3.1. Optimization of Electromagnetic Parameters of Magnetic Materials for Basal Layer
Accurate determination of electromagnetic parameters is crucial and fundamental for achieving precise design and optimization of the absorber. However, direct measurement of electromagnetic parameters in the P-band often suffers from inaccuracies. For instance, the reflectance value calculated from the measured electromagnetic parameters exhibits a significant disparity compared to the experimental value in terms of trend and magnitude, as depicted in
Figure 1d. This study is based on the relationship between reflectivity and electromagnetic parameters within the theoretical model [
25], which can be found in
Supplementary Materials. A GA was employed to iteratively optimize four key parameters (ε’, ε”, μ’, μ”) until the calculated reflectance value matches the experimental value.
Four arguments need to be discussed in a GA: population size, mutation probability, number of iterations, and objective function value
. First, we defined
, where
,
,
, and
represents the difference between calculated and measured reflectance values for magnetic materials with thicknesses ranging from 1 mm to 4 mm, respectively. Moreover,
,
R denotes reflectivity; through mathematical processing, the negative
R value was converted into numbers ranging from 0 to 1 to facilitate comparison and enhance convergence of the solution. Taking the case at 0.4 GHz as an illustrative example,
Figure 1b illustrates the electromagnetic parameters obtained by solving for different population sizes and mutation probabilities, and it is evident that for population sizes below 500, the solutions obtained from different mutation probabilities exhibit significant variation. However, as the population size surpasses 1000, the solution set expands, thereby augmenting the probability of discovering global minima. The resulting solution values are stable and yield superior optimization outcomes. The variation in the objective function value with the number of iterations at different frequencies is depicted in
Figure 1c. When exceeding 100 iterations, a convergence trend can be observed where the objective function value stabilizes below 0.5. Converted to reflectivity, the maximum difference observed in a single flat panel is merely 0.75 dB.
Based on the above discourse on arguments, the upper and lower limits (0, 100) of the unknown solution were selected, with a population size ranging from 1000 to 2000, 100 iterations, and a mutation probability of 0.1. The optimized electromagnetic parameters are depicted as the solid line in
Figure 1a, demonstrating consistent calculated reflectance values (shown as dot lines in
Figure 1d) with the experimental data. Furthermore, upon comparing the optimized electromagnetic parameters with the reflectivity, it can be inferred that higher ε’, μ’ within the frequency range of 0.3–0.6 GHz contribute to specific absorption performance observed with a thickness of 2 mm.
3.2. Selection of Material Type and Determination of Structural Type for Pattern Layer
The filtering frequency of the pattern layer is related to the electromagnetic parameters, position, and thickness of the basal layer. When magnetic materials are unilaterally loaded onto the pattern layer, its resonant frequency approximately corresponds to ; represents the resonant frequency of the pattern layer, and denotes the permeability (μ) or permittivity (ε) of the basal layer. At a frequency of 0.3 GHz, with approximate CIP composites being 92.63, there is a rightward shift in the resonant frequency of the pattern layer to 2.05 GHz.
After determining the resonant frequency of the pattern layer, transmission line theory [
26] can be applied to ascertain whether the pattern layer should exhibit capacitance or inductance. The transmission line theory states that when the terminal load is short-circuited, the input impedance
at the short circuit distance from the terminal is
,
β is the phase shift, and
.
Figure 2a depicts the variation of input impedance
Z with distance
l, showing that the capacitance and inductance of the input impedance cancel each other out at
. However, due to the long wavelength of the P-band, fabricating a thickness of
(75 mm) for the absorber becomes impractical in engineering applications. When the thickness is less than
, it leads to an overall increase in the resistance value of the absorber, necessitating the utilization of a pattern layer to introduce capacitive impedance and alleviate constraints on thickness [
27].
