## Abstract

The realization of ultrabroadband absorption is a fundamental part of a thermal emitter, especially in the application of radiative cooling. This study involved proposing and systematically analyzing a novel structure termed as an embedded metal-dielectric-metal (EMDM) structure. The results in the case of an individual resonator indicated that the EMDM resonator displayed a broader full width at half maximum (FWHM) that was 1.9 times that of the metal-dielectric-metal (MDM) resonator due to mode matching at the terminated end and enhanced scattering intensity. With respect to the case of periodic resonators, single-sized periodic EMDM resonators are employed to achieve a broader FWHM that is 3.8 times that of the MDM resonators. In addition, a strong coupling effect is confirmed between localized MDM and hybrid modes. An application of lossy-dielectric based periodic three-dimensional EMDM resonators indicated that an average absorptivity of 0.85 in the entire atmospheric window (8–13 μm). The results revealed the potential of EMDM structures for radiative cooling devices and other ultrabroadband absorbers.

© 2017 Optical Society of America

## 1. Introduction

Light–matter interaction is a universal research topic. Metamaterials are artificial materials that can be used in various potential applications such as optical camouflage [1], super/hyper lenses [2,3], holography [4], slow light [5], and perfect absorbers [6]. Among these applications, perfect absorbers have attracted considerable attention in plasmonic color palettes [7], plasmonic sensors [8], plasmon-induced transparencies [9], and thermal emitters [10]. A metal-dielectric-metal (MDM) resonator is a representative structure of a perfect absorber with advantages including spectral tunability, near-unity absorptivity, and ultrathin thickness [11]. However, single-sized periodic MDM resonators are not suitable for broadband absorption because the resonant-based structure results in a narrow absorption band. A few common strategies to expand an absorption band by overlapping resonances of resonators with different sizes involve a combination of multi-sized MDM resonators [12], multi-layer metal-dielectric resonator [13], and irregular shape MDM resonators [14]. The aforementioned structures do not involve any novel principles. Complicated geometries lead to non-trivial work with respect to the design and fabrication of broadband MDM absorbers.

One significant application of broadband MDM absorbers is passive radiative cooling, which recently attracts renewed interests in achieving sub-ambient temperature by high performance radiative cooling devices. Generally, such kind of devices can be divided into two main categories: night-time cooling and daytime cooling [15]. In night-time cooling, it requires a thermal emitter with an extremely high efficiency in an atmospheric window with the wavelength range of 8–13 μm; besides above requirement, daytime cooling also needs high reflectance in visible and near infrared frequency. Not limited in MDM structures, many remarkable results are achieved by various approaches, such as daytime cooling devices by multilayer photonic structures [16–19], glass-polymer hybrid metamaterial, and two-dimensional photonic crystals based on surface phonon polariton [20]; nighttime cooling devices made by silicon monoxide [21], white pigmented paints [22]. Compared with these approaches, employing MDM structures are feasible to achieve near-unity absorptivity or emissivity, excellent wavelength-selectivity, and ultra-thin thickness (0.1-1 μm) in atmospheric window, which is promising in application of high performance nighttime cooling (or under the condition of diffused sunlight). Due to the existence of higher order resonant modes in visible and near infrared frequency, MDM structure is not sufficient for daytime cooling, however, it is possible to overcome this dilemma by combination wide band dielectric mirror and MDM structures [23].

In the design of ultrabroadband MDM structures, the aforementioned strategies (i.e. multi-sized, multi-layers and irregular shape of resonators) were used by researchers to propose conical metamaterials [24], multi-layer and multi-band absorbers [25], and gradient-metasurface-based absorbers [26]. These studies were characterized by the achievement of excellent cooling performance (nighttime). However, it increases the difficulties in both design and fabrication, which may limit large scale cost effective production.

