Introduction

Metamaterial (MTM) is an artificial material with unique electromagnetic (EM) properties that are not available in nature1,2. Using the unusual EM properties of metamaterials, a lot of research has been done on antennas3,4, solar cells5,6, sensors7,8,9, and absorbers10,11 in the microwave to infrared frequency range. Metamaterial absorbers (MMAs) have received a lot of attention in the THz range because of their perfect absorption properties and potential applications in sensing, imaging, biotechnology, polarization conversion, and high-speed THz communications. Since Landy et al.9 demonstrated the first MMA, various types of MMAs have been proposed, including dual-12, triple-13, quad-14, penta-15, hepta-band7, and broadband16 absorbers. Among these, multi-band MMAs with high figure of merit (FOM) have received a lot of attention because they can be used to detect materials and hazardous gases17,18, and spectroscopic imaging19.

Several methods to increase absorption peaks have been proposed, such as stacking metallic resonators16,20,21 or incorporating resonators of various sizes in a single unit cell22,23. Recently, Wang et al.24 used four different sizes of metallic resonators to demonstrate quad-band MMAs. Additionally, dipolar resonance was used to propose triple-band, and quad-band absorbers based on triple-, and quad-square loop, respectively22,25. Wang et al. developed a triple-band THz MMA to attain multiband absorption at 0–3 THz26. In 2018, Janneh et al. presented a dual-band THz absorber consisting of a resonator on a polyimide spacer with a high Q-factor27. However, these approaches present manufacturing challenges at higher frequencies due to their larger size and thickness, making them impractical in practical applications22,26,27. Additionally, T-shaped28, ring-strip29, cave-ring13, and #-shaped30 MMAs also achieved multi-band absorption by integrating LC and dipolar resonances. The linewidth for these absorption devices is typically very large, ranging from one-tenth to one-fifth of the absorption frequency31,32,33,34. Since the absorption peak's linewidth is too wide, it cannot be used for practical sensor and detection applications. To achieve good sensing performance, the proposed absorption devices must have a narrow linewidth of the resonance, which is an important parameter when analyzing sensor applications31,32,33.

The proposed MMA consists of modified dual T-shaped resonators and ground plane spaced by a dielectric layer, resulting in multiple absorption peaks at 1.89, 4.15, 5.32, 5.84, 7.04, 8.02, and 8.13 THz. The electric field distribution is analyzed to understand the absorption mechanism, which suggests a physical mechanism of LC and crossed dipole resonance resulting in size reduction. Furthermore, to investigate the absorption dependence, parametric analysis was carried out using dielectric thickness and unit cell dimension. The proposed device has a simple structural design, less thickness, compact size, and narrow linewidth properties, suggesting sensing and detection applications. Table 1 compares the performance of nine relevant works with our work which clearly shows that the FOM and Q in our design outperform the previous work in Ref31,32,33,35,36,37,38,39,40,41,42. The fourth peak of absorption response has FOM of 44, full width at half maximum (FWHM) value of 0.05 THz, peak sensitivity of 2.2 THz/RIU, and Q-factor value of 117. Furthermore, Extreme randomized tree (ERT) model is used to predict absorption values for intermediate frequencies to reduce 60% of simulation time and resources. The proposed MMA has the potential to be implemented in a variety of biomedical sensing applications, including the detection of glucose and malaria due to its high Q-factor and narrow FWHM.

Table 1 Comparison of FOM and Q value of the proposed device with the previously reported terahertz MMA.
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Design and simulation

Figure 1a,b depict a perspective and top view of a three-layer hepta-band metamaterial **absorber. The top metallic layer is composed of two modified T-shaped resonators, while the bottom layer consists of a continuous metallic plane, and both layers are made of gold with a conductivity of 4.09 × 107 S/m. The middle layer is composed of polyimide with a dielectric constant of 3 (1 + j0.06). Table 2 illustrates the optimized dimensional parameters of a metamaterial structure. At the lowest operational frequency, the compactness and thickness of the absorber is 0.21λ and 0.061λ, respectively. The absorption is \(1-{\left|{S}_{11}\right|}^{2}-{\left|{S}_{21}\right|}^{2}\), where S11 and S21 represents the reflection and transmission coefficients, respectively. Here, \({\left|{S}_{11}\right|}^{2}={\left|{S}_{11,xy}\right|}^{2}+{\left|{S}_{11,xx}\right|}^{2}\) where S11,xy and S11,xx represent the reflection of cross-polarized and co-polarized EM waves, respectively8. The maximum absorption can be achieved when S11 and S21 are reduced to zero. The transmission coefficient can be reduced to zero (S21 = 0) since the thicknesses of the ground plane is greater than skin depth. Furthermore, geometric parameters such as the thickness and periodic dimension of the gold resonator must be optimized in order to reduce the reflection coefficient.

