWM of 3.07 GHz in S, C, X, Ku-bands.

  • The absorber requires a compact electrical size of 0.168λ0 × 0.168λ0.

  • The proposed absorber exhibits more than 85% absorption for all operating the bands.

  • The prescribed absorber is suitable for sensing and wireless communication applications operating within microwave S, C, X, Ku-bands.

2 Design and evolution of MMA cell

The unit cell for the MMA proposed in this work is shown with different views as depicted in Fig. 1. Copper as conducting material and the FR4 as the dielectric materials are employed in the fabrication of the absorber. Copper with an electrical conductivity of \(\sigma = 5.8 \times 10^7 \,{\text{Sm}}^{ - 1}\) and thickness of \(t = 0.035\,{\text{mm}}\) is used to make the patch and the ground layer. The FR4 with a tangent loss of 0.02, a dielectric constant of \(\varepsilon_r = 4.4\), and a thickness h = 1.6 mm is placed in the middle of the configured structure. The patch is produced by a combination of three resonators: a twisted square ring (Resonator-A), a modified cross (Resonator-B), and L-shaped chopsticks (Resonator-C). The parameters optimized for this MMA cell to achieve maximum absorption are detailed in Table 1.

Fig. 1
figure 1

Multiple perspectives of the suggested unit cell a bottom side, b side side, c various parameters of the patch, d upper view, e isometric view

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Table 1 Dimensions in (mm) required for unit cell patch
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The Ansys-HFSS-2022 software is used to design the suggested MMA unit cell under periodic boundaries. As shown in Fig. 2, primary and secondary boundary conditions have been applied in the x and y directions, while excitation is applied using the Floquet Port along the z-direction.

Fig. 2
figure 2

Configuration of the unit cell simulated in HFSS

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The absorption coefficient can be calculated at different frequencies by mathematical Eq. (1):

$$A\left( f \right) = R\left( f \right) - T\left( f \right)$$
(1)

where \(R\left( f \right) = \left| {S_{11} } \right|^2\) and \(T\left( f \right) = \left| {S_{21} } \right|^2\) are the coefficients of reflection and transmission, respectively. Since the background of this MMA structure is made of copper of thickness \(t = 0.035\,{\text{mm}}\) which is much larger than the skin depth, this allows to neglect the transmission of the EM wave through this structure, hence, the Eq. (1) is reduced as follows:

$$A\left( f \right) = 1 - R\left( f \right)$$
(2)

So, the coefficient of absorption is near to unity \((A = 1)\) if the coefficient of reflection equals zero \((R = 0)\).

Figure 3 illustrates the design progression steps of the unit cell, while Fig. 4 provides the absorption profile for each resonator. It is remarkable that Resonator-A exhibits three distinct absorption peaks within the S, X, and Ku frequency bands, achieving absorption rates of 98.83, 74.75, and 99.69% at frequencies of 2.80, 9.21, and 13.86 GHz, respectively. Resonator-B contributes to absorption in four frequency bands: S, C, X, and Ku, with absorption percentages of 96.83, 95.77, 98.25, and 84.80% at 2.98, 6.41, 9.59, and 14.18 GHz, respectively. Furthermore, Resonator-C gives dual-band absorption in the Ku-band at 12.71 and 14.04 GHz, with absorption rates of 72.72 and 99.96%, respectively. The two Resonators-B and -C are therefore placed inside the Resonator-A to produce the suggested MMA unit cell whose absorption characteristics are shown in Fig. 5. It appears that absorber offers a total of seven absorption peaks distributed across in the S, C, X, and Ku frequency bands. Because of the capacitive coupling between these three resonators, the absorption peaks are displaced, diminished, increased, and disappeared. In S-band the coupling between two Resonators A and B gives rise to an absorption peak at 3.15 GHz with 93.65% and a low peak of 9% at 2.35 GHz. The absorption peak of 85.75% at 5.96 GHz is provided by Resonator-B alone in the C-band, while the two Resonators A and B offer two absorption peaks of 94.14 and 99.51% at 8.73 GHz and 9.36 GHz, respectively in the X-band. Therefore, the three Resonators A, B, and C is responsible for the appearance of three peaks in the Ku-band at 13.22, 13.71, and 14.45 GHz with an absorbance of 90.55, 97.87, and 98.68%. Moreover, a low peak has appeared at 11.98 GHz with an absorbance of 30%. A major advantage of this suggested MMA structure is a large width at half maximum (FWHM), especially in the Ku-band, which is close to 2.04 GHz. Table 2 shows the detailed FWHM at each absorption band for this proposed absorber.

