1. Introduction

Electromagnetic metamaterial perfect absorbers can significantly enhance light-matter interaction at the nanoscale, which offer new opportunities to mold the flow of light, showing great applications in the fields of on-chip optical communications [1,2], all-optical switches [3,4], biosensors [5–7] and photodetectors [8–10]. For the common electromagnetic metamaterial absorbers, perfect absorption is realized by employing critically coupling under a single beam. This inevitably leads to limited and passive tunability of their absorption intensity. Consequently, coherent perfect absorption (CPA) has been introduced to actively tune absorption intensity via manipulating the properties of two coherent beams, which has received a steadily increasing attention [3,11]. Recently, various kinds of coherent perfect absorbers based on metamaterial structures have been proposed to actively modulate the optical absorption of coherent absorption absorbers, including thin film structure [3,12], grating structure [13], and rectangular array structure [14,15]. By simply manipulating the relative phase of two coherent beams without changing the device structural parameters, the absorption intensity of the absorbers can be dynamically adjusted from nearly 0 to 100% [16,17]. However, the above coherent perfect absorbers can only work at a single resonance wavelength, since the CPA system is illuminated coherently by the time reverse of the output of a lasing mode [18,19], which significantly limits the operation bandwidth of the absorbers. Such a narrow operation bandwidth severely hinders its broadband applications.

To solve this problem, two main strategies have been proposed to realize the broadband coherent perfect absorbers [20,21]. One strategy is to integrate multiple metamaterial structures with gradually varying geometric sizes into the same plane [20,22,23]. By simultaneously exciting multiple resonances to superpose several absorption peaks close together, the absorption bandwidth of coherent absorption absorbers can be broadened to 29.68% [24]. However, the absorber based on this structure endures out-of-control mode splitting, due to the mutual coupling between multiple metamaterial structures [25]. The other strategy is to vertically stack multiple layers of metamaterial structures [21,26]. The cross-shaped metasurfaces based on Dirac semi-metallic materials (BDS) and VO₂ are separated by dielectric layers with appropriate thickness. By simultaneously optimizing the Fermi level of BDS and the conductivity of VO₂ to tune the resonance modes of the two metasurfaces, the relative bandwidth of 30.16% can be realized [21]. Whereas, the size of multilayer composite structures is extremely large, and the fabrication of multiple aligned nanostructures is complex and time-consuming. Particularly, the performance of the absorbers based on above strategies extremely rely on the physical insight, expert experience, and trial-and-error optimization. This significantly reduces the design efficiency and results in limited geometry variety of metamaterial structure [27–29], which restricts the further broadening of the bandwidth of coherent perfect absorbers. Different from trial-and-error optimization strategy, inverse design methods based on genetic algorithm (GA) intelligent program have been demonstrated to realize high-performance metamaterial devices for all-optical modulator [30], all-optical polarizer [31] and optical chips [32]. GA inverse design can automatically search and optimize the target structure through the computer-aided design process, while ignoring the underlying physical mechanism [29,33]. More importantly, GA inverse design can develop metamaterials with rich geometric variety and degrees of freedom, which can realize high-performance devices beyond the traditional artificially designed devices [34,35], providing a brand-new route to realize the broadband coherent perfect absorbers.

Herein, we designed a tungsten-based metamaterial absorber by employing GA intelligent program to achieve broadband CPA effect. Compared with the precious metals (for instance gold, silver and copper), the tungsten has better thermal tolerance and relatively lower cost. Particularly, tungsten has a large inherent loss in the near-infrared band, making it an excellent material for broadband absorbers [36]. By optimizing the geometry of metamaterial via GA, the coherent absorption (>90%) was realized within a broad range from 1.32 to 3.28 µm, thereby achieving a relative bandwidth of 85.22%. The multiple closely wavelength-spaced resonances were excited to cover broadband absorption in the near-infrared region. The scattering matrix theorem is employed to declare the physical mechanism of CPA, and numerical simulations are applied to demonstrate the active tunability of proposed CPA metamaterial absorber. It is shown that the absorption intensity over a wide range can be continuously modulated via simply adjusting the relative intensity or phase difference of the two coherent beams. Moreover, the absorption performance at oblique incidence is studied. It is found that the coherent absorption maintains more than 90% over a broad range of incident angles for both TM and TE polarizations, exhibiting exceptional incidence angle insensitivity. The proposed tungsten-based CPA metamaterial absorber presents excellent broadband tuneable property, which is strongly anticipated to opens a new opportunity toward optical signal processing chip, all-optical modulators, and optical switchers.

