1. Introduction

Metamaterial absorbers(MAs) are one of the important applications of metamaterials, which are artificially constructed plasmonic nanostructures usually consisted of periodic subwavelength metallic and dielectric units, and possess exotic electromagnetic properties that cannot be realized by nature materials [1]. Due to the advantages of perfect absorption, incident angle insensitivity and thin thickness, MAs have been extensively investigated both theoretically and experimentally, which can be applied in photodetectors, microbolometers and thermal emitters [2–4].

In the infrared or visible region, the MAs usually consist of a tri-layer structure [3]. The top layer is periodically patterned metallic nanostructure, which is separated from the bottom metallic ground plane by a dielectric spacing layer [3]. The magnetic field is strongly localized in the spacing layer between the two metallic components and the electric field circulates around the spacer to form an induced current loop, which is explained by the magnetic polaritons(MPs) and predicted by a LC circuit model [5]. So far, the theory of MPs has been successfully employed to explain and predict the resonances in metallic grating-film structure [5–7], narrow slit arrays [8, 9], double-layer nanoslit arrays [10] and deep grating [11, 12]. As a matter of fact, MPs cannot only be used in one dimensional periodic structure, but also valid in two dimensional (2D) structures [13, 14]. In addition, the excitation of MPs can also be used to explain the responses for various materials, such as Silver, Aluminum, Tungsten [9], heavily doped semiconductor(Si) [15], and even for the phonon-mediated polar material SiC [11]. Importantly, the possibility of using MPs to explain the optical anomaly have been demonstrated in broad wavelength range extended from the near-infrared to terahertz range, as well as the microwave [15].

On the other hand, the single-band MAs are not suitable in some areas, such as the spectroscopic detection and phase imaging that require distinct absorption bands [16–18]. Therefore, the research on more advanced multi-band perfect MAs is necessary and now has become a new hot area. Up to now, however, all of the predictions based on LC circuit model are valid for single narrow band only. In this paper, according to the independent properties of the MPs which is elucidated by the electromagnetic field distributions at resonant wavelengths, we propose a parallel LC(PLC) circuit model to explain and predict the multi-band resonant absorption. To demonstrate the PLC circuit model, two benchmarks of dual-band absorption are presented. One is the multi-sized structure with identical dielectric spacing layer [18] and the other is the multilayer structure of the same strip width [19]. The influences of structure parameters on the absorption spectra predicted by PLC circuit model exhibit good agreements with those obtained from the RCWA calculation. To illustrate the power of the PLC circuit model further, we also design a broadband radiative cooling structure by merging several close peaks together. When the peaks are selected at the desired wavelength, all the corresponding structural and optical parameters are derived directly using the PLC circuit model with proper approximations. This radiative cooling application can also be omnidirectional and polarization insensitive by constructing a 2D periodic structure in the same design principle.

2. PLC circuit model of multi-sized structure

Actually, to obtain multiple absorption peaks, the absorbers are usually designed by laterally arranging various sized subunits into one unit cell [20–23] or vertically stacking multilayer structures [24–26]. Several distinct resonant peaks excited in different components all survive due to the independence of resonances. To simply illustrate the multi-band absorption, we firstly investigate the multi-sized dual-band absorber, as shown in Fig. 1(a). The structure is illuminated by a TM polarized plane wave. The metal-dielectric pairs of two different widths are periodically patterned on the metallic ground plane within a period Λ = 3.6 μm. The widths of each strip are w₁ = 1 μm and w₂ = 1.5 μm, respectively. The spacing distances between neighboring strips are kept identical of s = 0.55 μm. The thickness of the metallic strip and ZnS spacing layer are h = 0.07 μm and t = 0.13 μm, respectively. Within the considered wavelength range, ZnS is a lossless material with a refractive index of 2.2 [18, 27]. Silver is the material of the substrate and the metallic strips. The complex dielectric constants of silver (Ag) at wavelengths shorter than 10 μm are taken from Ref [28], while others out of that range are obtained by linear interpolation [29]. The thickness of the substrate is much larger than the skin depth, which blocks all light transmit through the whole structure. Thus, the absorptivity could be calculated by A = 1-R. The absorption spectra are calculated by the rigorous coupled wave analysis (RCWA) method. Figure 1(b) presents the absorption spectra of the multi-sized dual-band absorber at normal incidence. There are two distinct absorption peaks located at 5.59 μm and 8.19 μm with absorptivity of 98.17% and 99.36%, respectively. Furthermore, we simulated the structure with only one subunit of different width, as shown in Fig. 1(b). The red and blue curves represent the absorption spectra of only shorter or longer strip in one period, respectively. The comparison of simulation results show that the absorption spectra of the dual-band absorber are the superposition of those of each structure with one subunit only.

