1. Introduction

Metamaterials possess unique electromagnetic properties that cannot be found from existing materials in nature and can exhibit both electric and magnetic responses [1,2]. In the last decade, optical metamaterials have inspired many engineering applications, such as plasmonic sensors, thermophotovoltaic devices, solar energy harvesting, photodetector and radiation cooling [3–8]. In these application fields, much effort has been devoted to enhancing absorption at optical frequencies. In order to improve the optical absorption, most applications exploit the electric and magnetic resonances, such as surface plasmon polariton and localized surface plasmon [9–11], and magnetic polariton [12] using subwavelength-sized periodic structures. For example, Landy et al. [13] proposed to employ metamaterials to make a perfect absorber by controlling the electric and magnetic resonances at certain frequency. Since then, many researchers have reported theoretical analysis and new designs of perfect absorbers, such as metal-insulator-metal absorber [14], cross resonator [15] and flexible wide angle terahertz absorber [16]. However, these studies demonstrated only narrowband absorption in a certain frequency, which greatly hinders their practical applications in energy harvesting and photodetector. In order to remedy the resonance-originated narrow-band absorption by subwavelength-sized structures, more complicated structures were investigated. Liu et al. [17] demonstrated the multi-band absorption using different-sized cross-shape gratings at infrared range. Hu et al. [18] designed the metamaterial absorber that consists of three metallic layers separated by two dielectric layers for the polarization insensitive multi-band absorption in terahertz regime. Even though proposed multi-band absorbers successfully exhibited multiple resonance absorption peaks at the same time, each absorption peak was separated from each other, limiting the real-world applications.

One way of achieving broadband absorption is combining multiple resonance modes, such as magnetic resonance [19–21], surface plasmon, localized surface plasmon and dipole-dipole interaction [22, 23], and gap plasmon and Fabry-Perot resonance [24], in the spectral region of interests. Multiple phonon-polariton branches associated with higher-order diffracted evanescent waves on a two-dimensional grating could also result in broadband absorption [8]. Alternatively, hyperbolic metamaterials consisting of multiplayers of metal/dielectric films were also proposed for achieving broadband absorption by means of the slow light modes confined in the hyperbolic metamaterial [25, 26]. Recently, more detailed analysis of the broadband-absorption mechanism was provided based on the waveguide dispersion in the hyperbolic-metamaterial [27]. In addition to simply combining multiple resonance modes, if there exist geometry variations in the considered structure, one can also utilize localized surface plasmon as well as their interactions (i.e., plasmon hybridization modes) for achieving broadband absorption [4, 28]. However, little attention has been paid to the interaction of magnetic resonances, especially for vertically-aligned tandem structures [29].

In the present paper, we theoretically investigate the multiple electromagnetic resonance modes on a relatively simple two-dimensional tandem grating with emphasis on the interaction of magnetic resonances. The considered tandem structure comprises a fishnet structure made of Au/SiO₂/Au stack as a base, and a convex-type Au grating is then attached on top of it with or without additional SiO₂ spacer. The finite-difference time-domain (FDTD) method is employed in order to elucidate fundamentals of electromagnetic resonance modes and their interaction mechanisms. After understanding the multiple electromagnetic resonance modes on the proposed tandem grating structure, we also demonstrate a broadband absorber made of Au and SiO₂ in the visible spectrum for the potential application in solar cell.

2. Theoretical modeling

The considered tandem grating structures are depicted in Fig. 1. The metal and dielectric parts are selected as gold and SiO₂, respectively. Figure 1(a) shows the base fishnet structure that consists of an opaque Au substrate, a SiO₂ spacer and a concave-type Au grating (referred as base grating), forming Au/SiO₂/Au stack. Figure 1(b) describes tandem grating 1 (TG1) in which a convex-type Au grating (referred as topmost grating) is directly attached onto the base fishnet structure. On the other hand, for tandem grating 2 (TG2) as shown in Fig. 1(c), there exists an additional SiO₂ spacer between the topmost grating and the base grating such that two Au/SiO₂/Au stacks can be formed vertically by sharing Au grating element in the base fishnet structure. Geometric parameters defining the tandem grating structure are grating period (Λ), width of the metal ridge of the base concave grating (w_bg), width of the topmost convex grating (w_tg only for TG1 and TG2), thicknesses of the base grating (d_bg) and the topmost grating (d_tg only for TG1 and TG2), and thicknesses of the base spacer (d_bs) and the upper spacer (d_us only for TG2). For identifying the electromagnetic resonance modes associated with the proposed tandem gratings, we set Λ = 1000 nm, w_bg = 400 nm, w_tg = 600 nm, d_bs = 40 nm, d_bg = 40 nm, d_tg = 30 nm, and d_us = 30 nm. The Au substrate is set to be 300-nm thick (i.e., thicker than 20 times of the radiation penetration depth of Au) so that it can be considered as opaque in the considered wavenumber (ν) range from 2000 cm⁻¹ to 7000 cm⁻¹.

