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Abstract
In this study, we introduce a genetic algorithm (GA) into the catenary theory model to achieve automatic and inverse design for terahertz (THz) metasurface absorbers. The GA method was employed by seeking optimal dispersion distributions to achieve broadband impedance matching. A THz dual-metasurface absorber was designed using the proposed approach. The designed metasurface absorber exhibits an absorbance exceeding 88% at 0.21–5 THz. Compared to the traditional design method, the proposed method can reduce time consumption and find the optimal result to achieve high performance. The investigations provide important guidance and a promising approach for designing metasurface-based devices for practical applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, terahertz (THz) waves with frequencies ranging from 0.1 to 10 THz have become the focus of research owing to the development of THz sources and detectors [1]. THz electromagnetic (EM) waves possess the extraordinary advantages of nonionizing radiation, high sensitivity to weak interactions, strong penetration, and high contrast [2], which are excellent for imaging systems [3], nondestructive sensing [4–6], and short-distance communications [7]. Among these applications, THz absorbers play a critical role. Recently, artificially structured metasurfaces have shown flexible manipulation of amplitude, phase, and polarization of EM waves, which allows for enabling many extraordinary applications [8–11]. Owing to the unique properties of metasurfaces. THz absorbers with broad bandwidths and high-efficiency absorption have been realized in the past decades, which has significantly promoted the development of THz imaging systems, medical detection, and biosensors to meet the urgent requirements of miniaturization and integration [12–17].
Since Landy first introduced the concept of metamaterial (or metasurface) absorbers, different types of absorbers have been extensively studied by impedance matching with the surrounding environment [18]. However, because of the resonance characteristics [19], these perfect absorbers work in a narrow band, which is less than 20% of the center frequency. Considerable efforts have been made to expand the bandwidth of the THz absorber [20–23]. The superposition of multilayers or construction of composite supercells comprising antennas with different resonant frequencies are the most common methods. In 2014, Zhu implemented a metal-dielectric multilayer to realize a broadband THz absorber with a full width at half maximum of 127% [24]. In 2017, Kenny designed a planar THz absorber, which is made up of fractal crossed super cells, obtaining a broadband absorption of 2.82 to 5.15 THz [25]. In general, the EM responses of these types of metasurface absorbers are mainly achieved by iteratively solving Maxwell’s equations using the finite element modeling (FEM) or the finite-difference time-domain (FDTD) method [26,27]. Therefore, metasurface absorbers with specific EM properties are designed in a bottom-up manner, which depends on the continuous optimization of the geometric parameters in the process of trial and error by professional researchers. This traditional method is time-consuming and it is difficult to achieve high performance, which hinders the practical application of THz metasurface absorbers.
Thus far, many optimization methods for the quantitative calculation of the EM response on metasurfaces have been proposed [28,29]. Recently, it was shown that the EM field distribution and frequency dispersion of metasurface waves (M-waves) [30] can be well described by the catenary model, which provides a nearly accurate mathematical model for characterizing the EM properties of metasurfaces. Based on this model, a simple but powerful analytical design method is proposed to simplify the design of metasurface-based devices. However, the design of metasurface absorbers is constrained by the manual design process with limited geometric complexity and tedious parameter sweeping. This method cannot be directly employed for inverse designs with specific requirements, which is time-consuming.
In this study, a genetic algorithm (GA) is introduced into catenary field theory for the design of THz metasurface absorbers. The employment of GA can achieve an automatic design process and demand-oriented inverse design for high performance and a short time. As a proof-of-concept, a THz dual-metasurface absorber with absorbance exceeding 88% in the frequency range of 0.21–5 THz was designed using the proposed design method, which is consistent with the simulated results by full-wave simulation based on FEM under the same structural parameters. The inverse design program automatically outputs the optimized structural parameters and absorption spectrum. Moreover, the capability of the inverse design program to design a low-pass filter by adjusting the objective function was further proved. Compared to the traditional design method, the proposed design method can dramatically reduce time consumption and find the optimal result to achieve high performance. The investigations provide important guidance and a promising approach for designing metasurface-based devices for practical applications.
2. Theory and Design method
2.1 Catenary field theory
Owing to the diverse materials and patterns of metasurfaces, it is difficult to analyze the detailed interactions between EM waves and metasurfaces, which poses significant challenges to the design of metasurface-based devices. To date, the most common approach to acquire and analyze the EM response of metasurfaces is to use full-wave simulation based on FEM or FDTD methods, which is performed by constructing a 3D model and optimizing their materials and geometric parameters. This process is time-consuming, and the bottom-up design method relies on the experience of designers and cannot guarantee optimal device performance. To promote the practical application of metasurface devices, an automatic and inverse design method for design requirements is urgently needed.
