Efficient Design of a Terahertz Metamaterial Dual-Band Absorber Using Multi-Objective Firefly Algorithm Based on a Multi-Cooperative Strategy

Abstract

Terahertz metamaterial dual-band absorbers are used for multi-target detection and high-sensitivity sensing in complex environments by enhancing information that reflects differences in the measured substances. Traditional design processes are complex and time-consuming. Machine learning-based methods, such as neural networks and deep learning, require a large number of simulations to gather training samples. Existing design methods based on single-objective optimization often result in uneven multi-objective optimization, which restricts practical applications. In this study, we developed a metamaterial absorber featuring a circular split-ring resonator with four gaps nested in a “卍” structure and used the Multi-Objective Firefly Algorithm based on Multiple Cooperative Strategies to achieve fast optimization of the absorber’s structural parameters. A comparison revealed that our approach requires fewer iterations than the Multi-Objective Particle Swarm Optimization and reduces design time by nearly half. The absorber designed using this method exhibited two resonant peaks at 0.607 THz and 0.936 THz, with absorptivity exceeding 99%, indicating near-perfect absorption and quality factors of 31.42 and 30.08, respectively. Additionally, we validated the absorber’s wave-absorbing mechanism by applying impedance-matching theory. Finally, we elucidated the resonance-peak formation mechanism of the absorber based on the surface current and electric-field distribution at the resonance frequencies. These results confirmed that the proposed dual-band metamaterial absorber design is efficient, representing a significant step toward the development of metamaterial devices.

1. Introduction

Terahertz waves (0.1–10 THz), which lie between microwaves and infrared rays, have received increasing interest in recent years owing to their unique properties, such as penetrating non-conductive materials, low photon energy, and molecular fingerprint spectrum recognition [1,2,3]. Natural materials struggle to achieve efficient absorption or regulation in the terahertz band, while metamaterials overcome this limitation through artificially designed subwavelength structures and achieve customized electromagnetic responses [4,5]. The development of metamaterial-based terahertz devices, such as high-performance sensors and wavelength division multiplexers, has been driven by the unique properties of metamaterials, which have enabled highly sensitive sensing [6,7], polarization selection [8] of terahertz waves, and enhanced transverse beam shifting [9]. Owing to their simple structure and high absorption rate, terahertz metamaterial absorbers have broad application prospects in photovoltaic solar cells, thermal radiation devices, sensing and detection, stealth technology, surface-enhanced Raman scattering, and other fields [10,11,12,13,14]. Initially, single-band absorbers were explored extensively for their advantages of narrow-band high absorptivity, high sensitivity, and high signal-to-noise ratio [15]. However, single-band absorbers are usually optimized for a specific frequency and cannot meet multi-band or broadband requirements simultaneously, limiting their applicability. Compared with single-band absorbers, dual resonance peaks enable multi-point matching between the resonance frequency and the characteristic frequency of the measured substance, thereby increasing the information reflecting differences in the substances, broadening the spectral application range, and meeting the needs of multi-target detection and high-sensitivity sensing in complex environments [16,17].

At present, the design of dual-band terahertz metamaterial absorbers mostly follows a traditional process: structure design, electromagnetic simulation, parameter optimization, and performance improvement [18,19]. This process heavily relies on numerous electromagnetic simulations and continuous structural parameter optimization to achieve relatively excellent absorption performance. It is limited by low design efficiency and high time cost. To improve design efficiency, methods such as neural networks [20,21] and deep learning [22,23] have been used for terahertz metamaterial absorber design. Although these methods can optimize structural parameters according to requirements, they require large datasets for training and must be retrained when tasks change, resulting in high time costs. Therefore, the development of an efficient design method that can optimize structural parameters according to requirements must be prioritized.