 

Fig. 1 Schematic of one unit cell of the considered tandem grating structures: (a) base fishnet structure; (b) Tandem grating 1 (TG1); and (c) Tandem grating 2 (TG2).

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In order to calculate the spectral reflectance and local field distributions, FDTD simulation was performed with a commercial package (Lumerical, FDTD Solutions). The optical properties of gold and SiO₂ were obtained from Palik’s data [30] included in the package. The plane wave is incident from a free space to the normal to the structure. Here, only transverse magnetic (TM) polarization is considered with magnetic field oscillating perpendicular to the x–z plane (refer to Fig. 1). In the simulation, nonuniform meshes with a size of 5 nm in the x- and y-direction and 1 nm in z-direction were used. Perfectly matched layers are placed in the z-direction and periodic boundary conditions are applied in the x- and y-direction. The spectral reflectance is obtained by a frequency-domain power monitor placed above the plane wave source.

3. Results and discussion

Figure 2 compares the normal reflectance spectra of the fishnet structure, TG1, and TG2. Because the fishnet structure is the base structure for TG1 and TG2, let us consider first the resonance modes associated with the fishnet structure.

 

Fig. 2 Normal reflectance spectra of (a) fishnet structure; (b) TG1; and (c) TG2 for TM polarization. Geometric parameters are Λ = 1000 nm, d_bs = 40 nm, d_bg = 40 nm, and w_bg = 400 nm for the base fishnet structure, w_tg = 600 nm and d_tg = 30 nm for TG1 and TG2 only, and d_us = 30 nm for TG2 only.

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3.1. Resonance modes in the fishnet structure

As can be seen from Fig. 2(a), two remarkable dips exist in the reflectance spectrum at 4660 cm⁻¹ and 6180 cm⁻¹. The reflectance dip at 6180 cm⁻¹ is the fundamental mode of magnetic polariton (MP1). It is well known that the fishnet structure can support the magnetic polariton [1, 2]. The oscillating magnetic field induced by incident light generates the displacement currents that move in opposite direction in the base grating and in the Au substrate. These anti-parallel currents induce the diamagnetic response, which excites the magnetic polariton. Therefore, the magnetic field is strongly confined in the dielectric spacer sandwiched by the metal grating and the substrate [31]. In order to confirm the excitation of MP1 at 6180 cm⁻¹, we plot the time-averaged square of the magnetic field at the center of the base spacer normalized by that of the incident wave in Fig. 3. It is clear from Fig. 3(a) that at 6180 cm⁻¹ the enhanced magnetic field is localized within the metal grating region. It should be noted that in the fishnet structure, only small portion of the metallic ridge (i.e., x/Λ ∼ 0.2 and y/Λ ∼ 0.2) is involved for exciting MP1 mode, and rest of the metallic ridge area is responsible for the electric response of the fishnet structure [1, 2].

 

Fig. 3 Resonance modes associated with the fishnet structure: (a) time-averaged square of the magnetic field at ν = 6180 cm⁻¹ when MP1 mode is excited. The horizontal plane where the field distribution is plotted is depicted as an inset. The green lines indicate the position of metallic ridge in the base concave grating above the base spacer; (b) resonance condition of MP1 mode predicted by the LC-circuit model; (c) time-averaged square of the magnetic field at ν = 4660 cm⁻¹ when the coupled SPP is excited; and (d) SPP dispersion relation of the symmetric mode.