 

Fig. 1 (a) Schematic of the multi-sized dual-band absorber. (b) Absorption spectra of the multi-sized dual-band absorber at normal incidence. Insets are structure with only one subunit.

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In order to gain the physical mechanism of the multi-band absorption in multi-sized structure, we present the distributions of the electromagnetic field at two absorption peaks, as shown in Fig. 2(a) and 2(b). The contours represent the normalized magnetic field and the arrows indicate the electric field vectors. Clearly, the magnetic field is strongly confined under the shorter metallic strip and within the dielectric spacing layer at resonance wavelength of 5.59 μm. Meanwhile, the electric field circulates around the spacer to form an induced current loop. The electromagnetic field distributions indicate a diamagnetic response and excitation of the MPs. Besides, similar behavior for the magnetic and electric fields can be seen at resonance wavelength 8.19 μm, except that the resonance occurs under the longer metallic strip. It is evident that the incident electromagnetic waves are efficiently confined to the respective strip at the corresponding resonance wavelength. The fields in the neighbor pairs cannot affect each other as long as the separation between the two strips is large enough, which demonstrate the independence of the MPs.

 

Fig. 2 Distributions of the normalized magnetic field and electric field vector at the resonant peaks of (a) 5.59 μm and (b) 8.19 μm. (c) Schematic of the equivalent LC circuit model for grating-film structure. (d) Illustration of the PLC circuit model for multi-sized structure. Each subunit corresponds to one impedance and two impedances are connected in parallel.

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Recently, the equivalent LC circuit model has been widely used to predict the magnetic resonance condition for the single-band MAs [5, 11]. In the multi-sized structure, each metal-dielectric pair can be considered as an isolated unit, due to the electromagnetic field is strongly localized under the corresponding metallic strip without coupling to each other [30]. Moreover, the MPs are independent on the structure length along the direction of magnetic field. Therefore, the strip length has no effect on the prediction of LC circuit model, and the capacitance and inductance are given in per unit length. The employed equivalent LC circuit model is shown in Fig. 2(c), where arrows indicate the current flow directions [5–15]. Here, the parallel-plate capacitance $C_g={\epsilon }_{0}h/(\Lambda -w)$ is used to approximate the gap capacitance between the neighboring metallic strips, and the capacitance between the metal strip and substrate is given as $C_m=c_{\text{1}}{\epsilon }_d{\epsilon }_{0}w/t$, where ${\epsilon }_d$ is the spacing dielectric permittivity, c₁ is a numerical factor accounting for the non-uniform charge distribution at the metal surfaces. In addition, $L_e=\text{-}w/({\omega }^2\delta {\epsilon }_0)\cdot ({\epsilon }^{\prime }/({{\epsilon }^{\prime }}^2+{{\epsilon }^{″}}^2))$accounts for the contribution of the drifting electrons to the inductance and $L_m=0.\text{5}{\mu }_{0}wt$ represents the mutual inductance of the metallic strip and substrate, where ${\epsilon }^{\prime }$and${\epsilon }^{″}$ are the real and imaginary parts of dielectric constant of Ag. Furthermore, the power penetration depth $\delta =\lambda /4\pi \kappa$is frequency dependent, where $\kappa$ is the extinction coefficient of Ag. The impedance of one subunit can be expressed as

We investigate the relationship between the resonance wavelength and the refractive index of the dielectric spacing layer to prove the validity of the PLC circuit model. Figure 3 presents the absorption spectra of multi-sized structure as the refractive index varies from 1.6 to 2.8, while the geometry parameters are unvaried. It is seen that the two absorption bands both linearly move towards longer wavelength with the increasing refractive index. The predicted resonance wavelengths based on the LC circuit model are depicted as green filled triangles. The predicted resonance wavelengths of two absorption bands for several refractive index values both exhibit good agreement with those obtained by the RCWA calculation. However, there exists a little deviation for small refractive index of the shorter wavelength band. The reason lies that the free electrons distribute uniformly with relative shorter strip width, which needs a larger c₁ to provide better prediction. When we increase the cavity length, electrons tend to accumulate at the corners of metallic surface and a smaller c₁ is appropriate. Therefore, the non-uniform charge distribution factors are set to be 0.18 and 0.175 for the shorter and longer strip width, respectively. The refractive index dependence of the two resonant absorption peaks can be fully explained and predicted by the PLC circuit model.