Evolutionary algorithms are stochastic optimization methods that imitate the evolutionary process of biological populations. They fall within the category of adaptive algorithms and do not require pre-training data, thus offering high design efficiency. Several evolutionary algorithms have been used to optimize metamaterial parameters, including Particle Swarm Optimization [24,25,26,27,28], Genetic Algorithm [29,30,31,32], Artificial Bee Colony algorithm [33], Ant Colony Algorithm [34,35], and PSO-Fireworks hybrid algorithm [36]. However, most evolutionary algorithms employed in metamaterial design address single-objective optimization. Specifically, these algorithms use a weighted sum of absorptivity and quality factor (Q-factor) as the objective function. Although this simplified optimization approach appears to meet design requirements, the actual absorption and Q-factor values do not. Designing a dual-band absorber requires consideration of not only whether the Q-factor and absorptivity meet design objectives but also the balance between the two resonance peaks. Existing methods are not applicable for designing terahertz metamaterial dual-band absorbers. Therefore, introducing a multi-objective optimization strategy into the classical optimization algorithm is necessary to achieve parameter optimization through non-dominated sorting. Previous research has shown that the Firefly algorithm has the advantages of a simple concept, precise flow, fewer parameters, and ease of implementation. Combining non-dominated sorting with multiple cooperative strategies improves the overall performance of the Firefly algorithm, making it more suitable for dual-band absorber design.

This study adopts the Multi-Objective Firefly Algorithm based on a Multi-Cooperative Strategy to optimize absorber design. First, we propose a terahertz material absorber based on a circular split-ring resonator with four gaps (FCSRRs) nested in a “卍” structure. Using absorptivity and Q-factor as optimization objectives, parameters such as the geometric dimensions of the resonant structure and the dielectric material thickness were optimized with the Multi-Objective Firefly Algorithm based on a Multi-Cooperative Strategy algorithm. This approach achieves near-perfect absorption, greatly shortens iteration time, and realizes efficient optimization. Subsequently, electric field and surface current distributions were analyzed to elucidate the absorption mechanism, and absorber stability was discussed. This method improves the absorber design process, providing a more rapid and innovative approach for designing metamaterial absorbers and accelerating the development of metamaterial devices.

2. Structure

The unit cell of the proposed terahertz metamaterial absorber is illustrated in Figure 1a. The uppermost metal resonant layer and the lowermost metal reflective layer of the metamaterial absorber act as two parallel reflectors, with the dielectric layer in the middle forming a Fabry-Pérot cavity. When terahertz waves are incident perpendicularly, they reflect back and forth within the cavity, generating an interference phenomenon and achieving light confinement. Both the resonant pattern and the reflective layer are made of copper with conductivity σ of 5.96 × 10⁷ S/m. Polyimide (PI) materials with a relative dielectric constant ε_PI of 3.5 and a loss tangent tan δ of 0.0027 serve as the intermediate dielectric layer. The absorber can be fabricated using the following process: First, a 200 nm thick copper film is deposited on a silicon substrate to form the reflective layer, which has a dielectric constant ε_Si of 11.3. Then, a PI dielectric layer with thickness h_d is prepared on top of the reflective layer using spin coating. Finally, the top metal layer (200 nm) is patterned using electron beam etching and orthogonal photolithography. The fabrication process described in this section applies only to device fabrication; no experiments have been performed. The top resonant unit comprises a circular split-ring resonator with four gaps (FCSRRs) nested in a “卍” structure. The quadruple rotational symmetry of the resonator results in more stable absorption performance [37]. In Figure 1b, the top view of the proposed terahertz metamaterial absorber is demonstrated. In the illustration, P represents the period of the resonant unit; r_o indicates the outer radius; r_i denotes the inner radius of the ring; g refers to the gap of the split-ring; l and j represent the lengths of the long and short axes of the “卍” structure, respectively; and d_l signifies the line width of the “卍” structure.

During multi-objective optimization, six constraint conditions (Equations (m₁)–(m₆)) were applied to construct the unit cell absorber structure, as detailed in

Table 1. Constraint equations for the resonant element structure used during optimization.

Constraint Name	Constraint Equation	Range of Constraint
m1	$r_o+5-P/2$	m1 < 0
m2	$P/2-r_o-20$	m2 < 0
m3	$r_i+5-r_o$	m3 < 0
m4	$l/2+5-r_i$	m4 < 0
m5	$\sqrt{{({\displaystyle \frac{l}{2}})}^2+{(j-{\displaystyle \frac{d_l}{2}})}^2}+5-r_i$	m5 < 0
m6	$d_l-l/8$	m6 < 0

The dimensions of geometry in the table are expressed in micrometers (μm).