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The fundamental mode of magnetic resonance can be also verified by using the inductor-capacitor (LC) circuit model. The total impedance for the equivalent LC circuit is [20]

Another reflectance dip at 4660 cm⁻¹ in Fig. 2(a) is turned out to be caused by the coupled surface plasmon polariton (SPP) inside the base spacer. In the fishnet structure, there is a possibility that SPP occurs at two interfaces; that is, one at the interface between the base grating and the base spacer and the other at the interface between the base spacer and the Au substrate. Because the base spacer is only 40-nm thick, those SPPs can be coupled together, resulting in a symmetric mode and an asymmetric mode [32].

Figure 3(c) illustrates the enhanced magnetic field along the base Au grating to the x-direction. In contrast to the magnetic polariton, the propagating nature of the coupled SPP along the lateral direction is clearly observed. Moreover, the coupled SPP can also be verified using the dispersion relation [32]:

3.2. Resonance modes in TG1 and TG2

After understanding the coupled SPP and MP1 modes associated with the base fishnet structure, resonance modes in TG1 are now investigated. As shown in Fig. 2(b), TG1 also exhibits the key characteristics of the fishnet structure; that is, the coupled SPP and MP1 mode around 4500 cm⁻¹ and 6200 cm⁻¹, respectively. However, in contrast to the fishnet structure, the coupled SPP and MP1 modes associated with TG1 are branched out, resulting in two separate reflectance dips for each mode. This is because the topmost grating directly attached to the base grating introduces additional asymmetry in the geometry with respect to the base spacer.

It should be noted that both coupled SPP and MP1 modes require metallic elements above and below the base spacer (i.e., Au/SiO₂/Au stack). Because the width of the topmost grating (i.e., w_tg = 600 nm) is slightly wider than that of the metal ridge of the base grating (i.e., w_bg = 400 nm), the portion of 200 nm by 200 nm in the topmost grating is exposed out of the metal ridge and eventually affects formation of the coupled SPP and MP1 modes. As a result, TG1 exhibits additional branch in the coupled SPP and MP1 modes, as shown in Fig. 2(b).

In order to demonstrate effect of the topmost grating, Fig. 4 plots the time-averaged square of the magnetic field when the coupled SPP and MP1 modes occur. For the coupled SPP mode, Figs. 4(a) and 4(b) compare the magnetic field distributions at 4390 cm⁻¹ and 4560 cm⁻¹.Two magnetic field distributions look alike; however, the one at 4390 cm⁻¹ spreads more into the topmost grating region (see the region marked with the white arrow) than the other at 4560 cm⁻¹. By Comparing Figs. 4(a) and 4(b) with Fig. 3(c), we can infer that both cases at 4390 cm⁻¹ and 4560 cm⁻¹ are affected by the presence of the topmost grating to some extent, but the one at 4390 cm⁻¹ is affected slightly more than the other, suggesting that the geometric asymmetry caused by the topmost grating is responsible for double-branches of the coupled SPP mode in TG1.

 

Fig. 4 Resonance modes on TG1: (a) time-averaged square of magnetic field of coupled SPP normalized to that of incidence wave at ν = 4390 cm⁻¹; (b) time-averaged square of magnetic field of coupled SPP normalized to that of incidence wave at ν = 4560 cm⁻¹; (c) time-averaged square of magnetic field of MP1 normalized to that of incidence wave at ν = 5920 cm⁻¹; and (d) time-averaged square of magnetic field of MP1 normalized to that of incidence wave at ν = 6280 cm⁻¹. The horizontal plane where the field distribution is plotted is depicted as an inset. The green lines indicate positions of the base grating (solid line) and the topmost grating (dashed line).

