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Abstract
An ultra-broadband metamaterial absorber, consisting of 2D SiO2-Ti square bilayer grating on SiO2 film and Ti substrate, is proposed and designed by rigorous coupled wave analysis (RCWA) and genetic algorithm (GA) methods. The optimized structure shows an average absorption of 98.3% in the wavelength range of 300 nm to 2100 nm. Moreover, the metamaterial absorber is polarization-independent and also insensitive to incidence angle for both TM- and TE-polarized waves. The physical mechanisms responsible for nearly perfect broadband absorption, including the Wood's anomaly (WA), cavity resonance (CR), surface plasmon polaritons (SPPs) and the resonance of magnetic polaritons (MPs), have been analyzed clearly by finite-difference time-domain (FDTD) method and the inductor-capacitor (LC) circuit model. Overall, the proposed metamaterial absorber is a promising candidate in solar applications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metamaterial absorbers have attracted substantial research attention, because of their vast range of applications in color filters [1], solar–thermal energy harvesting [2], selective thermal emitters [3], photovoltaics (PV) [4] and sensors [5]. In general, strong light-matter interactions, such as cavity resonance (CR), surface plasmon polaritons (SPPs) and magnetic polaritons (MPs), are utilized to achieve large absorption enhancement in a broad or narrow wavelength range. The narrowband absorbers are of great importance for optical sensing and thermal emission applications. However, in some particular areas, such as solar energy harvesting and thermal emitters, it is highly desirable to achieve perfect absorption over a wide spectral band.
Hence, various structures have been proposed to obtain light absorption in a broad wavelength range, such as metallic gratings [6–8], photonic crystals [9,10], nanocomposites [11,12], quasi-periodic nanocones [13] and graphene nano-discs [14]. Mostly, the absorbers are composed of metallic gratings because of their simple design and cost-effective fabrication. For instance, Nguyen-Huu et al [15]. designed the solar absorber by employing simple tungsten (W) deep grating. Moreover, Chang et al [16]. have utilized W nanowires grating to achieve broadband absorption.
Furthermore, several methods can be employed to increase the broadband absorption of metamaterial absorbers with grating structures. For instance, the gratings with metal-insulator-metal (MIM) structure render ultra-broadband absorption. Lei et al [17]. employed a periodic array of titanium-silica (Ti-SiO2) grating deposited aluminum bottom film to achieve an average absorption of ∼97% in visible to near-infrared wavelength. However, the absorption drops below 90% around 1000 nm. Zhao et al [18]. numerically designed the metamaterial structure composed of W square gratings on SiO2 spacer and W substrate to obtain broadband absorption. However, the absorber has a relatively low average absorption (<90%) in the visible region.
Recently, some studies have pointed out that coating a dielectric film on the top metallic grating can additionally increase absorption. For instance, Silva-Oelker et al [19]. demonstrated the top HfO2 dielectric grating can induce resonance and enhance the broadband absorption of structure. Li et al [20]. obtained over 90% absorption in the spectral range of 570 nm to 3539 nm by placing the insulator-metal-insulator (IMI) grating on MIM film stacks. However, these absorbers are extremely sensitive to the polarization state.
In addition, multi-size grating or multilayer grating stack structure can also broaden the absorption spectrum. Cui et al [21]. proposed a broadband absorber based on an array of multi-sized gold nanostrips. Moreover, Liu et al [22]. demonstrated an ultra-broadband absorber by using a multilayer grating stack structure. However, these structures increase the design and fabrication complexities. In fact, for the absorber based on complicate nanostructures, minor changes in design parameters could change the absorption, and the optimization of high performance is difficult to be achieved. Yeng et al [23]. utilized a multi-level single-linkage method to optimize two-dimensional (2D) tantalum structure, filled with HfO2, and obtained ∼90% absorption in spectral range. Ghanekar et al [24]. optimized periodic SiO2/W gratings by using a genetic algorithm (GA) method and maximized the TPV cell output power. Therefore, optimization algorithm for parameter design is required. So far, it is a very meaningful but challenging work to design a simple grating-structured metamaterial absorber with the characteristics of polarization-independent ultra-broadband absorption.