The simplest capacitive pattern layer is a square resonant element, approximated as a capacitance C [
28,
29] in the equivalent circuit (
Figure 2c). The schematic diagram of metamaterial is shown in
Figure 2b, where the top layer consists of a square pattern with a side length 2a and a gap width 2b, while the lower layer comprises a magnetic substrate, and the bottom layer serves as a metal bottom plate reflector. The relationship between the size of structural units and the occurrence of standing wave resonance is expressed by the following equation:
where
represents the modulus of standing wave resonance,
n denotes the refractive index of the basal layer, and
is resonance frequency, GHz. In the case of resonance mode 1, it can be simplified as
. As the size of the pattern increases, there is a corresponding decrease in resonant frequency. When incorporating a substrate material, the unit size of the pattern is typically close to
, which approximately equals 3 cm.
The material of the pattern layer also has a significant impact on the absorption performance. In CST, a square pattern with dimensions of 50 × 50 × 0.05 mm was constructed to explore the effect of different conductivities [
30] (ranging from 10 to 10
8 S/m) of the pattern layer on the absorption performance. The lower substrate consists of FR4 with a thickness of 3 mm, a relative dielectric constant of 3.5, and a loss angle tangent of 0.025.
Figure 3a illustrates the reflectivity at various conductivity levels within the frequency range of 1–18 GHz. Weak absorption performance is observed at a conductivity level of 10 S/m. Broadband absorption occurs in the range of 8–16 GHz for the conductivities of patterns ranging from 10
2 to 10
3 S/m. On the other hand, narrowband and strong absorption occur in different frequency bands for conductivities between 10
6 and 10
8 S/m. Particularly when focusing on frequencies between 1 and 3 GHz (
Figure 3b), only materials with a conductivity exceeding 10
6 S/m exhibit absorption resonance peaks.
Figure 3c presents the results for both absorption rate and bandwidth at different conductivity levels in the pattern layer. As conductivity increases, both parameters initially increase until reaching their peak values at a conductivity level of 10
6 S/m; thereafter, they either decrease or remain stable.
This study focuses on the absorption of low-frequency waves, thus necessitating the selection of materials with high conductivity. However, it is challenging to obtain a material with the optimal calculated conductivity of 106 S/m in experiments. Therefore, copper, commonly used and easily obtainable with a conductivity of 5.8 × 107 S/m, is chosen as the material for the pattern layer.
3.3. The Influence of Metamaterial Structural Parameters on Absorption Performance
The electromagnetic parameters of the material were inputted into CST to calculate the reflection loss (RL) curves. Subsequently, the influence of metamaterial structural parameters (a, b, and h) on absorption performance was explored, and optimal parameters for P-band absorption were identified. The half-length ‘a’ of units ranged from 1 mm to 15 mm with an interval of 1 mm, while the half-gap ‘b’ ranges from 0.1 mm to 5 mm with an interval of 0.5 mm. The magnetic substrate thickness ‘h’ was set at values of 1 mm, 2 mm, 3 mm, and 4 mm. In
Figure 4, the dashed line represents the reference line for P-band absorption; structures exhibiting calculated curves below this line can be considered as possessing high-performance absorption.
In
Figure 4a, with h = 1 mm and b = 0.1 mm, narrow absorption peaks are observed around 0.65 GHz and 0.9 GHz when a = 2 mm. Resonance peaks appear at 0.3 GHz and 0.8 GHz when a = 4 mm. When the gap ‘b’ is increased to 5 mm, as depicted in
Figure 4b, no resonance absorption effect is observed for a = 2 mm. However, significant absorption resonance peaks are generated at 0.6 GHz and 0.8 GHz for a = 4 mm or 6 mm, respectively. For h = 1 mm (shown in
Figure 4a,b), no resonance is observed in the P-band for either a = 8 mm or 10 mm. When h = 2 mm, a marginal absorption effect is observed at 0.4 GHz for the single-layer magnetic substrate. Upon introducing square units in metamaterials with dimensions of b = 5 mm and a = 4 mm, resonance occurs at frequencies of 0.3, 0.5, and 0.65 GHz. These three absorption peaks are coupled, resulting in a reflection loss below −5 dB within the frequency range of 0.3–0.7 GHz. The effective absorption bandwidth measures 0.4 GHz, with a minimum reflection loss of −24.04 dB observed at a frequency of 0.48 GHz. When b = 0.1 mm, only when a = 2 mm does an absorption effect occur within the frequency range of 0.8–1 GHz; however, combining the pattern layer with other structural parameters yields negative effects. For h = 3 mm and b = 0.1 mm, similar absorption characteristics are observed for h = 2 mm (
Figure 4e). Finally, when h = 4 mm, absorbers with dimensions of a = 2 or 4 mm exhibit no significant absorption effect, whereas for patterns with dimensions of a = 6, 8, or 10 mm, the metamaterials combined with the substrate layer produce narrow absorption peaks within specific frequency bands.