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Regarding the MP1 mode of TG1, let us consider reflectance dips at 5920 cm⁻¹ and 6280 cm⁻¹ in Fig. 2(b). Similarly to the coupled SPP, MP1 mode can also be branched out on TG1 due to the presence of the topmost grating. First of all, the magnetic field distribution at 6280 cm⁻¹ in Fig. 4(d) is nearly the same as that of the fishnet structure in Fig. 3(b), confirming the excitation of MP1 mode. However, the magnetic field distribution in Fig. 4(c) looks quite different from those in Figs. 3(b) and 4(d). The most important difference is that the topmost grating shifts the location of anti-node of the magnetic field originally formed in the region of x/Λ ∼ 0.2 and y/Λ ∼ 0.2 in Fig. 3(b) to the region of x/Λ ∼ 0.2 and y/Λ ∼ 0.4 in Fig. 4(c). As a result, in Fig. 4(c) the strongly localized magnetic field also appears in the metallic ridge region that is in control of the electric resonance, but those two anti-nodes appeared in the region of y/Λ ∼ 0.8 are not important for the excitation of MP1. Field distributions in Figs. 4(c) and 4(d) reveal that the geometric asymmetry caused by the topmost grating is also responsible for double-branches of MP1 modes in TG1.

In addition to the coupled SPP (near 4500 cm⁻¹) and MP1 (near 6200 cm⁻¹) modes, there exists additional reflectance dip at ν = 3670 cm⁻¹ in TG1. This reflectance dip does not appear on the fishnet structure; hence, the presence of topmost grating on TG1 should be responsible for this dip. Figure 5(a) shows the time-averaged square of magnetic field at ν = 3670 cm⁻¹. We intentionally plot the magnetic field distribution at the topmost grating as well as at the base spacer. According to the field distribution at the topmost grating [i.e., left panel of Fig. 5(a)], the reflectance dip at 3670 cm⁻¹ is neither due to the localized surface plasmon nor due to the field localization effect because there is no noticeable field enhancement within or at the edge of the metal region. Instead, it is due to the magnetic polariton underneath the topmost grating (i.e., region of 0 < x/Λ < 0.6 and 0.4 < y/Λ < 1). Since the small portion of the topmost grating is exposed out of the metal ridge, additional magnetic polariton can be formed under the topmost grating, as noted by the field distribution at the base spacer [refer to right panel of Fig. 5(a)], which is quite different from the MP1 mode associated with the region of x/Λ ∼ 0.2 and y/Λ ∼ 0.4 in Figs. 4(c) and 4(d). Due to weak coupling of the incident wave to this MP mode, the resulting reflectance dip is not obvious (i.e., reflectance value is about 0.85) and the field distribution does not clearly show the strong anti-node like Fig. 4 under the topmost grating region. As a further evidence for this MP mode, we can refer to the LC-circuit model in Fig. 3(b) with the grating width of 600 nm (i.e., topmost grating). Notice that for TG1 the topmost grating is directly attached to the fishnet structure; thus, the thickness of Au film should be considered as d_tg + d_bg = 70 nm in the LC-circuit model, which will slightly alter the resonance condition. Nevertheless, it can be inferred that the resonance frequency for the grating of 600-nm-width will be around 4000 cm⁻¹, which fairly agrees with the reflectance dip at ν = 3670 cm⁻¹.

 

Fig. 5 Additional magnetic polariton in TG1: (a) time-averaged square of magnetic field at ν = 3670 cm⁻¹ normalized to that of incidence wave and (b) effect of the thickness of base spacer on the resonance modes in TG1.

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Figure 5(b) compares the reflectance spectra of TG1 when d_bs = 40 nm and 50 nm. For the coupled SPP, if the thickness of base spacer increases, the coupling of SPPs at two interface becomes weaker, leading to shifting the resonance frequency to higher region. For the MP, the LC-circuit model also predicts shift of the resonance condition to higher frequency for thicker d_bs. As clearly seen from Fig. 5(b), all the reflectance dips shift to higher frequency when d_bs increases to 50 nm, which qualitatively agrees well with the expected behavior. This also confirms that the reflectance dip at 3670 cm⁻¹ is due to the excitation of magnetic polariton under the topmost grating area.

As can be seen from Fig. 2(c), TG2 shows the richest features in the spectral reflectance. Similar to the fishnet structure, TG2 also supports the coupled surface plasmon and magnetic polariton near 4500 cm⁻¹ and 6000 cm⁻¹, respectively. On the other hand, around 3300 cm⁻¹, additional reflectance dips appear differently from the fishnet structure and TG1. We believe that reflectance dips at 3280 cm⁻¹ and 3510 cm⁻¹ are related to the coupled MP modes associated with two vertically aligned Au/SiO₂/Au stacks in TG2; that is, hybridization [29].