Herein, we propose an ultra-broadband, omnidirectional, metamaterial absorber based on Ti substrate, SiO2 spacer and 2D SiO2-Ti square bilayer grating. The top SiO2 grating acts as an anti-reflective protective coating and induces resonance, enhancing the absorption performance in the visible wavelength. The proposed absorber can achieve nearly perfect absorption with an average absorption about 98.3% in wavelength range of 300 nm to 2100 nm under normal incidence. In addition, the absorption remains higher than 90% over a wide incident angle range (up to 50°) for both TM- and TE- polarized waves. The wide-angle polarization-independent ultra-broadband absorption behavior indicates the potential of proposed absorber in solar systems and radiation thermal devices.
2. Structure design and optimization
The schematic diagram of the broadband metamaterial absorber is presented in Fig. 1(a), which consists of a 2D SiO2-Ti square bilayer grating arrays on the top, a thin SiO2 dielectric layer in the middle, and an opaque Ti substrate at the bottom. The square bilayer grating arrays are symmetric along the X and Y directions, indicating the polarization-independent optical response of proposed absorber. The top SiO2 grating induces resonance and acts as an anti-reflective protective coating, whereas the middle SiO2 layer acts as an excitation medium for magnetic resonance. Figure 1(b) presents the cross-section view of metamaterial absorber. The structural parameters of the period and width of the square grating are denoted as p and w, respectively. The thicknesses of SiO2-Ti-SiO2-Ti layers, from top to bottom, are denoted as $h_1$, ${h_2}$, ${h_3}$ and ${h_4}$, respectively. The thickness of the bottom Ti substrate (${h_4}$) is set at 300 nm, ensuring zero transmission. Therefore, the absorption can be calculated by A = 1-R, where R represents the reflection. Herein, the non-noble metal (Ti) is chosen as the metallic material, because Ti is highly lossy in visible and near-infrared (NIR) spectral regime, which enhances the ability to absorb sunlight. The optical constant of Ti is taken from the data of Palik [25]. In addition, SiO2 is assumed to be independent of frequency and the refractive index of SiO2 is set at 1.46. In practical application, although the simplified refractive indexes model of these materials may cause minor changes in the absorption spectrum, the conclusion obtained is basically the same [26]. With respect to the experimental fabrication, the standard deposition technique [27] and electron-beam lithography [28] can be applied to fabricate the proposed metamaterial structure.
Fig. 1. The schematic diagram of the broadband metamaterial absorber. (a) Three-dimensional schematic of the absorber. (b) The corresponding cross-section view.
The numerical calculations are carried out by using GA [29] and rigorous coupled wave analysis (RCWA) [30] methods, which are used to iteratively search the design parameters and calculate the absorption spectrum, respectively. Firstly, GA generates population of randomly structural parameters. Then, RCWA is used to calculate the absorption of each population in the discrete wavelength ranges of 300 nm to 2100 nm. The ideal value of absorption is set to be 1 at the same discrete wavelength range. Both the calculated and ideal values are substituted into a fitness function. If the fitness function satisfies the stop criterion, the optimization program is terminated and the best individuals are extracted. But if not, the GA applies selection, crossover, and mutation operations to generate new population (a better quality). The optimization procedure is repeated until the stop criterion is satisfied. More details of the GA optimization can be found in [15,31].
Though the structural parameters are optimized under normal incidence, the absorption is still large when the incident angle is non-zero, which will be shown later. The optimum geometric parameters are fixed as follow: $p$ = 470.6 nm, $w$ = 240.5 nm, ${h_1}$ = 70.1 nm, ${h_2}$ = 205.9 nm, ${h_3}$ = 41.3 nm and ${h_4}$ = 300 nm, as shown in Table 1.