 

Fig. 3 Absorption spectra of the multi-sized structure with varying the refractive index of the spacing layer, while keeping the geometry parameters unchanged. Green triangles indicate the resonance wavelength calculated from the LC circuit model.

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3. PLC circuit model of multilayer structure

We present another multilayer dual-band absorber to further demonstrate the validity of the PLC circuit model. Figure 4(a) show the schematic of multilayer structure and propagation configurations. Two pairs of vertically cascaded metal-dielectric stacks are periodically arranged on the metallic ground plane with a period of Λ = 2.3 μm. The width of the pairs are w = 1.6 μm. The dielectric spacing layers are Al₂O₃ and ZnTe for the upper and lower pairs with the same thickness of t = 0.11 μm. The thickness of each metallic(silver) layer is h = 0.08 μm. In the simulation range, the refractive index of ZnTe is 2.6 [28]. Dielectric constants and loss tangent of Al₂O₃ are 2.28 and 0.04, respectively [19]. The absorption spectra of the multilayer dual-band absorber at normal incidence are shown in Fig. 4(b). Two distinct absorption peaks can be observed at 6.36 μm and 10.63 μm with absorptivity of 98.41% and 98.35%, respectively. Moreover, we calculated the structure with only one metal-dielectric pair of different spacer, as shown in the inset of Fig. 4(b). The red and blue curves indicate the absorptivity of a single Al₂O₃ or ZnTe dielectric spacing layer, while the other geometry parameters are unchanged. The calculation results reveal that the two absorption peaks in the dual-band absorber are contributed by two different dielectric spacing layers, respectively.

 

Fig. 4 (a) Schematic of the multilayer dual-band absorber. (b) Absorption spectra of the multilayer dual-band absorber at normal incidence. Insets are structure with a single Al₂O₃ (n₁) or ZnTe (n₂) dielectric spacing layer.

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To get insight into the nature of the multi-band absorption in multilayer structure, we present the distributions of the electromagnetic field at two resonance wavelengths, as shown in Fig. 5(a)and 5(b). It is found that there exists strong magnetic field confinement in the Al₂O₃ spacer between the upper and lower metallic strips at 6.36 μm. The electric field vector distributes on both sides of spacer resulting in an induced eddy current. This electromagnetic field patterns also exhibit a diamagnetic response and excitation of the MPs. The magnetic and electric fields similarly distribute between the lower metallic strip and the substrate at 10.63 μm. It is identified that light of different wavelengths accumulate at different dielectric layer of the multilayer structure. Moreover, the field in the upper and lower dielectric spacing layer cannot couple to each other, due to the thickness of the metallic strips is larger than the skin depth. Different from the multi-sized structure, the structure and mechanism of the upper metal-dielectric pair are more like the double-layer nanoslit arrays rather than the grating-film structure [10], as shown in Fig. 5(c). Thus, we use the double layer model for the upper pair and the impedance of the double-layer arrays can be expressed as

 

Fig. 5 Distributions of the normalized magnetic field and electric field vector at the resonant peaks of (a) 6.36 μm, and (b) 10.63 μm. (c) Schematic of the equivalent LC circuit model for double-layer arrays. (d) Illustration of the PLC circuit model for multilayer structure. The impedances of two subunits are still connected in parallel.

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Similarly, we study the influence of the strip width on the absorption performance in multilayer dual-band absorber to further demonstrate the validity of the PLC circuit model. Figure 6 shows the absorption spectra of multilayer structure as a function of strip width ranging from w = 1.1 μm to w = 2.1 μm with the other parameters unaltered. It can be observed that the two resonance wavelengths are linearly redshift as we augment the strip width. The predicted resonance wavelengths of two absorption bands both match well with the RCWA results with varying the strip width. Though the multilayer structure possess the same strip width, the upper and lower dielectric spacing layers are two materials with different refractive index. Thus, the non-uniform charge distribution factors are still different with 0.18 and 0.17 for the upper and lower pair, respectively. The agreement between the RCWA calculation and LC circuit model prediction on the resonance wavelength further confirms the PLC circuit model.