Figure 1
figure 1

(a) Perspective view and (b) Top view of the proposed metamaterial absorber (MMA).

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Table 2 Geometrical parameters of a metamaterial structure.
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The proposed MMA is simulated using the ANSYS High Frequency Structure Simulator (Version 2022 R1) software with Floquet periodic port in the z-direction and periodic boundary conditions in x- and y-directions. During simulation, the EM waves that are polarized along the y-axis are often incident on the surface of the absorber along the z-axis. The fabrication of the MMA can be achieved using the following steps. First, the middle dielectric layer of the metamaterial absorber could be created by evenly spin coating the polyimide material with an optimized thickness on the Au film. Second, the middle dielectric layer for the top Au patterned array could be transferred using the spin-coating, exposure, development, evaporation, and lift-off technologies43.

Results and discussions

The proposed MMA exhibits multiple absorption peaks at 1.89 (f1), 4.15 (f2), 5.32 (f3), 5.84 (f4), 7.04 (f5), 8.02 (f6) and 8.13 THz (f7) with absorption coefficients of 98.75, 90.39, 80.40, 93.03, 91.2, 97.23 and 98.25%, respectively as shown in Fig. 2a. The FWHM bandwidth at f4 is 0.05 THz and quality factor (Q = fc/FWHM, where fc represents the resonance's center frequency) is 117, which is exceptional to already reported absorbers44. The frequency mode f4 has a Q value of 117, which is approximately 8.7, 7.3, 3.3, and 2.8 times greater than the other frequency modes f1, f2, f3, and f5, respectively. The Quality factor is known to be a critical parameter for evaluating frequency mode performance, with a higher Q value indicating that the device is more effective in sensing applications.

Figure 2
figure 2

(a) Absorption spectra with its inset of f6 and f7 resonating frequencies and (b) Real and imaginary normalized impedance of the proposed MMA.

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The MMA impedance must be equal to the free space impedance to achieve perfect absorption, and for that, the real and imaginary normalized impedances should be close to one and zero, respectively10,26. The obtained Z is 1.02 + j0.23, 1.02-j0.66, 0.42 + j0.28, 0.83-j0.47, 0.87-j0.59, 1.22-j0.26 and 1.22 + j0.19 at 1.89, 4.15, 5.32, 5.84, 7.04, 8.02 and 8.13 THz, respectively, as demonstrated in Fig. 2 (b). Effective permittivity (εeff) must equal effective permeability (µeff) to achieve unity impedance, as \(Z=\sqrt{{\mu }_{\mathrm{eff}}/{\varepsilon }_{\mathrm{eff}}}\). Real values of εeff and µeff must equal imaginary values in order to obtain perfect absorption, which is an ideal but challenging condition to achieve. As a result, the MMA's absorptivity is less than 100% since the normalized impedance will never be the same as the characteristic impedance. Figure 3a,b show the εeff and µeff of the proposed MMA, which were calculated using the S parameter retrieval method. Table 3 shows that the real and imaginary values of εeff and µeff are found to be equal because structure impedance equals free space impedance at perfect absorption.

Figure 3
figure 3

(a) Real parts of εeff and μeff and (b) Imaginary parts of εeff and μeff of the proposed absorber.

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Table 3 Comparison of absorption rate, normalized impedance and EM parameters.
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Figure 4a depicts the absorption response at various polarization angles, demonstrating that the asymmetric structure of the absorber makes it polarization sensitive. Such properties have a wide range of potential applications, including sensing, detection, and optoelectronics. Furthermore, the proposed MMA is demonstrated with different incident angles, implying that the proposed absorber is sensitive to incidence angles as shown in Fig. 4b, similar to previously reported work44. The e-field distribution is analyzed at 1.89, 4.15, 5.32, 5.84, 7.04, 8.02, and 8.13 THz to better understand the absorption mechanism, as illustrated in Fig. 5. The e- field is primarily accumulated on the inner side of the resonator which is attributed as LC or fundamental resonance, as shown in Fig. 5a,d,e. The e-field distributions are focused at the resonator's edges, which is related to the structure's crossed dipole or higher-order resonance occurring at higher frequencies because the size of the proposed structure is greater than a multiple of a half-wavelength of the resonant modes45, as shown in Fig. 5b,c,f. The e-field distribution can be found inner and outer sides of the resonator due to the structure's combination of LC and crossed dipole resonance as shown in Fig. 5e,g. These results suggest a novel method for designing a multi-band MMA that incorporates various resonance modes.