Table 1. Optimum geometric parameters of the metamaterial absorber.
3. Results and discussion
Based on the optimized geometry, Fig. 2 presents the absorption spectra of the metamaterial absorber under normal incident light. Clearly, the absorber exhibits an absorption larger than 94% in the spectral range of 300 nm to 2100 nm. The average absorption in this range is calculated to be 98.3%. Moreover, extremely high absorption (>99%) is obtained in the range of 545 nm to 635 nm and 1371 nm to 1822nm. To verify the accuracy of the RCWA method, the absorption spectra calculated by finite-difference time-domain (FDTD) method is compared with the RCWA method. In the FDTD simulation, a plane wave polarized along the x direction is used as incidence. Periodic boundary conditions are applied in the x and y directions under normal incident light. Perfect matched layer (PML) boundary conditions are applied in the z direction, to eliminate the boundary scattering. The mesh size is set to 0.5 nm in all directions to ensure the accuracy of the calculated results. From this figure, it can be found that two methods show negligible differences, which confirm that our numerical calculation here is reliable.
Fig. 2. The absorption spectra of the metamaterial absorber under normal incident light based on the optimized geometry.
A comparison of the spectral absorption performance for different kinds of absorbers is shown in Fig. 3. In Fig. 3(a), spectral absorption of flat Ti film absorber is greater than 75% in the start wavelength and its spectral absorption abruptly decreases to less than 50%. Compared with flat Ti film absorber, the absorption of flat IM film absorber enhances ∼14% due to the utilization of SiO2 anti-reflective film. In Fig. 3(b), it is clear that grating structure plays a critical role in enhancing the spectral absorption. The IMIM absorber shows superior optical absorption as compared to MIM absorber in the visible spectral region. This is due to the fact that the topmost SiO2 grating can allow multiple electromagnetic resonance modes, such as CR and SPPs, to be excited on the grating structure in the visible spectral region. Therefore, its absorption value is expected to be higher than the MIM absorber.
Fig. 3. The absorption spectral of different absorbers. (a) Flat Ti film absorber and flat IM film absorber. (b) MIM absorber and IMIM absorber.
In this absorber, the physical mechanism of absorption enhancement in visible wavelength range is main due to the excitation of WA, CR and SPPs. The resonance wavelength that causes an increase of absorption can be predicted via analytical solutions as described below.
When a diffraction order (m, n) appears or disappears at the grazing angle, the WA occurs, which causes abrupt changes in the absorption spectra. For a 2D periodic structure, WA can be predicted by the dispersion relation, as given below [32]:
As shown in Eq. (1), the WA mainly depends on the grating period (${\Lambda }$) at normal incidence ($\theta = {0^0}$), and the expression can be simplified as below:
In addition, for a 2D structure with cavities in the plane perpendicular to the vertical direction (z), the resonance wavelength of CR can be calculated by the following equation [33]:
Moreover, SPPs can be excited in periodic gratings because large in-plane wavevectors can be matched for diffracted waves according to the grating, which can significantly enhance the absorption of gratings. Under normal incidence, the resonance wavelength for SPP mode can be theoretically obtained by solving the following equation [35,36]:
The electromagnetic fields and electric field vectors distribution of the 2D structure in the x-z plane at wavelengths of 336 nm, 456 nm, and 769 nm, are calculated and depicted, as is shown in Fig. 4 and Fig. 5. Figures 4(a) and 5(a) show that most of the electromagnetic fields extend far away from the grating surface and concentrates above the grating. In addition, a weaker electric field is confined in the gap between two adjacent gratings. This is ascribed to the coupling between high-order WA and cavity mode. In order to verify the resonance wavelength of WA and cavity mode, we employ the Eqs. (2)–(3). The resonance wavelengths are 332.7 nm when (m, n) = (1, 1) for WA and 345 nm when (${i_1}$, ${j_1}$, ${k_1}$, $\eta $) = (0, 1, 1, 0.95) or (1, 0, 1, 0.95) for CR, which is basically consistent with the absorptance peak at λ = 336 nm.