In this study, a novel structure noted embedded metal-dielectric-metal (EMDM) structure is proposed, and the structure is appropriate for ultrabroadband absorption and especially the application of radiative cooling devices. With respect to an individual resonator, the full width at half maximum (FWHM) of the EMDM resonator is 1.9 times that of the MDM resonator and results from mode matching at the terminated end and enhanced scattering intensity. With respect to periodic resonators, single-sized periodic EMDM resonators are employed with an FWHM 3.8 times that of the MDM. Additionally, these resonators avoid near-field coupling between multi-sized resonators. In addition to conventional MDM resonator, the strong coupling effect is confirmed between localized MDM and hybrid modes. The existence of a hybrid mode enables EMDM resonators to significantly expand the absorption band. In an application of periodic 3D EMDM resonators, an average absorptivity of 0.85 was achieved in the entire atmospheric window (8–13 μm) by lossy-dielectric based single-sized ultra-thin EMDM resonators (0.6 μm).

The study is organized as follows. In Sec. 2, modified Fabry–Pérot (F–P) model is employed to analyze resonant characteristics of the individual MDM and EMDM resonators. The systematic analysis of reflection coefficient (*r*) at the terminated end [27,28] is used to clarify the underlying physics of broadband absorption in an individual EMDM resonator with top and side illuminations. In Sec.3, based on mode characteristics of dielectric-loaded metal waveguide (DLMW), the unusual change of *r* is attributed to the increased field confinement of DLMW mode and mode matching between EMDM mode and the first order DLMW mode within EMDM structures. In Sec.4, the single-sized periodic 2D and 3D EMDM resonators are comprehensively analyzed and it is confirmed that broadband absorption originates from the strong coupling between localized mode and hybrid mode. Finally, it is possible to further improve the absorption performance of EMDM structures by lossy dielectric materials.

## 2. Fabry-Pérot model in an individual resonator

#### 2.1 Individual MDM resonator

In an individual MDM resonator, standing wave modal profiles are formed due to multiple reflections by the terminated ends. From a mode analysis viewpoint, the MDM resonator supports same modes as the MDM waveguide. In mid infrared frequency, the MDM structures only support a highly confined symmetrical TM-mode due to the sub-wavelength size (neglecting surface mode on the top surface of the MDM structures). In this study, the symmetrical TM-mode is termed as the “MDM mode”. Two-dimensional (2D) structures (*y*-invariant) are discussed for the purposes of simplicity of the analysis. In contrast, three-dimensional (3D) structures only result in more mathematical complexity although new mechanisms do not appear. In Fig. 1(a), schematic geometry of an individual MDM resonator is shown in which copper (Cu) as a requirement of radiative cooling is utilized as metal layers with advantages of high thermal conductivity and complementary metal–oxide–semiconductor (CMOS) process compatibility [29]. In optics, Cu is an approximate perfect electric conductor in mid-infrared frequency with very large optical constant (both real and imaginary parts). The sub-wavelength thick gap layer is composed of amorphous silicon (a-Si) that includes the advantages of an almost constant high refractive index in mid-infrared frequency. Furthermore, it is noteworthy a-Si and Germanium (see Sec. 2.2) are also lossless materials (dielectric) in mid-infrared frequency, in contrast, many widely-used dielectric materials in visible frequency, such as SiO_{2}, Al_{2}O_{3}, and TiO_{2}, are lossy in mid-infrared frequency due to the effect of Reststrahlen band. In calculations, the optical constants of Cu and a-Si are taken from [30] and [31], respectively.

As shown in Fig. 1(a), specifically, *r* = *|r|e ^{iϕ}* denotes complex reflection coefficient at a terminated end including reflection amplitude

*|r|*and phase retardation

*ϕ*, which are the key parameters in this study. Additionally,

*s*and

*t*represent the thicknesses of a-Si layer and Cu strip, respectively. This was followed by the modified F–P model proposed by Barnard

*et al.*that analyzed the resonant properties of an individual metallic antenna [27], which could also be employed in the case of the MDM resonator. Resonant peak positions are calculated as follows:

*n*,

_{eff}*w, λ*, and

_{0}*k*denote the real part of effective refractive index, the width of resonator, the wavelength in free space, and the in-plane wave vector of the mode, respectively. Additionally,

*m*denotes the order of the resonance (

*m*= 1, 2, 3...).