Fig. 3
figure 3

Development stages of the prescribed unit cell design

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Fig. 4
figure 4

The absorption profile for Resonator-A, Resonator-B, Resonator-C

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Fig. 5
figure 5

The absorption profile for the proposed unit cell

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Table 2 Simulated result of the suggested unit cell
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3 Verification of the metamaterial behavior of this absorber

Equation (3) describes the mathematical relationship between the reflection coefficient \(r\left( f \right)\) and the input impedance of the MMA structure \(Z_a (f)\) which depends on the geometry of the patch, and the impedance of the free space \(Z_0\):

$$r\left( f \right) = \frac{Z_a \left( f \right) - Z_0 }{{Z_a \left( f \right) + Z_0 }}$$
(3)

where the input impedance can be expressed by the following relationship:

$$Z_a \left( f \right) = Z_0 \sqrt {{\frac{\mu_r (f)}{{\varepsilon_r (f)}}}}$$
(4)
$$Z_0 = \sqrt {{\frac{\mu_0 }{{\varepsilon_0 }} }} = 377\,\Omega$$
(5)

where \(\mu_r (f)\) is the relative permittivity as a function of frequency, \(\mu_0\) and \(\varepsilon_0\) are two physical constants, respectively the permeability and permittivity of the vacuum. It is already mentioned, the absorber is perfect if the reflection coefficient equals to zero. Figure 6a depicts the variations in normalized input impedance for the 2–16 GHz frequency range. Additionally, for each resonant frequency, Table 3 summarizes the values of the imaginary and real parts of the normalized input impedance. It indicates strong absorption at these frequencies by showcasing close to unity real parts and close to zero imaginary parts.

Fig. 6
figure 6

a Normalized impedance, b relative permittivity, c relative permeability, d refractive index for the unit cell

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Table 3 Values of permittivity, permeability, refractive index and normalized impedance
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To verify the metamaterial property of this absorber, permittivity, and permeability are (Nicolson–Ross–Weir) process according to the Eqs. (6–8) (Luukkonen et al. 2011). Due to the ground plane, the transmission of the electromagnetic wave is negligible, so that \(S_{21} = 0\). The \(S_{11}\) values are therefore taken for the TEM mode at normal incidence. The Permittivity (\(\varepsilon_r\)), permeability (\({\upmu }_r\)), and the refractive index (\(n_r\)) can be found by using the MATLAB codes developed for the referred relationships. The results indicate that the imaginary parts of the permittivity and permeability are simultaneously negative, which is not consistent with the Lorentz model generally used to study MTMs. These results therefore contradict the fundamental principles of physics. It is possible that these results are due to approximation and truncation and round off errors (Woodley and Mojahedi 2010). So, the imaginary components have not been taken in to account. The real parts of the permittivity, permeability and the relative index is plotted in Fig. 6b–d. For more details. Table 3 shows the values of these physical quantities at each resonance frequency. It is clear that the absorber under consideration exhibits a single negative (SNG) characteristic, the permeability or permeability is negative. In addition, the refractive index for each resonance frequency is close to zero or negative, confirming the metamaterial behavior of this absorber.

$$\varepsilon_r = \frac{2}{jk_0 h}\frac{{1 - (S_{21} + S_{11} )}}{{1 + (S_{21} + S_{11} )}}$$
(6)
$${{\upmu }}_r = \frac{2}{jk_0 h}\frac{{1 - (S_{21} - S_{11} )}}{{1 + (S_{21} - S_{11} )}}$$
(7)
$$n_r = real\left( {\frac{c}{j\pi fh}\sqrt {{\frac{{\left( {S_{21} - 1} \right)^2 - (S_{11} )^2 }}{{\left( {S_{21} - 1} \right)^2 + (S_{11} )^2 }}}} } \right)$$
(8)