Fig. 5. The magnetic field distribution in the x-z plane at different wavelengths. The contour represents the magnitude of magnetic field and the arrows indicate the electric field vectors.
Compared with the resonance at 336 nm, the electromagnetic field at a resonance of 456 nm becomes much stronger in the same region, indicating the excitation of first-order WA and cavity mode, as shown in Fig. 4(b) and Fig. 5(b). Therefore, the calculated resonance wavelengths are 470.6 nm at (m, n) = (1, 0) or (0, 1) for WA and 457 nm at (${i_1}$, ${j_1}$, ${k_1}$, $\eta $) = (0, 1, 0, 0.95) or (1, 0, 0, 0.95) for CR. It can be seen that there is a good agreement between the calculated resonance wavelength and absorption peak.
For the wavelength at 769 nm, the high absorption is mainly due to the excitation of the SPPs. In Fig. 4(c), the strong localized electric field starts to concentrate at the surface and sidewalls of the Ti grating, leading to a much stronger coupling effect. The coupling effects give rise to the enhancement of absorption. In addition, the magnetic field is tightly confined within the topmost SiO2 grating and rapidly decays away from the interfaces into the air, as shown in Fig. 5(c). These field patterns characterize the existence of the SPP resonance [37]. The resonance can be predicted by using the Eq. (4), and the calculated wavelength is 689 nm, which seems far apart from the absorptance peak at λ = 769 nm. This is due to the coupling of the SPPs with other resonances, such as magnetic resonance.
The metamaterial absorber supports the excitation of MPs in the gap of Ti grating and SiO2 spacer, referred as gap MPs and spacer MPs, respectively. Moreover, the corresponding magnetic field distributions are calculated at the resonance wavelength of 1016 nm and 1648 nm, respectively. Figure 6(a) illustrates the x-z cross-section of the magnetic field distribution at the resonance wavelength of 1016 nm. The arrows indicate the electric field vectors. Owing to the existence of SiO2 layer, most of the magnetic field in the air gap of adjacent gratings is transferred to the SiO2 layer under the air gap. Thus, the magnetic field is strongly confined in the middle SiO2 spacer layer and top SiO2 grating. In addition, the electrical field vectors mainly circulate around the spacer and air gap to form two induced current loops, which may induce the diamagnetic response and eventually excite MPs [38,39]. The position of both current loops represents the excitation of high-order spacer MPs and first-order gap MPs, respectively.
Fig. 6. The analysis of the resonance wavelength at 1016 nm. (a) The x-z cross-section view of the magnetic field distribution. The contour represents the magnitude of magnetic field and the arrows indicate the electric field vectors. (b) The equivalent LC circuit model for the gap MPs.
The inductor-capacitor (LC) circuit model [40], which stem from the charge distributions at magnetic resonance wavelength, has been employed to successfully predict the resonance condition for grating-based metamaterials. Figure 6(b) is the equivalent LC circuit model for the gap MPs. The capacitance between two adjacent Ti gratings is ${C_{m1}} = ({c_1}{\varepsilon _0}{h_2}w)/(p - w)$, where the numerical factor c1 is the nonuniform charge distribution at the Ti grating side surface and is set to be 0.32 [41]. ${\varepsilon _0}$ is the permittivity of vacuum. ${L_m}_1 = 0.5{\mu _0}(p - w){h_2}/w$ represents the mutual inductance between two adjacent Ti gratings, where µ0 is the vacuum permeability and ${L_{\textrm{e}}}_1 = ( - {h_2}\varepsilon ^{\prime})/[w{\varepsilon _0}{\omega ^2}\delta ({\varepsilon^{\prime}}^{2} + {\varepsilon^{\prime\prime}}^{2})]$ represents the contribution from the drifting electrons to the inductance, where $\delta \textrm{ = }\lambda /(4\pi k)$ is the power penetration depth, $k$ is the extinction coefficient of Ti, $\varepsilon ^{\prime}$ and $\varepsilon ^{\prime\prime}$ are the real and imaginary parts of the dielectric constant of Ti, respectively. Hence, the total impedance of LC circuit can be expressed as:
The resonance condition of gap MPs can be obtained by using $Z{1_{total}} = 0$. The LC model predicts the gap MPs to occur at the wavelength of 1134 nm. The relative error is 11.6%. Note that, the LC model is based on several approximations and could not consider the interaction between MPs and other modes, which may account for the discrepancy.