We revisited the concept of M-waves from a microscopic field of view and concluded that M-waves do not necessarily propagate along the macroscopic surface. The interfaces inside the metasurface also support the propagation of interfacial M-waves [31]. From a microscopic point of view (as shown in Fig. 1(a)), there are an infinite number of interfaces in the metasurface, which support M-waves when the dimension is in the deep-subwavelength scale. In this case, the gaps between the adjacent metal patches can be regarded as capacitance, and the metal patch can be regarded as an inductance, as illustrated in Fig. 1(b). Moreover, the electric field distributions and frequency dispersion of M-waves in these metallic structures were demonstrated to be well-characterized by the catenary model; that is, the EM properties of various metasurfaces can be described by a mathematical model. This model establishes a relationship between the structural parameters and their corresponding equivalent impedance.
The catenary-shape function F can be expressed as [32]:
Here, θ is the angle of incidence, λ is the wavelength, and G is a correction term [33]. p and s represent the structural parameters of the metasurface. Consequently, an analytical method derived from the catenary EM field theory was used to simplify the analysis and design.
Fig. 1. (a) M-waves in the microscopic regime. (b) The electric fields in a thin slit gap in metal, which can be regarded as capacitor (top panel), and the electric fields in a thick patch in metal, which can be regarded as inductor (bottom panel).
2.2 Genetic algorithm
The GA is a classical algorithm in deep learning [34]. It is a computational model that simulates the natural selection and genetic mechanism of Darwin's biological evolutionary theory [35]. The computational process includes crossover, selection, and mutation. It provides a problem-solving framework that is suitable for different scenarios. When solving complex combinatorial optimization problems, it can provide an optimal solution for the objective function. The use of a GA in the design of metasurfaces can help us achieve rapid structural parameter optimization and find the EM response most in line with the design requirements. It is expected to provide an automatic inverse design method to reduce time consumption. The flowchart of the algorithm is shown in Fig. 2.
Based on the above theoretical analysis of the catenary field model, we established an efficient and robust mapping relationship between the geometric parameters and their corresponding EM responses. Subsequently, with the help of a GA, we performed an automatic and inverse design for a broadband THz dual-metasurface absorber. In this method, the mapping relations are described by functions that efficiently calculate the reflective and absorptive spectra at each sampling point during the mutation process. MATLAB programming based on catenary field theory was used in this study. For the optimal process, a server with an Intel Core CPU I9-10900 K @3.7 GHz (10 cores, 20 threads) and 64 GB memory was employed. During the optimal GA process for the THz dual-metasurface absorber design, we observed consumption of 20 GHz for the CPU and 2 GB of memory.
3. Analysis and Discussions
For the dual metasurface in this study, we utilized a dual-metasurface absorber to apply automatic and inverse design methods. The pattern of the dual metasurface is shown in Fig. 3(a) and (b), whose geometric parameters are also illustrated in the figure. The metasurfaces are made up of chromium with a conductivity of 2.2×105 S/m, and the insulator comprises SU-8 photoresist with a permittivity of 2.79 + i 0.3. The low cost and stability of chromium and the easy availability of SU-8 are the main reasons for us to choose them, and their feasibility has also been experimentally verified in our previous work. First, we describe the EM properties of a dual metasurface using a mathematical model derived from the catenary field theory. Based on the above interpretation, the bottom layer metasurface is equivalent to the circuit model comprising resist R1, capacitor C1, and inductor L1, as depicted in Fig. 3(d). The EM features of the metasurface can be described by the equivalent impedance, and the equation can be expressed as [36,37]:
The detailed expressions of R1, C1, and L1 can be found in our previous article [32]. Among the equations, the impedance dispersion (capacitance and inductance at different frequencies) can be described using a mathematical model, such as the catenary-shape function, which builds the linkage between structural parameters and their corresponding equivalent impedance. Similarly, we can deduce the equivalent circuit model of the upper-layer metasurface, which comprises resist R2, two inductors Ls1, Ls2, and two capacitors Cs1 and Cs2. The corresponding equivalent impedance can be expressed as
Fig. 3. Geometric parameters and the corresponding circuit model derived from catenary field theory for upper-layer metasurface (a) and bottom-layer metasurface (b). (c) Schematic of the transmission line model for broadband THz absorption. (d) Equivalent circuit model of the THz dual-metasurface absorber.
After obtaining the equivalent impedance of each layer, the EM response of the entire metasurface absorber can be obtained using the transmission-line model. When the plane wave normally impinges on the dual-metasurface absorber, owing to reflections occurring at the metasurface and background, both forward-scattering waves exist in the dielectric spacer and surrounding space, as depicted in Fig. 3(c). Thus, we can obtain the equivalent circuit model of the dual-metasurface absorber, as shown in Fig. 3(d). Therefore, the reflection coefficient of the dual-metasurface absorber is expressed as follows:
where Zin and Z0 represent the input and free-space impedances, respectively. The absorbance spectra of the dual-metasurface absorber were then calculated using $A(\omega ) = 1 - R(\omega )$. Because the dual-metasurface absorber is a reflective device, its transmittance can be ignored. The input impedance can be calculated using the following iterative formula [38]:The calculations for some variables (such as βd and Zdi) can be found in our previous papers [32]. Thus far, we have established a mapping relationship between the EM responses (reflectance and absorbance spectra) and the structural parameters of a dual-metasurface absorber based on catenary field theory.