In order to elucidate the hybridization of MP modes, the normalized instantaneous magnetic field distributions inside the base spacer and the upper spacer are plotted in Figs. 6(a) and 6(b), respectively, at an arbitrary reference time t_AR. At a certain time t_AR, the magnetic fields around x/Λ ∼ 0.4 and y/Λ ∼ 0.7 in the base spacer and in the upper spacer are out of phase in Fig. 6(a) at 3280 cm⁻¹ (i.e., anti-symmetric MP). On the contrary, they are in phase in Fig. 6(b) at 3510 cm⁻¹ (i.e., symmetric MP). The hybridization of MP can be understood as the coupling of magnetic dipoles [29]. In case of the transverse coupling as in Fig. 6, laterally coupled dipoles in the anti-symmetric mode attract each other so that the restoring force decreases, which leads to lowering of the resonance frequency. On the other hand, for the longitudinal coupling, opposite holds true such that symmetric MP appears at lower frequency [36]. It is interesting to note that when MP1 is excited on the fishnet structure the magnetic field is strongly localized near x/Λ ∼ 0.2 and y/Λ ∼ 0.2 [refer to Fig. 3(a)]. Because of the coupling of MPs in two vertically aligned Au/SiO₂/Au stacks in TG2, the region where the magnetic field is localized in the base spacer has moved to the location under the topmost grating, suggesting that the coupling of MPs in vertically aligned Au/SiO₂/Au stacks plays a critical role in hybridization. The time-averaged square of magnetic field distributions shown in Figs. 6(c) and 6(d) also confirm that magnetic polariton occurs both in the base and the upper spacers. In both symmetric and anti-symmetric modes, the magnetic polariton associated with the upper Au/SiO₂/Au stack is stronger than that associated with the lower Au/SiO₂/Au stack.

 

Fig. 6 Hybridization of the magnetic polariton on TG2. The Magnetic field distribution (a) for the anti-symmetric mode at ν = 3280 cm⁻¹ and (b) for the symmetric mode at ν = 3510 cm⁻¹. The time-averaged square of magnetic field (c) for the anti-symmetric mode at ν = 3280 cm⁻¹ and (d) for the symmetric mode at ν = 3510 cm⁻¹. The inset shows the plane where the field distribution is plotted.

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Finally, we investigate the reflectance dip at 2920 cm⁻¹ in Fig. 2(c). It turns out that the reflectance dip at 2920 cm⁻¹ is caused by the magnetic polariton only in the upper Au/SiO₂/Au stack of TG2. As shown in Fig. 7(a), strongly localized magnetic field is formed in the upper spacer region between the topmost grating and the base grating. In addition, the LC-circuit model is also employed to predict the resonance frequency of the fundamental magnetic polariton mode in Fig. 7(b). The reflectance dip position excellently agrees with the prediction by LC-circuit model, confirming that the reflectance dip at 2920 cm⁻¹ is indeed caused by the magnetic polariton only in the upper Au/SiO₂/Au stack of TG2. Because of the upper spacer, TG2 does not support the magnetic polariton like Fig. 5, but the upper Au/SiO₂/Au stack of TG2 can support its own MP.

 

Fig. 7 Magnetic polariton associated with the upper Au/SiO₂/Au stack of TG2: (a) time-averaged square of magnetic field at ν = 2920 cm⁻¹. The inset shows the plane where the field distribution is plotted; and (b) resonance condition predicted by the LC-circuit model.

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Although TG2 involves only four layers coated on the substrate (i.e., relatively simple in geometry), it can support multiple resonance modes: (i) surface plasmon and magnetic polariton associated with the base fishnet structure; (ii) hybridized magnetic polaritons associated with the vertically aligned Au/SiO₂/Au stacks; and (iii) magnetic polariton associated with the upper Au/SiO₂/Au stack.