Figure 7(a) illustrates the x-z cross-section view of the magnetic field distribution, corresponding to the absorption peak at 1648 nm. In this case, resonance of the diamagnetic response and strong enhancement of the magnetic field occurs not in the gap of the Ti grating, but in the SiO2 spacer. It can be seen that an electric current loop is formed in the SiO2 spacer between the Ti grating and substrate, and the magnetic field confined wherein is strongly enhanced. The phenomenon of gap magnetic resonance disappears, while the phenomenon of spacer magnetic resonance is up to the strongest.
Fig. 7. The analysis of the resonance wavelength at 1648 nm. (a) The x-z cross-section view of the magnetic field distribution. The contour represents the magnitude of magnetic field and the arrows indicate the electric field vectors. (b) The equivalent LC circuit model for the spacer MPs.
The LC circuit model for spacer MPs to predict the resonance wavelength is illustrated in Fig. 7(b), where ${C_{m2}} = {c_2}{\varepsilon _0}{\varepsilon _d}{w^2}/{h_3}$ represents the parallel-plate capacitor between top Ti grating and metallic substrate, ${\varepsilon _d}$ is the real part of dielectric function of SiO2, ${c_2}$ is the non-uniform charge distribution at metallic surface and is taken as 0.25 in the calculation from Ref [41]. ${\textrm{C}_{g2}} = {\varepsilon _0}{h_2}w/(p - w)$ is the gap capacitor between neighboring gratings. ${L_{m2}} = 0.5{\mu _0}{h_3}$ and ${L_e}_2 = \textrm{ - }\varepsilon ^{\prime}/[{\varepsilon _0}{\omega ^2}\delta ({\varepsilon^{\prime}}^{2} + {\varepsilon^{\prime\prime}}^{2})]$ are the parallel-plate inductor and kinetic inductor, respectively.
Therefore, the total impedance of the electric current loop can be expressed as:
The MPs resonance wavelength can be obtained to be 1682 nm by zeroing the total impedance of the circuit, which is in good agreement with the wavelength calculated by RCWA method.
To further understand the interaction of this structure, the effects of geometric parameters on the metamaterial absorber are also investigated. Figure 8 presents the effect of structural parameters, such as period (p), width (w), thickness (${h_2}$) of Ti grating and thickness (${h_3}$) of SiO2 layer, on absorption behavior in the spectral range of 300 nm to 4000 nm.