2. Structure and principle

The schematic diagram of the proposed CPA metamaterial absorber is shown in Fig. 1(a), which is composed of the vertically stacked dielectric/metal/dielectric structure. In practice, such device may be fabricated by the following steps [34]. Firstly, the optimized binary-pattern tungsten-based metamaterial structure is obtained by GA intelligent program. Subsequently, the metallic tungsten is deposited on the dielectric silicon nitride (Si₃N₄) by radio frequency sputtering. Then, the customized binary-pattern metamaterial is fabricated with focused ion beam (FIB) milling on the tungsten layer. Finally, the Si₃N₄ layer is deposited on the top of tungsten/Si₃N₄ structure by radio frequency sputtering. Compare to the perfect absorbers based on precious metals, the proposed tungsten-based metamaterial absorber has better thermal tolerance and relatively lower cost. The optimized binary-pattern tungsten-based metamaterial with the top view is illustrated in Fig. 1(b). The binary-pattern in the unit cell consists of a 12 × 12 pixel array filled with tungsten or air, where the pixel width w = 100 nm. The period P of nanostructure is 1200 nm, and the thicknesses of tungsten and Si₃N₄ are 45 nm and 60 nm, respectively. To investigate the CPA characteristics of the proposed metamaterial absorber, the finite difference time domain (FDTD) method is employed to perform the simulation calculations. In the simulations, the periodic boundary conditions are applied in the x- and y-directions, and the perfect matching layer is applied in the z-direction. Non-uniform mesh setting is implemented to save storage space and computation time. The minimum element mesh size in tungsten is set to 5 nm along the z-directions, and gradually increases outside the tungsten. The element mesh size is set to 10 nm and 10 nm along the x- and y-directions, respectively. The permittivity parameters of tungsten and Si₃N₄ are taken from Palik [37] and Kischkat et al. [38], respectively. To realize CPA condition, the two beams of coherent light (I₁ and I₂) with the same intensity illuminate the metamaterial absorber from opposite z-direction. As shown in Fig. 1(c), when the phase difference of coherent beams Δφ = 2nπ (n = 0, 1, 2…), they are symmetrically distributed on both sides of the absorber, which is called even modes. Due to the destructive interference effect of the two coherent beams, the scattering lights (O₁ and O₂) are suppressed, and the coherent absorption is enhanced. Meanwhile, when the phase difference Δφ = 2(n+1)π (n = 0, 1, 2…), the odd modes are formed. Due to the constructive interference effect of the two coherent beams, the scattering light is enhanced, resulting in a decrease in the coherent absorption. The coherent absorption close to 1 is called CPA effect, while the coherent absorption close to 0 is called coherent perfect transmission (CPT) effect.

 

Fig. 1. (a) Schematic diagram of the proposed tungsten-based CPA metamaterial absorber. The inset shows the cross view of the CPA metamaterial absorber, where the thickness of Si₃N₄ layer and tungsten layer are d_Si3N4 = 60 nm and d_w = 45 nm, respectively. (b) The binary-pattern tungsten metasurface of the unit cell, where the period is P = 1200 nm, the width is w = 100 nm. (c) The coherent absorption spectra at the phase difference of coherent beams Δφ = 2nπ and 2(n+1)π (n = 0, 1, 2…).

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To better understand the propagation of coherent beams in the proposed tungsten-based CPA metamaterial absorber, the scattering matrix S can be derived. When two coherent beams (I₁ and I₂) irradiate the metamaterial absorber, defined as coherent illumination, the output beams (O₁ and O₂) can be described as [11]

The coherent absorptivity A_co is related to the relative intensity α and phase difference Δφ, indicating that A_co can be continuously modulated by adjusting α and Δφ from 0 to 1. When the relative intensity α of two coherent beams is equal to 1, Eq. (3) can be written as