. Equations (m₁)–(m₃) limit the dimensions of the inner and outer radii of the FCSRR, ensuring the outer radius r_o does not exceed the period P and is larger than the inner diameter r_i. The period P was optimized within the range 150–200 μm. Equations (m₄)–(m₆) describe the constraints of the “卍” structure in terms of lengths and dimensions of the long and short axes l and j, respectively, ensuring that the inner radius r_i of the FCSRR structure is not exceeded.

The absorptive properties of metamaterial absorbers are characterized by absorptivity (A_ω) and the Q-factor. Absorptivity can be derived using the following equation:

$$A_{\omega }=1-R_{\omega }-T_{\omega }=1-{\left|S_{11}\right|}^2-{\left|S_{21}\right|}^2$$

(1)

where R_ω represents the reflectivity, T_ω represents the transmissivity, and ω denotes the frequency of the incident terahertz wave. Reflectivity and transmissivity are expressed as $R_{\omega }={\left|S_{11}\right|}^2$ and $T_{\omega }={\left|S_{21}\right|}^2$, respectively. S₁₁ and S₂₁ represent the reflection and transmission coefficients. Absorber parameters, including structure, size, and material, can be evaluated based on absorptivity. Higher A_ω values indicate better absorption of the incident wave. The absorptivity, A_ω, reaches its maximum when both R_ω and T_ω equal zero, corresponding to near-perfect absorption of electromagnetic waves. According to the skin depth calculation formula (${\delta }_s=1/\alpha =\sqrt{2/\omega \mu \sigma }$), transmittance approaches 0 when the reflective layer thickness (0.2 μm) is considerably larger than the copper skin depth (0.0595–0.1458 μm) in the 0.2–1.2 THz frequency band. Therefore, the absorptivity can be given as follows:

$$A_{\omega }=1-R_{\omega }=1-{\left|S_{11}\right|}^2$$

(2)

The degree to which the resonance peak diminishes is reflected by the Q-factor, which is calculated as follows:

$$Q={\displaystyle \frac{f_0}{X_{\mathrm{FWHM}}}}$$

(3)

where f₀ denotes the resonant frequency and X_FWHM indicates the full width at half maximum of the resonant peak. As X_FWHM decreases, the resonance peak becomes sharper, resulting in a higher Q-factor and lower resonance loss.

3. Method

The Firefly Algorithm [38] (FA) is an intelligent optimization method inspired by the behavior of firefly swarms. Optimization is achieved through mutual attraction, where brighter fireflies attract dimmer ones. The brightness of a firefly corresponds to its fitness value within the objective function. If the absolute brightness of firefly F_i is greater than that of firefly F_j, then F_j moves toward F_i. The attraction of firefly F_i toward firefly F_j is defined as follows:

$${\beta }_{ij}(r_{ij})={\beta }_0{e}^{-\gamma r_{ij}^2}$$

(4)

where β₀ represents the maximum attraction, which is the attraction of the firefly at r = 0; γ denotes the light absorption coefficient. The firefly’s attraction increases as the distance decreases. r_ij represents the distance between firefly F_i and firefly F_j and is calculated as follows:

$$r_{ij}=‖x_i-x_j‖=\sqrt{{\displaystyle \sum _{k=1}^n{(x_{ik}-x_{jk})}^2}}$$

(5)

When firefly F_j is attracted to firefly F_i, its position is updated as follows:

$$x_j(t+1)=x_j(t)+{\beta }_{ij}(r_{ij})(x_i(t)-x_j(t))+\alpha {\epsilon }_j$$

(6)

where t represents the number of iterations of the algorithm; x_i(t) and x_j(t) denote the positions of fireflies F_i and F_j at time t, respectively; α signifies the step factor, a uniformly distributed random number in the interval [0, 1]; and ε_j is a vector of random numbers sampled from a Gaussian or uniform distribution; the term αε_j introduces stochastic perturbations.