It is important to select a proper incident light source to further analyze the resonant characteristics of a MDM resonator. As shown in Fig. 1(a), the resonator is top illuminated by a normal incident plane wave polarized along the *z* axis, and this configuration is termed as “top illumination”. In a top illumination, the enhanced electric field can be evaluated by summing the contributions of multiple reflections at the terminated end of sub-wavelength sized gap (*E _{end}*) where |

*E*| is defined as the amplitude of

_{end}*E*at the center of the terminated end of the gap [Fig. 1(a)]. According to the F–P model, the dependence of |

_{end}*E*| is given by a previous study [27] as follows:

_{end}*E*| is a representative quantity that can be directly connected with the resonant characteristics of resonator such as the peak position and line-shape and scattering or absorption cross section.

_{end}In contrast, as shown in Fig. 1(a), “side illumination” is also considered in which a single mode source (MDM mode) is placed at a terminated end of gap layer and propagates along the *z* axis. In the case of the side illumination, it directly establishes a connection between |*E _{end}*| and

*r*that is independent of the coupling of free space waves. Additionally, the side illumination is suitable for the analysis of sub-radiant or dark modes due to the asymmetrical distribution of dipole moments. Given the equation proposed by Miyata

*et al.*[28], the dependence of |

*E*| is given as follows:

_{end}*|r|*and

*ϕ*play vital roles in the resonant characteristics of individual MDM resonator, i.e., a small value of

*|r|*expands the peak width and a large value of

*ϕ*shifts the position of the resonant peak. Additionally, when light propagates between media of different refractive indices, then

*|r|*and

*ϕ*can be analytically derived by Fresnel equations. However, Fresnel equations lose validity and always underestimate

*|r|*and

*ϕ*in sub-wavelength structures due to an abrupt mode mismatch at the terminated end. Instead, a finite-difference time-domain (FDTD) calculation is performed by using a commercial software from Lumerical FDTD Solutions to evaluate

*|r|*and

*ϕ*in an individual MDM resonator with geometrical parameters corresponding to

*s*= 0.05 μm,

*t*= 0.6 μm and

*w*= 1.25 μm. The details of the FDTD calculation and mode source settings can be found in previous studies [27,28].

As shown in Fig. 1(b), near-unity *|r|* (red line) and near-zero *ϕ* (blue line) are found, resulting from tiny values of *s* (*s/λ _{0}* <<1) and the approximation of Cu as a perfect electric conductor in mid-infrared frequency. It should be noted that the frequency dependence of

*|r|*and $\varphi $ is relatively weak.

In Fig. 1(c) and (d), based on FDTD results of *|r|* and *ϕ*, Eq. (2) and (3) are employed to evaluate spectra of |*E _{end}*| under top and side illuminations, respectively. In order to examine the validity of the F–P model, the spectra of |

*E*| are directly calculated by the FDTD method and compared with those from the equations. With respect to the predictions of the peak position of |

_{end}*E*|, the results of Eq. (2) and (3) are acceptable with a considerably small deviation of resonant wavelengths (

_{end}*Δλ/λ*≈0.03) when compared with that of the FDTD results. With respect to predictions of line-shape of |

_{0}*E*|, Eq. (2) and (3) provide accurate results of the peak value (as confirmed from the peak at

_{end}*λ*= 6.1 μm in Fig. 1(c)) albeit with a relatively narrow line width. The observed deviation is mainly attributed to the scattering of radiation modes from the free space at the terminated end although the validity of the F–P model is confirmed.

_{0}From Fig. 1(c) and (d), the first order resonant peaks (*m* = 1) at *λ _{0}* = 12.4 μm (FDTD results) are observed and are radiant in both illuminations and are also termed as asymmetric MDM or bright modes in a previous study [32]. Additionally, the results of FDTD as well as that of the equations exhibit almost similar peak positions, which imply a relatively weak frequency-dependent coupling efficiency between the free space and MDM resonator. Furthermore, in the case of side illumination, the second order resonant peak (

*m*= 2) is observed at

*λ*= 6.1 μm in Fig. 1(c) (based on the FDTD result). In contrast, it is predicted that this type of peak will not appear in the case of top illumination (Fig. 1(d)) because the dipole moment is cancelled out by the symmetrical dipole distribution (dark mode).