To improve the design efficiency of the metamaterial for broadband CPA, the GA intelligent program is used to search and optimize the coherent absorption. As a gradient-free algorithm, the GA can easily escape local optima and efficiently find high performance designs from large discontinuous solution spaces [33,39]. The flowchart of GA inverse design combined with FDTD is shown in Fig. 2(a). Firstly, before running the GA program, we need to define basic design parameters (the encoding method of tungsten-based metamaterial) and the fitness function. The geometry of tungsten-based metasurface in a unit cell is pixelized into the chromosome with binary numbers, where “1” represents tungsten and “0” represents free space, as illustrated in Fig. 2(b). The code of the green region is generated by the program each time, and the gray region code is generated by the mirror symmetry of the red region. In order to realize the polarization-independent optical response, the codes of green region and the gray region are regarded as a whole, and counterclockwise rotations of 90°, 180°, and 270° are performed to the code in a unit cell. The fitness function is defined as $F\textrm{ = }\sum\nolimits_{{\lambda _i}} {\sum\nolimits_{{\theta _j}} {{A_{TM}}{{(\lambda )}^2}} } + {A_{TE}}{(\lambda )^2}$ in the 1-4 µm to evaluate the coherent absorption properties of each pixelized structure, where A_TM and A_TE are absorptivity under TM-polarized and TE-polarized coherent beams illumination, respectively, λ_i and θ_j are the wavelengths and incident angles, respectively. The incident angles θ_j are chosen to be 0, 10°, 20° and 30°. Subsequently, after running the program, the GA program will randomly generate 30 initial populations, corresponding to 30 random geometries of metasurface in FDTD. These populations are sequentially passed to the FDTD to calculate the coherent absorption spectra. Next, the GA program selects high-quality individuals from the 30 populations and removes low-quality individuals, where the high-quality individuals refer to metasurface with higher F values. Individuals are then optimized through cross-over and mutation operators, as shown in Fig. 2(c). Here, the cross-over operation refers to selecting two high-quality individuals as parents, and replacing and recombining parts of their codes to generate new individuals. Mutation operation refers to changing the parts of codes in an individual, that is, the binary value changes from 0 to 1 or from 1 to 0. After cross-over and mutation operations, the new generation population is passed to the FDTD for the next iteration optimization. Optimization process continues until the 150th iteration, and the maximum value and the mean value of fitness F in each iteration are depicted in Fig. 2(d). It is noting that as the iteration increases, F_max of populations keeps rising until it reaches to the optimal result (the maximum value). F_mean of the population follows the F_max rising, and eventually fluctuates. The individual with F_max in the 150th iteration is selected as the final optimal metamaterial structure.

 

Fig. 2. (a) Flowchart of GA program combined with FDTD for broadband CPA optimization. (b) Binary-pattern in the unit cell consists of a 12 × 12 pixel array filled with tungsten or air. (c) Schematic diagrams of cross-over operation and mutation operation. (d) Max fitness and mean fitness as the iterations increases.

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3. Results and discussion

To illustrate the formation condition of the broadband CPA in the GA program optimized tungsten-based metamaterial structure, comparison simulations were performed under the inherent and coherent light illumination. The spectra of the absorber under the incoherent light illumination are shown in Fig. 3(a), and the corresponding phases of transmission and reflection coefficients are depicted in Fig. 3(b). It can be seen that the same reflection and transmission intensities can be simultaneously obtained at 1.78 µm, 2.55 µm and 2.65 µm. Additionally, the phase differences of transmission and reflection are close to π only for these three wavelengths, which satisfy the conditions of coherent absorption: |t´| = |r´|. When two coherent beams with the same intensity and a relative phase difference of 0 are simultaneously irradiated onto the metamaterial structure, the interference of coherent beams can suppress the scattering of light, resulting in an increase in the absorption intensity. When the condition of coherent absorption is satisfied, the coherent absorptivity of 99.52%, 99.89% and 99.99% can be achieved at 1.78 µm, 2.55 µm and 2.65 µm, respectively, as presented in Fig. 3(c). The relative bandwidth can be defined as (λ_max−λ_min)/λ₀, where λ_max, λ_min, and λ₀ are the maximum, minimum and center wavelength, respectively [40]. It is noting that the high coherent absorption (>90%) can be achieved within abroad range from 1.32 to 3.28 µm, and a broad relative bandwidth of 85.22% can be obtained, which is strikingly larger than that of the CPA metamaterial structure [19,21,22]. Figure 3(d) shows the difference phase of reflection and transmission coefficients. It is further proved that the phase differences are closely to 0, satisfying the condition of coherent absorption: |t´| = |r´|. The electric field distributions |E| on the top of tungsten-based metamaterial structure at 1.78 µm, 2.55 µm and 2.65 µm are displayed in Fig. 3(e). It is found that the electromagnetic energy is distributed in certain locations of the metamaterial structure at different resonance wavelengths. The multiple plasmonic resonances supported in the metamaterial structure generate the broadband coherent absorption. Furthermore, since tungsten is regarded as a non-magnetic dispersive medium in the proposed absorber, the time-averaged dissipated power density is ${Q_h} = \frac{{{\varepsilon _0}\omega \varepsilon _m^{^{\prime\prime}}(\omega ){{|E |}^2}}}{2}$, where $\varepsilon _m^{^{\prime\prime}}(\omega )$ represents the imaginary part of the dielectric constant of tungsten [41]. The absorbed electromagnetic energy is finally converted to heat dissipation owing to the optical loss in tungsten.