The influence of a single parameter ‘a’ on the overall absorption performance is not monotonic but rather intricately intertwined with the other two parameters. For instance, when h = 1 mm and b = 0.1 mm, a smaller value of ‘a’ will generate resonance at a certain frequency point. Conversely, when h = 4 mm and b = 5 mm, a larger value of ‘a’ results in resonance occurring at certain frequency points.
Based on the results shown in
Figure 4, the optimal results in terms of effective absorption bandwidth and maximum reflection loss for the metamaterial are observed when a = 4 mm, b = 5 mm, and h = 2 mm among the aforementioned schemes. To investigate the impact of pattern layer gap on the absorption performance of metamaterials, we fixed a = 4 mm and h = 2 mm while varying the value of ‘b’ from 0.1 to 5 mm.
In
Figure 5a, it can be seen that within the frequency range of 0.3–0.6 GHz, an increase in gap size leads to a gradual decrease in the RL value of the absorber, accompanied by distinct and sharp resonance peaks at 0.34 GHz and 0.49 GHz. The absorption peak at 0.34 GHz shifts towards higher frequencies with increasing gap size, whereas the absorption peak at 0.49 GHz shifts towards lower frequencies as the gap increases. Additionally, an absorption peak is observed in the frequency range of 0.6–0.8 GHz, which also exhibits a downward shift with increasing frequency values. Simultaneously, there is a gradual increase in maximum reflection loss for the absorber within the range of 0.3–0.8 GHz as the gap increases. On the other hand, within the frequency range of 0.8–1 GHz, an upward trend can be observed in terms of RL value for larger gaps.
In transmission line theory, the capacitance of a square resonant unit is ; as the unit size increases, the gap between square rings decreases, resulting in a decrease in equivalent capacitance and a shift of the absorption frequency band towards lower frequencies. Additionally, when a square resonant unit is superimposed on a magnetic material, mutual resonance occurs, resulting in multiple absorption peaks where both parameters (a and b) are coupled together. Therefore, the effect of gaps on the shift of absorption peaks and the maximum absorption loss exhibits relatively intricate behavior. Specifically, when b = 0.1–1.1 mm, the RL value exceeds −5 dB within the frequency of 0.3–0.8 GHz but falls below −5 dB within 0.8–1.0 GHz; when b = 2.6–5 mm, an opposite absorption effect is observed with the RL value less than −5 dB in 0.3–0.8 GHz and no absorption effect in 0.8–1.0 GHz. Only at b = 1.5 or 2.1 mm does the RL value remain below −5 dB across the entire frequency range of 0.3–1.0 GHz.
Furthermore, we fixed the dimensions of a = 4 mm and b = 1.5 mm while varying the thickness h of the magnetic substrate from 1 to 4 mm in order to investigate its impact on the absorption performance of the metamaterial absorbers (
Figure 5b). At a thickness of h = 1 mm, the metamaterial exhibits a maximum reflection loss of −9.4 dB at a frequency of 0.62 GHz, with RL values ranging from 0.36 to 0.75 GHz being less than −5 dB. The effective absorption bandwidth is measured as 0.39 GHz. When h increases to 2 mm, three absorption peaks are observed at frequencies of 0.32 GHz, 0.49 GHz, and 0.68 GHz, respectively, with a maximum reflection loss value reaching up to −11.3 dB. Furthermore, an extended effective absorption bandwidth covering the entire P-band is achieved under these conditions.