4 Performance analysis of designed MMA

This section explores the effects of electromagnetic (EM) wave polarization and oblique incidence on the absorption profile of the suggested MMA. Figure 7 shows the absorption spectrum for two polarization modes, TE and TM, under conditions of normal incidence, with polarization angles (Φ) ranging from 0° to 180° with an increment of 30°. It is observable that for all values of the polarization angle, the absorption profile behavior is not affected for both modes. This is due to the structural symmetry of the three resonators. Further, the absorption behavior of various oblique incidence angles (ϴ) which ranges from 0° to 50° with a step of 10° for both polarization modes TE and TM is also presented in Fig. 8. It can be seen that the first four peaks of absorption of the S, C, and X bands are greater than 70% up to the incidence angle of 50° for both modes. Whereas for the Ku-band, when the angle of incidence exceeds the value 15°, several absorption bands appear. It is also remarkable that the two spurious absorption bands appear for large incidence angles at 6.6 GHz and 10.6 GHz for TE mode. On the other hand, another spurious absorption band at 11.4 GHz increases and shifts towards the lower frequencies for TM mode. This spurious absorption is probably due to the cross-coupling, between the resonators, the electrical excitation for TM polarization, and the magnetic excitation for TE polarization, which may because of the diminution of absorptions for high incidence angles (Chaurasiya et al. 2016). Furthermore, in accordance with Eqs. (9) and (10), the reflection coefficient for TE and TM modes depends on both the angle of incidence (\(\theta_i\)) and the angle of transmission (\(\theta_t\)) (Balanis 2012). Clearly, as the \(\theta_i\) varies, the reflection coefficient fluctuates intrinsically, leading to a decrease in the level of absorption.

$$\Gamma_{TE} = \frac{Z_a \cos \theta_i - Z_0 \cos \theta_t }{{Z_a \cos \theta_i + Z_0 \cos \theta_t }}$$
(9)
$$\Gamma_{TM} = \frac{Z_a \cos \theta_t - Z_0 \cos \theta_i }{{Z_a \cos \theta_t + Z_0 \cos \theta_i }}$$
(10)
Fig. 7
figure 7

The impact of polarization angle variations on the absorption spectrum a TE mode, b TM mode

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Fig. 8
figure 8

The impact of angle of incidence variations on the absorption spectrum a TE mode, b TM mode

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One of the criteria for considering an absorber to be perfect is its ability to absorb EM waves for both co-polarized and cross-polarized modes (Kong et al. 2019). Similarly, no EM waves pass through this MMA structure because of the copper ground plane. Moreover, the reflection coefficient depends on two types of EM waves, as shown by the following equation:

$$A\left( f \right) = 1 - \left| {S_{11xx} } \right|^2 + \left| {S_{11xy} } \right|^2$$
(11)

where \(R_{xx} = \left| {S_{11xx} } \right|^2\) and \(R_{xy} = \left| {S_{11xy} } \right|^2\) are the corresponding reflection coefficients for co- and cross-polarized EM waves respectively. The \(S_{11xx}\) and \(S_{11xy}\) parameters for TEM mode under normal incidence are displayed in Fig. 9a. As the cross-polarized \(S_{11xy}\) parameter is of very low values, so the absorption of the designed MMA is affected by the cross-polarized waves as represented in Fig. 9b.

Fig. 9
figure 9

a \(S_{11xx}\) and \(S_{11xy}\) parameters for TEM mode under normal incidence, b absorption for co-polar and both co and cross-polar

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The Ampere and Faraday Maxwell equations (12, 13) as well as the two medium Eqs. (15, 16) may be utilized to explain how the EM wave interaction with the absorbing metamaterial (Paul 2009, Wartak et al. 2011)

$$\nabla \times H = J + \frac{\partial D}{{\partial t}}$$
(12)
$$\nabla \times E = - \frac{\partial B}{{\partial t}}$$
(13)

where

$$\nabla = \left[ {\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}} \right]$$
(14)
$$D\left( t \right) = \varepsilon \cdot E(t)$$
(15)
$$B\left( t \right) = \mu \cdot H(t)$$
(16)

To comprehend the absorption mechanism, it is crucial to analyze the EM field distribution on each part of each resonator. In this section, a detailed discussion is delivered for the results shown in Fig. 10. The electric field is identified on the right and left sides of both Resonators A and B at 3.15 GHz. This corresponds to the fact that the first absorption band is caused by the combined contribution of these two resonators. For 5.96 GHz frequency, the E-field is focused on the highest and lowest sides of Resonator B, so the generation of second absorption band is due to presence of this resonator. At 8.73 GHz, the stronger E-field appears at the top of the left and right square parts of the Resonator-A, and at the top and bottom of the linear parts of the Resonator-B. At 9.36 GHz, the E-field appears to converge towards the center and extends to both the left and right sides of Resonator-B, and further on the two square parts upper left and right of the Resonator-A indicating that the third and fourth absorption peaks within the X-band are due to the contribution of Resonators A and B. In Ku-band, strong E-fields are observed on the various areas of three resonators. This confirms that the three peaks of this band are generated by the coupling of three Resonators A, B, and C. At 13.22 GHz, a significant E-field strength appears on the four internal edges L-shaped resonator and on the middle of corners Resonator-B, and on two square parts at upper left and right of the Resonator-A. On the four inner and outer edges of the L-shaped resonator, the electric field is much centered at 13.71 GHz. Finally, for the last absorption band at 14.45 GHz, the E-field is located on the upper most left and right sides of square part Resonator-A, as well as along both the exterior and interior edges of Resonator-C. According to the Maxwell equations as referred, two electric and magnetic fields are coupled. This means that the variation of the E-field as a function of time creates a curl of the H-field. Figure 10 also shows the distribution of the H-field at each resonance frequency. It can be observed that the H-field is near to zero when the E-field constant, and the presence of the H-field is also noticed when the E-field changes. Furthermore, the comparison of the distributed surface current and the H-field pattern reveals that the intensified H-field is present in the region of the high current density.