3.3. Application for broadband absorption in the visible spectrum

Because TG2 can support multiple resonance modes, it would be suitable for achieving broadband absorption. Here, we demonstrate a broadband absorber in the visible spectrum based on TG2 made of Au and SiO₂. As discussed with Fig. 5(b), the resonance condition of magnetic polariton and coupled SPP can be shifted to higher frequency region by changing the geometry. For example, the resonance frequency of MP mode will move to higher frequency if the grating width decreases or the spacer thickness increases. For the coupled SPP, increasing the spacer thickness will change its resonance condition to higher frequency region. Although the dielectric function of Au can be very different in the visible spectrum from that in the near or mid-infrared region, the general trends of how the resonance condition shifts with respect to the geometry would still be valid in the visible spectrum. Based on the aforementioned physical insight, the broadband absorber are designed in the visible spectrum with Λ = 450 nm, d_bs = 30 nm, d_bg = 30 nm, w_bg = 250 nm, d_us = 50 nm, d_tg = 30 nm, and w_tg = 200 nm.

Figure 8(a) shows the effect of polarization angle ϕ (refer to the inset of Fig. 8(a) for its definition) at normal incidence. When ϕ = 0°, the incident magnetic field oscillates along the y-axis. Because the considered structure is opaque, the absorptance is simply one minus the reflectance. As can be seen from Fig. 8(a), the spectral absorptance value is above 0.9 in the wavelength ranges from 350 nm to 550 nm. As the wavelength increases up to 650 nm, the absorptance value decreases to 0.75. At wavelengths above 650 nm, the absorptance shows a steady decreasing tendency. It is interesting to note that the spectral absorptance rarely changes with respect to the polarization angle, which provides a great advantage of TG2 as a polarization-independent broadband absorber. Although the structure is made of a noble metal, we can achieve strong broadband absorption from 300 nm to 700 nm, demonstrating that multiple resonances supported on TG2 can be controlled by changing geometric parameters for a specific application. It should be noted that higher-order resonance modes, such as magnetic polariton and diffraction-induced surface plasmon [8, 33] can also appear in the higher frequency region. Consequently, in the visible spectrum, surface plasmon, magnetic polariton, and some of their higher-order modes are expected to be combined together, leading to broadband absorption. Furthermore, Au behaves lossier in the visible spectrum than in the infrared region, which may contribute to broadband absorption as well.

 

Fig. 8 Spectral absorptance of the designed broadband absorber: (a) for different polarization angles at normal incidence; and (b) for different incidence angles. The geometric parameters of the broadband absorber are Λ = 450 nm, d_bs = 30 nm, d_bg = 30 nm, w_bg = 250 nm, d_us = 50 nm, d_tg = 30 nm, and w_tg = 200 nm.

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We now discuss the effect of incidence angle (θ) in Fig. 8(b) that shows the absorptance spectra at θ = 30° for each polarization states. At θ = 30°, the spectral absorptance does not significantly vary in the wavelength ranges from 300 nm to 600 nm and remain above 0.75 for for both polarizations. There exist some variations in the spectral absorptance in 650 ∼ 800 nm, however, overall absorption characteristics are not significantly altered with respect to the incidence angle. Although it is not shown, we also checked the robustness of the broadband absorption with respect to the geometrical parameters, especially for the misalignment of the topmost grating to the base fishnet structure. It was found that lateral misalignment of the topmost grating within 40 nm in the x- and y-directions does not alter the absorptance spectrum when λ < 650 nm. Therefore, the proposed broadband absorber has a great potential in solar cell application [4, 22, 37].

4. Concluding remark

We have numerically investigated the electromagnetic resonances on two-dimensional tandem grating structures with the fishnet structure being a base of them. For TG1 in which the topmost Au grating is directly added onto the base fishnet structure, it was found that the coupled surface plasmon and magnetic polariton in the base structure are branched out due to the geometric asymmetry. If additional SiO₂ spacer is added between the topmost Au grating and the base fishnet structure (i.e., TG2), new magnetic resonance modes appear due to hybridization of MP modes. With the understanding of multiple electromagnetic resonance modes on tandem grating structures, we successfully designed a Au-based broadband absorber whose absorptance does not significantly vary with the polarization and incidence angles. The results obtained in this study will facilitate the development of engineered nanostructures for real-world applications related with the renewable energy conversion, such as photovoltaic solar power generating systems, where the broadband absorption is important.