The design objective of the dual-band metamaterial absorber is to obtain two perfect resonance peaks with the highest possible Q-factors within a specified frequency range. Due to inherent limitations, the Firefly Algorithm cannot directly solve multi-objective optimization problems. Therefore, a multi-objective optimization strategy based on non-dominated sorting is employed to identify solutions that satisfy the design goals. Non-domination is defined by the relationship between two points, x₁ and x₂, in the objective space. x₁ and x₂ are considered mutually non-dominant if an objective function exists for x₁ greater than that for x₂ and an objective function exists for x₂ greater than that for x₁. If x* is not dominated by any other solution in the space, then x* is considered a Pareto non-dominant solution. During the optimization of absorber parameters, multiple Pareto optimal solutions are generated. The Pareto front is updated through non-dominated sorting to guide the population toward the optimal frontier.

The Multi-Objective Firefly Algorithm based on Multiple Cooperative Strategies (MOFA-MCS) [39] has been proposed to enhance the Firefly Algorithm’s performance in solving complex multi-objective optimization problems. The workflow of the terahertz metamaterial absorber design using MOFA-MCS is illustrated in Figure 2. The firefly population size is set to 15, with a maximum iteration number N = 15. The dimension of the fireflies depends on the number of parameters to be optimized (seven in this study). The population must be initialized before the optimization process begins. Relying on a random initial population in most Multi-Objective Firefly Algorithms can lead to cases where the population is overly concentrated. Such concentration risks trapping the algorithm in local optima, preventing it from reaching the desired search objectives. To mitigate this, MOFA-MCS divides the search space into multiple uniform subspaces and randomly distributes fireflies within each subinterval. This approach ensures uniform population distribution and maintains randomness in individual firefly placement.

The reflection coefficient (S₁₁) and the absorption-transmission coefficient (S₂₁) corresponding to the structural parameters of each firefly’s absorber were calculated using electromagnetic simulation software. Absorptivity and the Q-factor were then calculated using Equations (2) and (3). The algorithm terminates if the design objectives are satisfied (A₁ and A₂ are greater than or equal to 98%, and Q₁ and Q₂ are greater than or equal to 30) or if the maximum number of iterations is reached (N = 15). Otherwise, a Pareto solution is randomly selected from the current Pareto set as an elite solution through roulette selection. This elite solution guides the fireflies’ movements. When neither firefly dominates the other, both move toward the elite solution. If a dominance relation exists, for example, F_i > F_j, firefly F_j moves toward both the elite solution and F_i. The objective function values are recalculated after all firefly positions in the population are updated. The process reiterates until the termination condition is satisfied.

To balance local and global search capabilities, MOFA-MCS replaces the perturbation term “ε_j” in Equation (6) with Lévy flights derived from non-Gaussian stochastic processes. This allows some fireflies to explore regions near the current optimum while others explore distant regions. The Lévy flight and elite solution strategies produce two forms of the position update equation:

(1)

When firefly F_j has no dominance relationship with others, the update equation is as follows:

$$x_j(t+1)={\omega }_0{x}_j(t)+(1-{\omega }_0){\beta }_{gj}(r_{gj})(x_g(t)-x_j(t))+\alpha \otimes s$$

(7)

(2)

When firefly F_j is dominated by firefly F_i, the update equation is as follows:

$$x_j(t+1)=x_j(t)+{\omega }_0{\beta }_{ij}(r_{ij})(x_i(t)-x_j(t))+(1-{\omega }_0){\beta }_{gj}(r_{gj})(x_g(t)-x_j(t))+\alpha \otimes s$$

(8)

where ω₀ represents a uniformly distributed random number defined in the interval [0, 1], r_gj denotes the distance between elite individual F_g and firefly F_j, s signifies a random perturbation of Lévy flights, and $\otimes$ denotes the inner product operation.

4. Result Analysis

4.1. Analysis of Absorption Spectral Performance

To evaluate the influence of different algorithms on the performance and design efficiency of the absorber, the optimization results of the MOFA-MCS and Multi-Objective Particle Swarm Optimization (MOPSO) algorithms [37,40] were compared, as shown in

Table 2. Absorber performance and design efficiency comparison.