Fig. 10
figure 10figure 10

Magnetic (H) field, electric (E) field, and surface current distribution presentations for the unit cell MMA

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Additionally, to investigate the absorption properties of the MMA, the equivalent circuit model (ECM) of the designed MTM unit cell is presented in Fig. 11. Each RLC circuit functions as a resonator for each resonance frequency, the inductance (L) represents the ability of the MMA to store magnetic energy, while the capacitor (C) is used to represent the ability of an MMA to store electrical energy. Resistance (R) simulates energy loss within the absorber. The RLC \((R_8 ,L_8 ,C_{15} )\) circuit functions as a resonator for the copper solid masse. Then for the capacitors \((C_{2,4,6,8,10,12,14,16} )\), its role is to provide isolation between two adjacent resonances. To obtain the desired S11, all circuit element values are fine-tuned on ADS (Hannan et al. 2022a, c): L1 = 109.6 nH, L2 = 12.1 nH, L3 = 14.9 nH, L4 = 16.51 nH, L5 = 7.611 nH, L6 = 10.4181 nH, L7 = 10.7536 nH, L8 = 0.5 nH, C1 = 0.025578 pF, C2 = 0.798 pF, C3 = 0.1001 pF, C4 = 1.11 pF, C5 = 0.02607 pF, C6 = 0.318 pF, C7 = 0.0192355 pF, C8 = 0.03683 pF, C9 = 0.03856 pF, C10 = 0.137 pF, C11 = 0.010614 pF, C12 = 0.21181343 pF, C13 = 0.014351 pF, C14 = 0.564 pF, C15 = 0.01175 pF, C16 = 0.3 pF, R1 = 58 Ω, R2 = 36 Ω, R3 = 29.5 Ω, R4 = 36 Ω, R5 = 9.54 Ω, R6 = 25.995 Ω, R7 = 22.8 Ω, R8 = 25 Ω. The impedance for the two surfaces (patch and ground) equals the impedance of the free space (377 Ω). Figure 12 shows the S11 values obtained by both ADS and HFSS software. It is remarkable that for first four resonance frequencies, the values of the S11 obtained from HFSS and ADS are matched perfectly. For the Ku band, slight variations in S11 values are noted as obtained by ADS software, which may be due to the presence of the broad absorption band that characterizes the Ku band.

Fig. 11
figure 11

ECM of the propounded hepta-band MMA

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Fig. 12
figure 12

S11 simulation results using HFSS and ADS

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5 Parametric structural analysis and performance comparison

The geometry of the metallic components that frames the patch of the MMA structure affects the effective inductance and capacitance. When the length and width of a metal element in the patch vary, this leads to variations in its effective inductance. Similarly, changes in the gap between the metal elements of the patch influence its effective capacitance. These changes in the dimensions of the structural parameters affect both the effective inductance and capacitance, which can have an impact on the coupling between the patch elements and lead to changes in the absorption profile. In this section, the effects of variations in the geometrical parameters of this proposed MMA are investigated and discussed. Several parameters for this MMA can be varied to obtain a strong resonance. The length ‘a’ of the square part of the Resonator-A has been varied from 0.41 to 1.06 mm, with all other parameters held constant. The impact of parameter ‘a’ on the absorption profile is illustrated in Fig. 13. When the parameter ‘a’ increases in its value, the magnitude of fourth absorption band increases but the third peak decreases and shifts slightly toward the low frequencies, while the magnitudes of first two bands remains unchanged. For the Ku band, the first absorption peak decreases, and the peak magnitudes of last two bands are increased. The parameter ‘w’ varies from 9.1 to 9.9 mm, and its influence on the absorption spectrum is plotted in Fig. 14. As ‘w’ increases, the first and third bands are not affected, while the second bands lightly alternate between increased and decreased values. The fourth band increases and shifts slightly towards lower frequencies, as do the peak absorption of the Ku band, which also exhibit a slight shift towards lower frequencies. Furthermore, Fig. 15 shows the effect of the variation in Resonator-B’s parameter ‘u’ from 0.12 to 0.52 mm on the absorption profile. With an increase in ‘u’, all of the absorption bands shift towards the lower frequencies, so the first and third absorption peaks increase, while the second band fluctuates between a minimum to maximum absorption values.