 

Fig. 3. The (a) spectra and (b) phase of reflection and transmission under a single beam illumination. (c) Absorption and scattering intensity under two coherent beams illumination. (d) Phase difference of reflection and transmission coefficients under two coherent beams illumination. (e) Corresponding electric field distributions at the resonance wavelengths λ = 1.78 µm, 2.55 µm and 2.65 µm.

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Then we investigate the phase modulation property of proposed CPA metamaterial absorber. The coherent absorption mapping with varying phase difference of two coherent beams Δφ is shown in Fig. 4(a). It can be seen that the coherent absorption of metamaterial absorber can be periodically modulated by Δφ, and the corresponding period is 2π. As Δφ varies from 0 to π, the coherent absorptivity in the 1-4 µm continuously decreases from the maximum value to the minimum value, resulting in a decrease in the relative bandwidth. When Δφ continues to increase to 2π, the coherent absorptivity increases from the minimum value to the maximum value. To be more intuitive, the extracted coherent absorption is a function of Δφ at 1.78 µm, 2.55 µm and 2.65 µm, as depicted by the solid line in Fig. 4(b). The three curves have the same variation trends, and the coherent absorptivity at 1.78 µm, 2.55 µm and 2.65 µm can be actively and continuously tuned from 99.52% to 0.61%, 99.89% to 0.31%, and 99.99% to 0.30%, respectively. Thus, coherent absorptivity varies with increasing Δφ from 0 to π, enabling the transition from the CPA to the CPT. To quantitatively describe the modulation phenomenon, we compared the extracted coherent absorptivity with theoretical calculated by Eq. (4). It is indicated that the coherent absorptivity modulation followed the cosine law, and the theoretical calculated results exhibit good consistency with the simulation results. Figure 4(c) shows the magnetic field distributions of H_z component at Δφ = 0 and π at 1.78 µm, corresponding to CPA and CPT. When Δφ = 0, the magnetic field localizes around the metamaterial structure due to the destructive interference of scattered beam, thus generating a CPA effect. As for Δφ = π, a large proportion of energy escapes from the metamaterial structure due to the constructive interference of scattered beam, thus generating the substantial reduction of absorption in CPT effect. We further show that such continuous modulation of absorption intensity by simply controlling Δφ can be realized for both TM and TE polarizations, as shown in Fig. 4(d). With the help of active tunability, the proposed absorber is expected to be utilized in optical modulators and switchers.

 

Fig. 4. (a) Coherent absorption mapping map as Δφ increases from 0 to 4π. (b) Extracted coherent absorptivity from (a) at 1.78 µm, 2.55 µm and 2.65 µm. The solid lines and circles represent simulation and theoretical calculation results, respectively. (c) Corresponding magnetic field distributions of H_z component with Δφ = 0 and π at 1.78 µm. (d) Coherent absorptivity as a function of Δφ at 1.78 µm, 2.55 µm and 2.65 µm for TM and TE polarizations.