As shown in Fig. 3, the parameters [l1, w1, g1], [l3, w2, g2, w3, g3], and [d1, d2, d3] influence the absorption of the THz absorber. Therefore, the geometric parameters of the dual-metasurface absorber are regarded as the population that evolves for ultra-broadband THz absorption. In the written program, the initial population comprises [l1, w1, g1], [l3, w2, g2, w3, g3], and [d1, d2, d3], and the range of the parameter is depicted by Eq. (13). The range of these parameters is determined by their relationship in actual dimensions and the results in our previous work. Moreover, the range was adjusted continuously with trial and error of running programs. The period of the single loop and double loop can be calculated by Eq. (14) and Eq. (15), respectively.
Objective function:
Here, Ai is the absorbance of each sampling point in the desired frequency range of 0.1–5 THz and AD is the desired absorbance. The total number of sample points was 1001, thus, the frequency range was divided into 1001 segments. n1 and n2 are the numbers of sampling points of the upper and lower boundaries of the desired frequency range, respectively. Based on the catenary field theory, we can establish a direct linkage between the structural parameters and the corresponding equivalent impedance. Through the iterative formula, the EM response (reflectance, transmittance, and absorbance) of metasurface-based devices can be obtain. Therefore, the absorbance of THz absorber to be designed can be calculated quickly based on this method. The genetic algorithm was employed in this method to seek for optimal parameters to achieve broadband impedance matching. The structural parameters of metasurfaces constitute the population of genetic algorithms, and their equivalent impedances are regarded as fitness of population. When we want to design a THz absorber with a desired absorption in specific frequency range, the corresponding equivalent impedance (i.e. dispersion distribution) of every layer metasurface can be calculated. The population with higher fitness can be calculated by genetic algorithm, and its electromagnetic response can be substituted into the objective function for judgment. To design an ultra-broadband and highly efficient THz absorber, we set the AD to 1, which represents the desired absorbance in the designed frequency range of 100%. n1 and n2 are set to 1 and 1001, respectively; that is, the desired frequency range is 0.1–5 THz. Figure 4 is the flow chart of inverse design method based on GA. The number of populations and individuals was set as five to ensure that the running time of the program was shortened under the condition of high accuracy of the results. The dimensions of the variables are consistent with the number of variables in the initial population. The crossover probability range was set as 0.7–0.9, and the mutation probability range was set as 0.001–0.05. The objective function is set to the maximum value of (16). The algebra whose optimal value remains unchanged is the retention algebra. The basis for judging whether the retention algebra changes is whether the current optimal value is greater than the previous optimal value. The minimum retention algebra of the optimal individual was set to five, and the initial retention algebra was set to zero. When the retention algebra exceeds five, the cycle ends, and the program outputs the absorbance curves that meet requirement with optimal structural parameters. When the retention algebra is less than five, the program can automatically calculate each population in turn. When the calculation of the offspring generation objective function is completed, the offspring generation objective function will be insert into the current population. It will replace the calculated value of objective function and returns the result population. This is the detailed process of merge offspring and father in the Fig. 4.
Thereafter, the written program automatically outputs the optimal structural parameters and the corresponding absorption spectra. The optimal absorbance is shown in Fig. 4(a) (black solid line) and the optimal geometric parameters are listed in Tables 1–3. The absorbance of the designed dual-metasurface absorber exceeded 88% in the range of 0.21–5 THz. In particular, the absorbance exceeds 98% at lower frequencies (below 0.5 THz), which is rare in previous studies. The output geometric parameters shown in the table are in line with the actual situation. To further verify the feasibility of the inverse design, a commercial software package based on the FEM (CST Microwave Studio) was used to simulate absorbance under the same parameters. In the simulations, the unit cell boundary conditions were applied in the x and y directions, whereas the top and bottom boundaries normal to the z-axis were set as open. Due to the symmetry of the structure, we applied the normally incident TM-polarized plane wave with the polarization of the electric field along the x direction. The simulation results are depicted in Fig. 5(a) using a red solid line. It can be observed that the red line is consistent with the black line. Although there are slight differences between these two lines, the overall results prove the feasibility of the automatic and inverse design methods. The introduction of the GA enabled us to quickly determine the optimal parameters and desired characteristics. The combination of an inverse design program and full simulation based on the FEM or FDTD can quickly and accurately realize the design of metasurface devices. This design strategy can significantly reduce time consumption and improve design efficiency.