Fig. 13
figure 13

The impact of parameter ‘a’ on the absorption profile

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Fig. 14
figure 14

The impact of parameter ‘w’ on the absorption profile

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Fig. 15
figure 15

The impact of parameter ‘u’ on the absorption profile

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Table 4 compares the multiband MMA presented in this paper with some other recent MMAs reported in the literature. The performance analysis has been executed and presented in terms of the electrical dimension of the unit cell, the frequency band, the number of resonances with their absorption percentages, and the bandwidth FWHM. The suggested MMA achieves septuple absorption peaks within the microwave S, C, X, and Ku bands and has the largest absorption bandwidth FWHM as compared to other absorber designs as projected in comparison table. Furthermore, when compared to previously published research articles, this absorber is competitive in terms of its dimension.

Table 4 Comparative study of the proposed multi band metamaterial (MTM) absorber with other reported absorber designs
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6 Experimental validation

To validate the simulation results, the prototype of absorber’s unit cell is fabricated on an FR4-Epoxy substrate (tangent loss of 0.02, dielectric constant of \(\varepsilon_r\) = 4.4). Figure 16 shows the prototype from two different perspectives, top and bottom views with a ruler showing its measurements.

Fig. 16
figure 16

Top and bottom faces of the manufactured unit cell

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The reflection coefficient of the prototyped MMA is measured using vector network analyzer (VNA) and on the basis of these measurement results the absorption coefficient (\(A\left( f \right) = 1 - \left| {S_{11} } \right|^2 )\) and the absorption coefficient is displayed in Fig. 17. The manufactured prototype of the suggested absorber exhibits seven absorption peaks at 3.14, 6.06, 8.69, 9.38, 13.15, 13.68, and 14.49 GHz with an absorption level of 90.47, 85.11, 92.78, 98.99, 92.24, 97.30, and 98.79%, respectively. For the four application bands, the corresponding measured FWHM bandwidths are 0.21 GHz (3.36–2.98 GHz) for the S-band, 0.19 GHz (6.13–5.94 GHz) for the C-band, 0.35 GHz (9.01–8.66 GHz) and 0.44 GHz (9.64–9.20 GHz) for the X-band, and 2.14 GHz (14.92–12.78) for the Ku-band. The measurement results are compatible with those of the simulation. However, the presence of slight differences between measurement and simulation results are obvious, due to the small size of the manufactured prototype and alignment imperfections, fabrication and measurement tolerances.

Fig. 17
figure 17

Comparison of measurement and simulation results a S11 parameters, b absorption coefficient

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7 Conclusion

A novel polarization-insensitive multi-band metamaterial (MTM) absorber is presented in this proposed article. The unit cell of the optimally designed metamaterial based absorber consists of three resonators, which are designed on a low cost FR-4 substrate with an overall size of 13.8 mm × 13.8 mm. The absorption mechanism is examined by analyzing the distribution of electric and magnetic fields, as well as surface currents. The extraction of the permeability and permittivity parameters indicates that this structure exhibits a single negative (SNG) response. The design and simulation studies were performed using a high frequency structure simulator (HFSS). The equivalent circuit model (ECM) based analysis for the unit cell of the proposed absorber was also investigated. The results analyzed through the simulation tool (HFSS) and equivalent circuit model (ECM) shows good agreement. To confirm the reliability of simulation results, the prototype model was fabricated and experimentally tested. The measurement results validate the specific design approach of this proposed metamaterial absorber (MMA). The suggested absorber exhibits seven absorption peaks at 3.14, 6.06, 8.69, 9.38, 13.15, 13.68, and 14.49 GHz with absorption rates of 90.47, 85.11, 92.78, 98.99, 92.24, 97.30, and 98.79%. The proposed MMA finds widespread uses in sensing, radar, and satellite communications in microwave frequency regions.