 

Fig. 6 Absorption spectra of the multilayer dual-band absorber with various strip lengths while keeping the other parameter fixed. Green triangles indicate the resonance wavelength predicted from the PLC circuit model.

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4. Preliminary design using the PLC circuit model

The ability to finely tune the resonance wavelength by varying the refractive index or the strip width can be applied to effectively broaden the bandwidth of perfect absorption. It is verified that several closely positioned resonant peaks are merged in the absorption spectra by incorporating different patterns of metallic elements [27, 29], which can be explained by the independence of MPs and PLC circuit model. Here we present a broadband structure covering 8-12 μm that coincides with peak thermal radiation wavelength at terrestrial temperature to realize radiative cooling [31, 32], which can be preliminarily designed by using the PLC circuit model.

The first suggested one dimensional radiative cooling structure consists of two different width subunits, each of which is two pairs of metal-dielectric stacks with the same lateral size but different refractive index spacer, as shown in the inset of Fig. 7(a).The incident light is TM polarization as the same as the aforementioned cases. We fix four distinct absorption peaks at λ₁ = 8.5 μm, λ₂ = 9.5 μm, λ₃ = 10.5 μm and λ₄ = 11.5 μm for the radiative cooling structure in the preliminary design process. Then the structure is preliminarily designed by using the PLC circuit model. Firstly, the spacing distances between neighboring subunits should not be much smaller than the total thickness of the radiative cooling structure, due to the cavity mode may be excited if the aspect ratio is too large. Actually, the coupling between the neighboring subunits is extremely weak for large gaps, thus the contribution of C_g can be neglected because it is less than 1% of C_m. In this case, both the Eq. (1) and Eq. (4) can be simplified as

 

Fig. 7 (a) Absorption spectra of the radiative cooling structure for the TM polarization. Inset is schematic of the radiative cooling structure. (b) Absorptivity as a function of wavelength and angle of incidence.

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We use all the predicted structure parameters in the numerical calculation and the absorption spectra is presented in the Fig. 7(a). It is obvious that four distinct absorption peaks are realized at the wavelength 8.47 μm, 9.49 μm, 10.47 μm and 11.55 μm with absorptivity of 98.71%, 99.99%, 99.87% and 99.95%, respectively. The absorption band of the radiative cooling structure is very broad as a result of merging four close peaks together. The full width half maximum (FWHM) is expanded up to 4.38 μm, which is about three times larger than that of a traditional single band absorber. We also investigate the absorption spectra of the radiative cooling structure as a function of the incident angle, as shown in Fig. 7(b). It is observed that the four distinct absorption peaks still remain very high and will not affect each other at the angle of incidence of 65°. Beyond this, the four absorption peaks monotonically decrease with the increasing angle of incidence. Moreover, the exciting wavelength of surface plasmon polaritons(SPPs) move towards longer wavelength as the angle increases, which strongly deteriorate the magnetic resonances. This deterioration makes the absorption peaks to decrease and couple to each other. Nevertheless, the results reveal that the proposed radiative cooling structure still can operate quite well over a wide range of incident angle for TM polarization.

Furthermore, it is also expected that a polarization-independent radiative cooling structure can be achieved for the practical application [21–23]. Thus, we extend the mechanisms of multi-band absorption to the 2D periodic structure and construct a polarization-independent radiative cooling structure, as shown in Fig. 8(a).In the same preliminary design steps, we substrate the assumed first resonance wavelength λ = 8 μm and the refractive index of upper spacing layer n₁ = 2.9 to Eq. (6), which is a part of the total impedance and also one item of Eq. (2). On condition that the thickness of metallic layer and upper dielectric spacing layer are h = 0.06 μm and t₁ = 0.3 μm, the width of the smallest patch w₁ = 1.38 μm can be obtained. Thus, the four-layer subunits of four different sizes w₁ = 1.34 μm, w₂ = 1.42 μm, w₃ = 1.5 μm and w₄ = 1.6 μm are given with proper differences. The four subunits are incorporated in one unit cell, with the two widest patches diagonally placed. Each subunit is in the middle of a square sublattice of 2.6 μm. Thus, the unit cells repeat themselves with the period of 5.2 μm in both lateral dimensions. The structures are laterally symmetric, which ensure the polarization insensitivity. For the last resonance wavelength 12 μm and the widest patch w₄ = 1.6 μm, the lower dielectric spacing layers is estimated at n₂ = 4 with thickness of t₂ = 0.5 μm, which is adjusted to be impedance-matched. Figure 8(b) presents the absorption spectra of the 2D radiative cooling structure for different polarization angles from 0° (TM polarization) to 90° (TE polarization) in a step of 10°. It is obvious that eight absorption peaks are all independent to the polarization angle of the incident wave, which corresponds to the eight metal-dielectric patches. The polarization insensitive property is originated from the fourfold rotational symmetry of the 2D structure, which can be found in the neighboring nine sublattices.