Method	f0 (THz)	A (%)	Q-Factor	Population Size	Iterations	Total Time (h)
MOPSO	0.617	99.92	33.11	15	14	12.22
	0.869	98.26	30.13
MOFA-MCS	0.607	99.14	31.48	15	8	6.98
	0.936	99.42	30.09

. Both methods were executed under identical hardware conditions, with the same objective requirements (A₁, A₂ ≥ 98%) and an initial population size of 15. Both algorithms successfully optimized the absorber structure according to the design targets. The absorber optimized using MOFA-MCS achieved absorption peaks exceeding 99%, approaching perfect absorption. Notably, MOFA-MCS required only eight iterations to complete the optimization, with an iteration time approximately half that of MOPSO. To improve design efficiency, MOFA-MCS was adopted for structural parameter optimization in this study. Figure 3 depicts the absorption spectrum corresponding to the optimized parameters (P = 155 μm; d_l = 7 μm; d_r = 9 μm; g = 11 μm; h_d = 9 μm; j = 31 μm; l = 100 μm). Two resonance peaks were observed within the 0.2–1.2 THz range, located at 0.607 THz (Mode 1) and 0.936 THz (Mode 2), with absorptivity values exceeding 99% and Q-factors of 31.48 and 30.09, respectively.

4.2. Analysis of Electromagnetic Characteristics

The degree of matching between free-space impedance and the intrinsic impedance of the absorber determines the reflection and transmission of incident electromagnetic waves at the interface. To verify the absorption mechanism, numerical analysis based on impedance matching theory [41] was conducted.

$$R_{\omega }={\displaystyle \frac{Z_1-Z_0}{Z_1+Z_0}}$$

(9)

$$T_{\omega }={\displaystyle \frac{2Z_1}{Z_1+Z_0}}$$

(10)

where Z₀ and Z₁ represent the wave impedances of electromagnetic waves in free space and in the absorber, respectively. For metamaterial absorbers, the reflection of electromagnetic waves must be minimized while ensuring their complete entry into the absorber to achieve efficient absorption. This necessitates matching the input impedance Z₁ of the absorber with the wave impedance Z₀ in free space. In impedance matching theory, the wave impedance is given as follows:

$$Z=\sqrt{{\displaystyle \frac{{(1+S_{11})}^2-S_{21}^2}{{(1-S_{11})}^2-S_{21}^2}}}$$

(11)

To achieve perfect absorption of terahertz waves by the metamaterial absorber, the equivalent impedance Z₁ must match the free-space impedance Z₀, that is, Z = 1. The equivalent impedance of the metamaterial absorber at Mode 1 and Mode 2 is shown in Figure 4. At 0.607 THz and 0.936 THz, the real part of the equivalent impedance Re (Z) approaches 1, and the imaginary part Im (Z) approaches 0, indicating that the equivalent impedance Z ≈ 1. This confirms that the proposed absorber achieves impedance matching at both resonant frequencies, which explains the near-complete absorption of the incident terahertz wave.

To further clarify the formation mechanism of the characteristic resonant peaks of metamaterial absorbers in the terahertz frequency band, simulation analyses were conducted on the overall structure of the absorber, including the electric field and surface current distributions at the two resonant modes, Mode 1 (0.607 THz) and Mode 2 (0.936 THz), as shown in Figure 5. The absorber adopts a three-layer metal–dielectric–metal structure, forming a Fabry–Pérot cavity. Electromagnetic waves are reflected multiple times within the cavity to form standing waves, enabling the excitation of magnetic dipole resonance. The current distributions at the top and bottom layers of the absorber at 0.607 THz are shown in Figure 5a,b. In Mode 1, the current direction at the top layer (resonant layer) is from top to bottom and is opposite to that at the bottom layer (reflective layer), indicating the formation of a loop current between the two metal layers—i.e., a magnetic dipole. When the frequency of the incident electromagnetic field matches the natural frequency of the magnetic dipole, strong magnetic dipole resonance is excited. A similar loop current and magnetic dipole resonance are also observed in Mode 2, as shown in Figure 5d,e.