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It is further found that the absorption intensity can also be actively modulated by tuning the relative intensity of coherent beams. The coherent absorption mapping as a function of the relative intensity and wavelength at the phase difference of 2nπ (n = 0,1,2…) is depicted in Fig. 5(a). It is found that coherent absorption of metamaterial absorber can also be adjusted by relative intensity α. As α varies from 0 to 1, the coherent absorptivity increases from the minimum value to the maximum value. Figure 5(b) shows the extracted coherent absorptivity as α varies from 0 to 1, where the lines and symbols represent the simulation and theoretical calculation results, respectively. When the relative difference Δφ of 2nπ (n = 0,1,2…) is fixed, the Eq. (3) can be written as ${A_{co}} = 1 - \frac{1}{2}\frac{{{{(1 - \alpha )}^2}}}{{1 + {\alpha ^2}}}$. It can be observed that when α = 0, the coherent absorptivity of nearly 50% can be achieved at 1.78 µm, 2.55 µm and 2.65 µm, corresponding to the single incoherent light illumination. As α varies from 0 to 1, the coherent absorptivity increases from nearly 50% to nearly 100% due to the interference of coherent beams. This reflects that the simulation matches the theoretical calculation result well. Therefore, it is feasible to actively modulate the coherent absorptivity by adjusting the relative intensity of coherent beams, exhibiting the flexible tunability of the proposed absorber.

 

Fig. 5. (a) Coherent absorption spectra as a function of wavelength and relative intensity of two coherent beams. (b) Coherent absorption intensity with the relative intensity varying from 0 to 1 at 1.78 µm, 2.55 µm and 2.65 µm extracted from (a). Lines and symbols represent simulation and theoretical calculation results, respectively. (c) Coherent absorptivity and (d) Relative bandwidth with varying incident angles for TM and TE polarizations. (e) Coherent absorption spectra as a function of wavelength and thickness of Si₃N₄ layer. (f) Corresponding modulation depth with different thickness of Si₃N₄ layer.

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The polarization angle is another key factor for coherent absorption modulation of proposed CPA metamaterial structure. The coherent absorptivity and the relative bandwidth with varying incident angles from −90° to 90° for TM and TE polarizations are displayed in Figs. 5(c) and 5(d), respectively. As for both polarization directions, the coherent absorptivity maintains greater than 90% within a wide angle range from −60° to 60°, and then decreases sharply as incident angle continues to increase, since the incident electromagnetic field cannot effectively support electromagnetic resonance [16]. Besides, the relative bandwidth maintains greater than 73.12% within a wide incident angle range from −40° to 40°, and the maximum value of 102% and 86.84% can be achieved for TM and TE polarization, respectively. These features suggest that the proposed modulator can effectively operate over a wide range of incident angles for both TM and TE polarizations.

We finally investigated the effect of the thickness of the Si₃N₄ layer on the coherent absorption performance of proposed tungsten-based CPA metamaterial structure. The coherent absorption spectra under fixed Δφ of 0 and α of 1 with different d_Si3N4 are shown in Fig. 5(e). It can be seen that as the d_Si3N4 increases from 50 nm to 90 nm, the resonance wavelengths of coherent absorption are all red-shifted, and the coherent absorptivity at resonance wavelengths keeps greater than 99%. To quantitatively assess the modulation capability of CPA metamaterial structure, the modulation depth of tungsten-based CPA metamaterial structure with different d_Si3N4 is depicted in Fig. 5(f). The modulation depth is defined as M(λ) = max(SI)/min(SI), where SI represents the normalized total output intensity, denoted by SI = |r + t|² [40]. When d_Si3N4 varies from 50 nm to 90 nm, the modulation depth maintains more than 3.38 × 10³ and the maximum modulation depth of 4.54 × 10⁴ at 2.55 µm can be achieved. Therefore, the proposed CPA metamaterial structure possesses excellent modulation capability with high modulation depth.

4. Conclusion

In conclusion, we presented the GA intelligent program enabled design and optimization of the tungsten-based broadband CPA metamaterial absorber. The broadband coherent absorption properties are contributed from the multiple closely wavelength-spaced resonances. Simulations showed that a high coherent absorption (>90%) can be achieved within a wide range from 1.32 µm to 3.28 µm, and a broad relative bandwidth of 85.22% can be obtained. By actively adjusting the relative intensity or phase difference of the two coherent beams, the absorption intensity can be continuously modulated, and the maximum modulation depth of 4.54 × 10⁴ can be realized at 2.55 µm by proper selection of the thickness of dielectric layer. Moreover, the relative bandwidth maintains greater than 73.12% within a wide incident angle range from −40° to 40° for both TM and TE polarizations. The scattering matrix theorem is applied to explain the physical mechanism of CPA, and the analytical results exhibit good consistency with the numerical calculations. This result provides an efficient approach to design tungsten-based CPA metamaterial absorber with rich geometric variety, which offers an effective way to achieve the broadband light absorption modulation, accelerating the development of active optoelectronic devices.