Figure 4
figure 4

Simulated absorption response at various (a) polarization angles (ϕ) from 0 to 60° and (b) incident angles (θ) from 0 to 60°.

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Figure 5
figure 5

Electric field distribution on the top plane at 1.89 (a), 4.15 (b), 5.32 (c), 5.84 (d), 7.04 (e), 8.02 (f) and 8.13 THz (g).

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As shown in Fig. 6a,b, parametric analysis is performed with the substrate thickness (hs) and unit cell dimension (a) to determine their effect on the absorption response. The inverse effects of hs on resonance frequencies can be estimated using transmission phase:

$$\alpha =\frac{4{h}_{s}\sqrt{{\varepsilon }_{r}-{sin}^{2}\theta }}{\uplambda }$$
(1)

where εr and λ is the dielectric constant and wavelength, respectively. The ratio hs/λ is remain fixed with constant value of εr, α, and θ. As a result, the relationship between the resonating frequency and the substrate thickness is inversely proportional. However, a and λ are directly proportional to each other resulting in inverse effects on resonance frequencies. As a result, the absorption response shows red shift phenomenon as the hs and a increases15. Table 4 compares previously reported studies10,11,13,15,28,46,47,48 with proposed MMA in terms of unit cell size, thickness, and resonating frequencies, indicating that the proposed MMA has seven absorption peaks with compact metallic structure, which is superior to previously reported MMA.

Figure 6
figure 6

Absorption response at different (a) substrate thickness (hs) from 9.6 to 9.9 μm and (b) unit cell dimension (a) from 34 to 34.4 μm.

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Table 4 Comparison of the previously reported THz MMAs with the proposed MMA.
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Refractive index sensing

In comparison to the other resonance peaks, the fourth resonance peak has a narrow FWHM and a high Q, as previously discussed. In Fig. 7, the frequency dependence is demonstrated by changing the refractive index (n) of the surrounding environment from 1.00 to 1.10. The total frequency shift of f1, f2, f3, and f5 were found to be approximately 0.23, 0.28, 0.22, and 0.32 THz, respectively, as shown in Fig. 7a. The dielectric constant (ɛr) is proportional to the n with ɛr = n2, thus increase in n increases the capacitance of the over layer, which is inversely proportional to resonance frequency49. As a result, Fig. 7b depicts the inverse relationship between the resonance frequency (f4) and the refractive index. The sensitivity (S) is Δf/Δn (THz/RIU) where Δn and Δf represent change in the refractive index and frequency, respectively49,50,51. The S values for frequency modes f1, f2, f3, f4, and f5 were found to be 2.3, 2.8, 2.2, 2.2, and 3.2 THz/RIU, respectively. FOM is also an important parameter to compare sensing performance for different types of sensors which is calculated as S/FWHM. The FOM values of the resonating frequency f1, f2, f3, f4, and f5 are 15.33, 9.33, 12.22, 44, and 20, respectively. The FOM of resonating frequency f4 is approximately 2.9, 4.7, 3.6, and 2.2 times that of the other resonating frequency, f1, f2, f3, and f5, respectively. It is clear from the results that proposed absorber has potential applications in a variety of fields, including solar energy, stealth technology, sensing, and detection.

Figure 7
figure 7

(a) Absorption response with different refractive index (n) and (b) relation between frequency and refractive index.

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Application in biochemical sensing

Glucose is a vital biochemical that powers essential metabolic functions in the human body. Therefore, we can assess the suitability of the proposed structure as a biochemical sensor by simulating two different scenarios for detecting the presence of bio-molecules. Water has a refractive index of nw = 1.3198 and 25% glucose in water has a refractive index of ngw = 1.359441,52. The fourth peak has a sensitivity of 2.1 THz/RIU in water and 4.7 THz/RIU in water with 25% glucose, ensuring the MMA's enhanced detection capabilities, as shown in Fig. 8a. Future MMA fabrication could pave the way for more sensitive biomolecule detection53.

Figure 8
figure 8

Effect on absorption response with the sensing of (a) Glucose and (b) Malaria.