The excitation of Modes 1 and 2 is not solely due to magnetic dipole resonance. The presence of metal microstructures alters the propagation path and mode of the electromagnetic waves, thereby inducing additional resonant modes. The surface electric field distributions of Modes 1 and 2 are shown in Figure 5c and 5f, respectively. In Mode 1 (Figure 5a,c), the electric field is primarily concentrated in the internal “卍”-shaped structure. The movement of charges from the top to the bottom under the influence of the electric field generates surface current, which is primarily concentrated along the metal lines of the internal “卍” structure. The current forms a standing wave pattern with smaller amplitudes at both ends and a maximum at the center, characteristic of dipole resonance. In Mode 2 (Figure 5d,f), the electric field is concentrated on the left and right arcs of the external FCSRR, and the resulting current directions are opposite. The alternating electric field causes the dipole to oscillate, exciting dipole resonance. Based on this analysis, the resonant peak of Mode 1 is generated by the coupling of the dipole resonance in the top “卍” structure and the magnetic dipole resonance of the overall structure. In contrast, the resonant peak of Mode 2 results from the coupling of the dipole resonance in the top FCSRR structure and the magnetic dipole resonance of the overall structure.

4.3. Stability Analysis

The proposed absorber’s absorption performance in practical scenarios with variations in incident wave was assessed. Figure 6 illustrates the effects of polarization mode, polarization angle, and incidence angle on absorption performance. As shown in Figure 6a, the absorption spectra under TE and TM polarized terahertz waves exhibit substantial overlap, indicating that the absorber’s performance is independent of the polarization direction of the incident wave.

Figure 6b,c present the effects of polarization angle and incidence angle, respectively. The absorption characteristics of Mode 1 and Mode 2 remain unchanged as the polarization angle increases from 0° to 90°, demonstrating polarization insensitivity. As the incidence angle increases from 0° to 75°, an additional resonance peak appears near 1.0 THz when the angle exceeds 30°. However, this does not significantly affect the resonant frequencies or absorptivity of the primary modes, and the absorber maintains good absorption stability. As the angle of incidence exceeds 60°, the absorption of Mode 1 decreases to about 85%. Meanwhile, the absorption of Mode 2 changes less, remaining at a high level. Although Mode 1 and Mode 2 absorption decrease significantly at an angle of incidence of 75°, this angle is too large and generally does not occur in practical applications. In summary, the absorber has relatively high angular stability and can meet the requirements of conventional applications.

In the actual manufacturing process, the parameters of the absorber must be precise to the order of one hundredth of a micrometer. However, limitations in manufacturing technology can lead to inevitable errors in the geometric dimensions of the resonant structure and the thickness of the dielectric material. To enhance the feasibility of practical application, the influence of manufacturing tolerance on the absorption performance of the device was studied. Based on the design parameters (P = 155 μm; d₁ = 7 μm; d_r = 9 μm; g = 11 μm; h_d = 9 μm; j = 31 μm; l = 100 μm, PO), geometric parameters (PA1–PA4) were randomly generated within an error range of ±4%, and the resulting absorptivity curves are shown in Figure 7a. The resonant frequency and absorptivity of the absorber vary with the error; however, the overall absorption trend remains consistent. The corresponding random-error absorption performance and frequency shift are shown in Figure 7b. The resonant frequency shift is relatively small, with the maximum shift of Mode 1 at 2.97% and that of Mode 2 at 5.88%. The absorptivity changes for both Mode 1 and Mode 2 are less than 1%. The proposed metamaterial absorber exhibits good fault tolerance. Even with manufacturing errors, it maintains stable absorption performance.

5. Conclusions

This paper proposed the Multi-Objective Firefly Algorithm based on Multiple Cooperative Strategies to design a terahertz metamaterial dual-band absorber. The design features a circular split-ring resonator with four gaps nested in a “卍” structure, which was optimized by the MOFA-MCS algorithm. The absorption rates at the two resonance points of 0.607 THz and 0.936 THz exceeded 99%, achieving “perfect absorption” of the incident terahertz wave. Compared with the traditional MOPSO method, the MOFA-MCS algorithm required fewer iterations and only half the duration. In addition, the mechanism of resonance peak generation was explained by combining the surface current and electric field distributions. Finally, the stability of the designed dual-band absorber was verified by changing the polarization direction, polarization angle, and incident angle of the incident terahertz wave. In summary, the efficient design features of the MOFA-MCS algorithm demonstrate strong potential for developing dual-band absorbers based on terahertz metamaterials for practical applications such as biomolecule detection.