## Abstract

We perform a comprehensive analysis of multi-band absorption by exciting magnetic polaritons in the infrared region. According to the independent properties of the magnetic polaritons, we propose a parallel inductance and capacitance(*PLC*) circuit model to explain and predict the multi-band resonant absorption peaks, which is fully validated by using the multi-sized structure with identical dielectric spacing layer and the multilayer structure with the same strip width. More importantly, we present the application of the *PLC* circuit model to preliminarily design a radiative cooling structure realized by merging several close peaks together. This omnidirectional and polarization insensitive structure is a good candidate for radiative cooling application.

© 2014 Optical Society of America

## 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
*w _{1}* = 1 μm and

*w*

_{2}= 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.

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.

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, *c*_{1} is a numerical factor accounting for the non-uniform charge distribution at the metal surfaces. In addition, ${L}_{e}=\text{-}w/({\omega}^{2}\delta {\epsilon}_{0})\cdot ({\epsilon}^{\prime}/({{\epsilon}^{\prime}}^{2}+{{\epsilon}^{\u2033}}^{2}))$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${\epsilon}^{\u2033}$ 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

*PLC*circuit model to explain and predict the dual-band resonant absorption peaks, as shown in Fig. 2(d). When several components are connected in parallel, the total impedance can be expressed as

_{tot}= 0.

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
*c _{1}* to provide better prediction. When we increase the cavity
length, electrons tend to accumulate at the corners of metallic surface and a smaller

*c*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

_{1}*PLC*circuit model.

## 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 Al_{2}O_{3} 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 Al_{2}O_{3} 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
Al_{2}O_{3} 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.

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
Al_{2}O_{3} 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

*LC*circuit model is the same with Eq. (1) of the grating-film structure. As shown in Fig. 5(d), we connect the two independent subunits in parallel and the total impedance can be express as

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.

## 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 *λ*_{1} = 8.5 μm,
*λ*_{2} = 9.5 μm, *λ*_{3} =
10.5 μm and *λ*_{4} = 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

*s*= 0.6 μm and

*h*= 0.06 μm, respectively, which have little influence on the resonance wavelength as long as they are large enough. Importantly, the refractive index of the upper spacing layer is given as an initial value

*n*

_{1}= 2.6. To determine the two strip widths, we investigate the upper metal-dielectric pairs in the first place. Thus, we take the designed resonance wavelengths

*λ*

_{1}= 8.5 μm,

*λ*

_{2}= 9.5 μm, the refractive index of the upper spacing layer

*n*

_{1}= 2.6 and other parameters into the Eq. (6):

*L*

_{m},

*L*

_{e}and

*C*

_{m}, respectively. According to the

*PLC*circuit model, the two independent impedances of the upper spacing layer are connected as the Eq. (3)

_{upper}= 0, we can obtain the widths of two strips with

*w*

_{1}= 1.38 μm and

*w*

_{2}= 1.55 μm on condition the dielectric spacing layer is

*t*= 0.21 μm and the non-uniform charge distribution factors is

*c*= 0.19. Therefore, the period can be obtained by $\Lambda =2s+{w}_{1}+{w}_{2}$ = 4.13 μm. It is clear that the Eq. (9) has given the widths of two strips, thus the two strip widths are fixed. Then, we substitute the resonance wavelengths

_{1}*λ*

_{3}= 10.5 μm,

*λ*

_{4}= 11.5 μm and the two strip widths into the Eq. (6) to determine the refractive index of the lower spacing layer where the parameters

*A*and

*B*are the same with Eq. (7) and Eq. (8) except for the $C={c}_{\text{1}}{n}_{2}^{2}{\epsilon}_{0}$. In the same way, the two independent impedances of the lower spacing layer given by Eq. (10) and Eq. (11) are connected in parallel:

_{lower}= 0, there is only one unknown quantity

*n*

_{2}. The refractive index of the low spacing layer can be solved implicitly with

*n*

_{2}= 3.23. Note that, only the proper thickness of the spacing layer can make the four distinct absorption peaks near-unit.

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
*n*_{1} = 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 *t _{1}* = 0.3
μm, the width of the smallest patch

*w*

_{1}= 1.38 μm can be obtained. Thus, the four-layer subunits of four different sizes

*w*

_{1}= 1.34 μm,

*w*

_{2}= 1.42 μm,

*w*

_{3}= 1.5 μm and

*w*

_{4}= 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

*w*

_{4}= 1.6 μm, the lower dielectric spacing layers is estimated at

*n*

_{2}= 4 with thickness of

*t*= 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.

_{2}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.

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 *w*_{4} 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.

## 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.

## Acknowledgments

We thank Doctor Junmin Zhao for helpful discussions. The supports of this work by the National Natural Science Foundation of China (No.51121004, No.51306043, and No.11474077) and the Program for Innovation Research of Science in Harbin Institute of Technology(Grant No. A201411 and Grant No.B201407) are gratefully acknowledged.

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