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Application in detection of malaria

Rapid malaria screening is crucial because 250 million people worldwide are infected each year41,54. The healthy red blood cell (also known as an RBC) has a refractive index of 1.399. Furthermore, as malaria progresses, the refractive indices of infected RBCs are 1.373 and 1.383 in the schizont and trophozoite phases, respectively. The proposed sensor can detect various stages of malaria, as shown in Fig. 8b. The proposed sensor achieved a high sensitivity value of 2.5 and 2.7 THz/RIU for infected RBC-I and II, respectively. Recent metamaterial-based biosensors show good agreement between simulation and measurement results55. This latest study confirms that the proposed absorber/sensor will perform as expected if experimentally implemented in the future.

Prediction of absorption coefficient using machine learning technique

In this section, the purpose of regression models for the simulation process is briefly discussed, along with how regression models can reduce the time and resource requirement to simulate the effective absorber design by 40%, 50%, or 60%. Regression analysis is an effective approach used to determine dependent parameter values based on independent parameter values. In the present work, frequency is an independent parameter in absorber design, whereas absorption coefficient is a dependent parameter. The designing process of a complex structure takes more time and resources to simulate the experimental design. These issues can be effectively handled with the application of machine learning (ML) approaches for regression problems and can identify the missing parameter values as well. The following three stages can be utilized to fix this issue using ML-based regression analysis methodologies.

Step 1: Simulate the absorber's design by increasing the frequency's step size value.

Step 2: Using simulated data, train the ML-based regression model.

Step 3: Predict the absorptivity of intermediate frequencies using the trained regression model.

Extreme randomized tree (ERT) regression model

In the present study, the regression analysis has been done using ERT-Model which is constructed by a binary recursive partitioning mechanism to enhance the prediction accuracy. The motivation to utilize this technique is that the explicit randomization of the cut-off points and attribute is aggregated with ensemble averaging, thus it can reduce variance more strongly than the weaker randomization schemes. The technique has two vital parameters i.e., number of attributes randomly selected at each node (K) and the minimum sample size for splitting a node (nmin). Parameter K determines the attributes strength employed to predict the target, such that K = 1……p, where p is the number of independent parameters employed to predict target parameter. In regression problems, larger value of K is preferred for enhanced accuracy56. In this work, total six attributes i.e., substrate thickness (hs), unit cell dimension (a), incident angle (θ), polarization angle (ϕ), refractive index (n) and frequency (f) are chosen for predicting the absorption coefficient. For a wide range of regression applications, the value of nmin should range between 2 and 1056.

In ERT model, a collection of “m” unpruned regression trees (RT1, RT2….RTm) is constructed. The predictions of the trees are aggregated to yield the final prediction by computing arithmetic average which is used to amalgamate the outcomes of each regression tree as indicated by the given equation:

$$\mathrm{predicted\,output}=\sum_{j=1}^{m}{RT}_{j}(x)$$
(2)

where x denotes the independent parameter value and m denotes the number of trees. Several performance indices can examine the trained regression model accuracy such as R-Square Score (R2S) and Adjusted R Square Score (Adj-R2S), Integral absolute error (IAE), Mean Absolute Percentage Error (MAPE) and Mean Squared Error (MSE), etc. Out of these, R2S and Adj-R2S are often used metrics to assess how accurately a trained regression model predicts outcomes. These considered indices can be computed using Eqs. (3) and (4) as follows:

$${R}^{2}S=1-\frac{\sum_{i=1}^{N}{\left(Predicted\,Valu{e}_{i}-Actual\,Valu{e}_{i}\right)}^{2}}{\sum_{i=1}^{N}{\left(Actual\,Valu{e}_{i}-Average\,Target\,Valu{e}_{i}\right)}^{2}}$$
(3)
$${Adj-R}^{2}S=1-\frac{(1-{R}^{2}S)(N-1)}{N-p-1}$$
(4)

where ‘N’ is the number of samples used to test the regression model.

In this work, rigorous analysis has been carried out for predicting the absorption coefficient value using designed ERT regression model. The effectiveness of the ERT model prediction is assessed under three different test cases i.e. TC-40, TC-50, and TC-60. In TC-40, two separate subsets of the simulation-generated data are used. In one subset, 40% of the simulation records were chosen at random, and trained for the ERT regression model, while 60% of the simulation records were chosen for another subset for the model's predictive accuracy. Two equal-sized, non-overlapping subsets of simulation records are created from the data generated during the simulation of TC-50. From the simulation data, one subset comprises even numbers of rows, and another contains odd numbers of rows. Any partition can be used to train the ERT regression model, and others can be used to evaluate the model's predictive power. Additionally, the data produced during simulation in TC-60 is divided into two distinct categories. One subgroup contains 60% randomly selected simulation records, while the other contains 40%. One subset is utilized to train the ERT-Regression model, and the other subset is used to assess the model's predictive ability.