Figure 8(a) shows that MPs resonance peaks slightly shift to higher wavelengths with the increase of grating period. In the LC model, the grating period mainly affects the gap capacitor, which is two orders of magnitude smaller than the parallel-plate capacitor. Therefore, the grating period has little influence on the resonance of MPs. On the other hand, the SPP absorption peak is significantly influenced by the period and shifts to higher wavelengths with increasing period. The trend can be explained by the Eq. (4). As shown in Fig. 8(b), when the width w change from 150 nm to 350 nm, the absorption peaks at MPs resonance red shift, especially the spacer MPs peak. This is due to the fact that increase of the width w leads to the increase of capacitance and inductance and the trend of magnetic resonance wavelength can be described by the LC model. On the other hand, the SPP peak does not change due to the SPP resonance conditions only depending on the grating period rather than grating width. Figure 8(c) presents the influence of Ti grating thickness on spectral absorption. The SPP peak is nearly independent of ${h_2}$. Similar to width w, larger thickness ${h_2}$ also results in larger values for capacitance and inductance, thus increasing the wavelength of MPs resonance. In addition, compared Fig. 8(b) with Fig. 8(c), the effect of thickness ${h_2}$ on gap MPs peak is greater than width w, which is confirmed by the Eq. (5). For the effect of SiO2 layer thickness, obviously, the optimal of spectral absorption is observed at approximately ${h_3}$ = 45 nm ∼ 55 nm, as shown in Fig. 8(d).
Fig. 8. Spectral normal absorption contour plot for various (a) Period p, (b) Width w, (c) Thickness h2 of the Ti grating and (d) Thickness h3 of the SiO2 layer.
In order to illustrate the fabrication-tolerance, according to the spectral absorption in Figs. 8(a)–8(d), we calculate average absorption of metamaterial absorber in the wavelength (300 nm ∼ 2100 nm) with different fabrication errors. Here, the fabrication errors are defined as ±10 variations of structure parameters. Table 2 is the calculated results. It can be seen that the absorption performance is very insensitive to the small range variation of structure parameters. Thus, the proposed absorber has a large fabrication-tolerance. In addition, as the refractive index of SiO2 changes significantly in the large wavelength range, we calculate the absorption spectra with different SiO2 refractive indexes and the results are shown in Fig. 9. When the SiO2 refractive index increases from 1.42 to 1.49, the absorption almost no change, which indicates the refractive index of SiO2 can be simplified to 1.46 in the calculation.
Table 2. Average absorption of metamaterial absorber in the wavelength (300 nm ∼ 2100 nm) with different fabrication errors.
In order to investigate the angle dependence of the proposed absorber, we perform calculations and the results are depicted in Fig. 10. For both TM polarization and TE polarization modes, in the near-infrared wavelength, MPs resonance peaks are almost unchanged at oblique incidence angles. The angle independent is attributed to an intrinsic characteristic of MPs [42]. On the other hand, in the shorter wavelength, SPP peak is highly sensitive to incident angles and presents an inclined narrow-band. Although the position of the SPP peak is affected by the incident angle, the intensity of absorption is still very high. Therefore, the absorber can maintain a high absorption (>90%) under a high incident angle of 50° in the wavelength range of 300 nm to 2100 nm. These results indicate that the proposed absorber is large incident angle-insensitivity. In Fig. 11, the effect of polarization angles on the absorption efficiency of the absorber is also studied under normal incidence. The absorption performance does not have any change with the change of polarization angles, indicating the characteristics of polarization-independent. This is due to the high symmetrical structure of the proposed absorber.
Fig. 10. Spectral absorption contour plot at various incident angles (0∼80°) for (a) TM mode, (b) TE mode.
4. Conclusion
In summary, an ultra-broadband metamaterial absorber, consisting of 2D SiO2-Ti square bilayer grating on SiO2 spacer and Ti substrate, is proposed and analyzed by using RCWA and GA methods. The proposed structure exhibits an extremely high average absorption of 98.3% in the spectral range of 300 nm to 2100 nm. Moreover, the metamaterial absorber is polarization-independent and maintains a high absorption (>90%) in both TM and TE modes with an incident angle of up to 50°. A detailed analysis by FDTD method reveals that the coupling effect of WA, CR, SPPs, gap MPs and spacer MPs are responsible for the broadband absorption. In addition, the prediction of magnetic resonances wavelengths is in good agreement with the analyses of the equivalent LC models. The absorber has great potential for solar system applications due to the characteristics of wide-angle polarization-independent ultra-broadband absorption.
Funding
National Natural Science Foundation of China (61605169, 61775191).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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