Table 1. Optimal geometric parameters of the lower layer metasurface
Table 2. Optimal geometric parameters of the upper layer metasurface
Table 3. Optimal geometric parameters of the height of three layer insulators
To demonstrate the advantages of the proposed strategy in terms of design speed, the evolution process is described in Fig. 5(b). The program was run in MATLAB software, and the server configuration for running this software was as introduced above. As shown in the evolution process of the design program, the evolution algebra was repeated 20 times. According to the result of running the program, the time consumption for the program to evolve one generation is 51 s; that is, the entire design process only takes 1020 s. If we use the traditional design method of full-wave simulation based on parameter sweeping, the designers must optimize 11 parameters and continuously summarize the law of adjusting parameters to find the best set of parameters that can meet the design requirements. This design process will take at least several days or even longer. For example, using the CST optimization algorithm to optimize a generation takes an average of 15 minutes. While GA can almost run the entire design process with the same time. Therefore, the automatic and inverse design methods show a higher design efficiency than traditional methods.
Because the dual metasurfaces are symmetric, the EM responses are polarization-independent. Here, we only extracted the electric-field distribution in the TE mode to explain the mechanism of broadband absorption. Figure 6 depicts the instantaneous electric field distribution of the metasurface at the resonant frequencies (0.42, 1.20, and 2.76 THz) in the X-Y plane under normal incidence. The strong resonance at 0.42 THz occurs at the adjacent single loop of the lower layer metasurface and the metal patch of the upper layer metasurface, as shown in Fig. 5(a) and (b). At a resonant frequency of 1.20 THz, strong electric field confinement mainly occurs in the corner and center of the double loop. Simultaneously, the electric field is confined between adjacent loops of the bottom metasurface. For higher frequencies (2.76 THz), all resonances occurred in the gap between the upper and lower metasurface units, and a hybrid mode appeared between the two metasurfaces. The electric-field distribution illustrates the mechanism of broadband absorption.
Fig. 5. (a) Comparison of the absorption calculated by full wave simulation and circuit model. (b) Inverse design evolution process of the dual metasurface.
To further demonstrate the capability of the inverse design program, we employed low-pass filter spectral properties as the objective function for metasurface design. The objective function is the same as that described above, but we need to add a limited condition to obtain the specific absorbance. The condition is set to calculate the objective function when the number of sampling points is greater than 390(corresponding to 2 THz). The structural form of the metasurface was the same as that described above. The output optimal results are shown in Fig. 7(a), where the black line represents the absorbance and the red line represents the reflectance. The optimal structural parameters of the dual metasurface are indicated by the rectangular box in the figure. The absorbance is close to 1 in the range of 2–5 THz, and the corresponding reflectance is close to 0 in the same range. The optimal results reveal that the metasurface device allows only low-frequency THz waves to pass, and high-frequency waves are absorbed, showing low-pass filtering characteristics. The optimal results meet our design requirements. The result in Fig. 7(b) shows that the entire simulation process has only undergone 40 iterations, which proves that the design process is fast. The optimal spectral curve can be effectively controlled by the GA, which provides a promising way to optimize more powerful metasurface-based devices in the future.
Fig. 6. Electric field distribution in the lower layer metasurface at resonant frequency of 0.42 THz (a), 1.20 THz (c), and 2.76 THz (e). Electric field distribution in the upper layer metasurface at resonant frequency of 0.42 THz (b), 1.20 THz (d), and 2.76 THz (f).
Fig. 7. (a) Absorbance and reflectance of designed low-pass filter designed by inverse design method. (b) Inverse design evolution process.
4. Conclusion
We introduced the GA into the catenary theory model to propose an automatic and inverse design method for THz metasurface absorbers. The GA method was employed by seeking optimal dispersion distributions to achieve broadband impedance matching. As a proof of concept, a THz dual-metasurface absorber was designed using the proposed approach. The designed metasurface absorber exhibits an absorbance exceeding 88% at 0.21 to 5 THz, which is in good agreement with the results of the full-wave simulation based on the FEM. The inverse design program automatically outputs the optimal parameters and characteristics. Moreover, the capability of the inverse design program to design a low-pass filter by adjusting the objective function was further proved. Compared to the traditional design method, the proposed design method can dramatically reduce time consumption and find the optimal result to achieve high performance. The investigations provide important guidance and a promising approach for designing metasurface-based devices for practical applications.
Funding
National Natural Science Foundation of China (62105093); Natural Science Foundation of Hebei Province (F2020208005); Hebei Provincial Key Research Projects (19255901D, 20355901D); Science and Technology Project of Hebei Education Department (QN2020435); Doctoral Research Initializing Fund of Hebei University of Science and Technology (1181382).
Disclosures
The authors declare no conflicts of interest related to this study.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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