In all considered test cases TC-40, TC-50, and TC-60, machine learning models are used to determine how significantly simulation resource consumption can be minimized. If regression models are effective at predicting absorption values for TC-40, it shows that an ERT model trained with 40% of the simulation records will be able to predict absorption values for the remaining 60% of records, saving 60% of the simulation time and resources. Additionally, if the TC-60 regression model's prediction accuracy is high, it can be concluded that the ERT model trained with 60% of the simulation records can accurately predict absorption values for the remaining 40% of records, resulting in a 40% reduction in simulation time and resource needs.

Figure 9 shows the heat map indicating Adj-R2 Score of ERT model using various combinations of attributes where nmin varies from 2 to 10. Figure 9a–c shows the Adj-R2S for different values of hs for test cases 40%, 50%, and 60% respectively. Similarly, the impact of other attributes (a, ϕ, θ and n) on Adj-R2S have been examined under considered test cases of 40%, 50%, and 60% which is depicted in Fig. 9d–o. The Adj-R2 score values approach to unity indicates high prediction accuracy with minimum prediction error57,58. It is clear from heat maps that in all the cases the most preferable Adj-R2S values are obtained when the smoothing strength nmin = 2.

Figure 9
figure 9

Heat map indicating Adjusted R2 Score of ERT model using numerous combinations of (a–c) substrate thickness, (d–f) unit cell dimension, (g–i) incident angle, (j–l) polarization angle and (m–o) refractive index for different test case 40%, 50%, and 60%. Images were produced using MATLAB (R2022b, https://www.mathworks.com/).

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Figure 10a–e shows the scatter plots of predicted absorption values by ERT model vs simulated absorption values for varying substrate thickness from 9.6 µm to 10 µm for different values of nmin under test case TC-40. Under similar conditions of hs and nmin, the scattered plot of prediction vs true value for TC-60 is shown in Fig. 10f–j. Results show that the predicted value approaches to true value when nmin is 2. Moreover, Fig. 11a–e shows the scatter plots of predicted absorption values by ERT regression model vs simulated absorption values for varying refracting index from 1 to 1.1 for different values of nmin under test case TC-40. Moreover, for similar refractive index and nmin conditions (as in Fig. 11a–e). The scattered plot of prediction values vs true value for TC-60 is shown in Fig. 11f–j. Results show that in all the cases the most preferable predicted values are obtained when the smoothing strength nmin = 2.

Figure 10
figure 10

Values of absorption predicted by ERT model vs simulated values of absorption for various substrate thickness (hs) (a) nmin = 10 for test case (TC)- 40 (b) nmin = 8 for TC- 40 (c) nmin = 6 for TC- 40 (d) nmin = 4 for TC- 40 (e) nmin = 2 for TC- 40 (f) nmin = 10 for TC- 60 (g) nmin = 8 for TC- 60 (h) nmin = 6 for TC- 60 (i) nmin = 4 for TC- 60 and (j) nmin = 2 for TC- 60.

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Figure 11
figure 11

Values of absorption predicted by ERT model vs simulated values of absorption for various refractive index (n) values (a) nmin = 10 for TC-40 (b) nmin = 8 for TC- 40 (c) nmin = 6 for TC- 40 (d) nmin = 4 for TC- 40 (e) nmin = 2 for TC- 40 (f) nmin = 10 for TC- 60 (g) nmin = 8 for TC- 60 (h) nmin = 6 for TC- 60 (i) nmin = 4 for TC- 60 and (j) nmin = 2 for TC- 60.

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Conclusions

For terahertz applications, a hepta-band MMA comprised of modified dual T-shaped resonators deposited on polyimide is presented. Multiple absorption peaks with absorptivities greater than 80% can be found at 1.89, 4.15, 5.32, 5.84, 7.04, 8.02, and 8.13 THz. The hepta-band absorption is primarily caused by the combination of the metallic resonator's dipolar response and LC resonance, as explained by examining the e-field distribution. The proposed MMA is suitable candidate for bio-medical applications due to its Q value of 117, FOM of 44 and high sensitivity of 4.72 THz/RIU. Furthermore, a machine learning assisted ERT regression model is used to learn absorber behavior and predict absorption values for intermediate frequencies. The Adjusted R2 score was close to 1.0 for nmin = 2, demonstrating the prediction efficiency of the ERT model in estimating absorption values in various Test Cases. The experimental results show that using the ERT model to simulate absorber design can reduce simulation time and resource requirements by 60%. The proposed MMA sensor designed with an ERT model is applicable in biomedical applications for the detection of malaria and glucose.