In summary, it can be concluded that optimal comprehensive absorption performance is obtained when employing dimensions of h = 2 mm, a = 4 mm, and b = 1.5 mm for the metamaterial absorber design scheme. However, when considering thicker substrates such as h = 3 or 4 mm, negligible significant absorption performance is observed across the entire P-band due to potential resonance peaks occurring below 0.3 GHz with increasing thickness of the magnetic substrate. Additionally,
Figure 5b presents a schematic diagram illustrating CST simulation for our metamaterial structure, where
Zmax represents the surface-emitting electromagnetic waves.
3.4. The Exploration of Absorption Mechanism
To investigate the absorption mechanism of the designed metamaterial absorber, we monitored the spatial distribution of the electric field, magnetic field (
Figure 6), and energy loss density (
Figure 7) on the surface of the metamaterial at P-band absorption peak frequencies of 0.32 GHz, 0.49 GHz, and 0.68 GHz. The color scale on the right side of
Figure 6 represents a cloud map ranging from blue to red, indicating increasing field strength. At a frequency of 0.32 GHz, the electric field is primarily distributed in both upper and lower parts of the metamaterial pattern layer (
Figure 6a), while the magnetic field is predominantly concentrated in its middle section (
Figure 6e). This complementary distribution between magnetic and electric fields serves as a prime example of
characteristic resonance properties [
31]. The concept of
resonance can be understood by analyzing the overlapping of transmitted waves and multiple reflected waves, resulting in the formation of a standing wave within a material. Assuming that along with incident direction, superimposed waves’ electric and magnetic fields are
and
, while the electric and magnetic fields of superimposed waves in the opposite direction are altered as
and
. Then, the mathematical formulation representing the standing wave field is provided:
and
. The expressions for the electric and magnetic fields of standing waves indicate a phase angle of 90 degrees between them, resulting in spatial separation. This characteristic can be utilized to detect the presence of resonance at
. At a frequency of 0.49 GHz, the distribution of electric and magnetic fields remains complementary and consistent with
resonant behavior. At 0.68 GHz, the electric field is distributed among structural units, demonstrating strong coupling and pronounced diffraction effects within the pattern layer.
The distribution of surface energy loss density in metamaterials (upper row) and the energy loss density between the top metal and intermediate basal layer (lower row) is illustrated in
Figure 7. The energy loss density distribution at 0.32 GHz exhibits a similar pattern to that of the electric field distribution map, primarily concentrated in the upper and lower regions of the pattern layer as shown in
Figure 7d. Therefore, the energy dissipation at this frequency mainly comes from electrical loss. The energy loss density distribution at 0.49 GHz also shows a close resemblance to the electric field distribution, indicating a strong correlation between them. Additionally, an evident increase in energy loss is observed at the periphery of the metamaterial pattern layer, particularly in the red area depicted in
Figure 7b, along with amplified losses between adjacent units. Therefore, it can be concluded that the predominant source of energy dissipation at this frequency primarily stems from electrical losses, augmented by the coupling effect between neighboring structural units induced by strong electric and weak magnetic fields, as well as edge scattering effects. At 0.68 GHz, the energy loss density distribution exhibits a prominent red hue within the interstitial regions between pattern layers. The dissipation of energy at this specific frequency primarily arises from the coupling effect between adjacent structural units and edge scattering effects. Overall, the appearance of characteristic absorption peaks can be attributed to three main factors:
resonance, coupling effect between adjacent structural units, and scattering effects. By connecting multiple frequency points, the design scheme enables the structure to achieve broadband absorption in the P-band.