 

Fig. 8 (a) Schematic of the two dimensional radiative cooling structure. (b) Absorption spectra with different polarization angles from 0° (TM) to 90° (TE) in a step of 10° at normal incidence.

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The influences of azimuthal angle on the absorption spectra are also investigated at a fixed incident angle of 25° for both TE and TM polarization, as shown in Fig. 9(a)and 9(b).The azimuthal angle φ is defined as the angle between the plane of incidence and the x axis of the coordinate. It is seen that the absorption spectra at the azimuthal angle of 45° is nearly same as that of 0° and 90° for the TE polarization. Similarly, the absorption band remains very high at different azimuthal angles for the TM polarization. The results show that radiative cooling structure is very robust to the azimuthal angle of the incident electromagnetic wave. In addition, the 2D radiative cooling structure is definitely angular insensitivity due to the exactly same absorption mechanism with the aforementioned case (as shown in Fig. 7(b)). Therefore, we can realize an omnidirectional and polarization insensitive broadband perfect absorption structure. These advantages make it a good candidate for radiative cooling application.

 

Fig. 9 Absorption spectra for different azimuthal angles of φ = 0°(red), and φ = 45°(green), and φ = 90°(blue) at a fixed incident angle of 25° for (a) TE and (b) TM polarization. Insets are propagation configuration.

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It is noted that there are some small peaks in the short wavelength. To investigate the origin of the small peaks at short wavelengths, we calculate absorption spectra for different incident angles 10° and 25° at a fixed azimuthal angle of 0° under the TM polarization, as shown in Fig. 10(a).It is clear that the small peaks at short wavelength are nearly unchanged, which verify that these peaks are not excited by SPP resonance [33]. Fig. 10(b) are the distributions of the magnetic field on the x-z cross-section in the middle of the biggest patch w₄ at the wavelength 7.63 μm. It is seen that the second harmonics modes occur at the lower layer of the biggest patch. Therefore, we consider that the small peaks at short wavelengths are due to the second harmonic modes, which can only be excited at oblique incidence. However, the fundamental modes and the second harmonic modes are excited simultaneously. Thus, the resonance wavelength is shift, and the peak cannot occur at about half of the wavelength of the fundamental modes.

 

Fig. 10 (a) Absorption spectra for different incident angles 10° and 25° at a fixed azimuthal angle of 0° under the TM polarization. (b) Distributions of the magnetic field on the x-z cross-section in the middle of the biggest patch w₄ at the wavelength 7.63 μm.

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5. Conclusion

In conclusion, we have presented two different dual-band absorbers to illustrate the multi-band resonant absorption which can be predicted by the parallel LC(PLC) circuit model proposed in this paper. One is multi-sized structure with the identical dielectric spacing layer, and the other is the multilayer structure of the same strip width. In both cases, the resonant wavelengths are analytically predicted by the PLC circuit model and agree well with the RCWA calculations with varying structure parameters. More importantly, based on the PLC circuit model, we preliminarily designed a radiative cooling application, which is realized by merging several close peaks together. All the structural and optical parameters are derived directly using the PLC circuit model with proper approximations. This radiative cooling application is very robust to the incident angle. Further, the radiative cooling application can also be omnidirectional and polarization-insensitive by constructing a 2D periodic structure in the same design principle. This structure possesses excellent performance for radiative cooling application and can be extended for band-stop filters, thermal emitters and other energy conversions application by tuning the operating wavelength.