_{0}#### 2.2 Individual EMDM resonator

The schematic geometry of an individual EMDM resonator is shown in Fig. 2(a) in which a conventional MDM cavity is embedded into the dielectric layer. Germanium (Ge) is employed as the lossless dielectric material due to the high constant refractive index (*n _{d}* ≈4 in mid infrared frequency with the negligible imaginary part [31],). A key geometrical parameter corresponds to the thickness of each layer, which is denoted as

*t*in this case, where

_{d}= t + s*t, s*and

*t*denote the thicknesses of the Cu strip, a-Si gap layer and Ge dielectric layer, respectively, and

_{d}*w*denotes the width of Cu strip.

Figure 2(b) depicts the spectra of *|r|* and *ϕ* calculated by FDTD methods with the same process as that in Fig. 1(b) under the condition that *t* = 0.6 μm, *s* = 0.05 μm, *t _{d}* = 0.65 μm, and

*w*= 0.8 μm. Curves of

*|r|*and

*ϕ*exhibit an unusual change with respect to the wavelength. Especially,

*|r|*

_{min}= 0.35 (10.1 μm) and

*ϕ*

_{max}= 0.41 π (11.7 μm) are observed, which implies unusual mode matching between the MDM cavity and embedded layer at the terminated end. In the MDM cavity, the single symmetrical MDM mode is still supported, in order to distinguish with the aforementioned MDM mode, this mode is termed as “EMDM mode”. Additionally, the adjoining embedded layer may support propagating mode (fields are mainly bounded in embedded layer), which is distinct from the counterpart of individual MDM resonator (only metal substrate without embedded layer). In detail, the underlying physics will be examined in next section. The results shown in Fig. 2(b) indicate that the EMDM structure can effectively tune

*|r|*and

*ϕ*in the mid-infrared frequency, which provides an opportunity to expand the resonant peak width and absorption band.

This is followed by discussing the spectra of |*E _{end}*| in an individual EMDM resonator under top and side illuminations. Note |

*E*| is the key result in Eqs. (2) and (3) of F-P model. As shown in Fig. 2(c), the result of Eq. (3) (blue line) is in agreement with the FDTD calculations (red line). Compared with Fig. 1(c), the prediction of peak width by Eq. (3) does not indicate a remarkable difference between MDM and EMDM resonators around

_{end}*λ*= 12.6 μm under side illumination [see Fig. 1(c)]. Although

_{0}*|r|*is significantly smaller in the EMDM structure, the expectable expanding effect does not exist due to the rapid change in

*|r|*and

*ϕ*[Fig. 2(b)]. Instead, a weak shoulder of |

*E*| (note this is not resonance) is observed approximately in the range of 7–10 μm. Additionally, with respect to the result of FDTD (red line) in Fig. 2(c), a broader peak width of |

_{end}*E*| is observed due to mode matching from embedded layer.

_{end}As shown in Fig. 2(d), with respect to the result of Eq. (2) (blue line), a line-shape similar to that in Eq. (3) [Fig. 2(c), blue line] and no obvious expanding effect relative to the MDM case [see Fig. 1(d)] are observed. It should note the frequency-dependent coupling from free space wave (plane wave) to EMDM mode is neglected in derivation of Eq. (2). In contrast, a remarkably broader resonant peak can be confirmed under top illumination in FDTD results (red line, including the coupling from free space wave), leading to large peak width of |*E _{end}*| in EMDM resonator. Besides the resonance of EMDM resonator [predicted by Eqs. (2) and (3)], this effect also originates from the matching from embedded layer and frequency-dependent free space wave.

As shown in Fig. 3(a), the coupling intensity from free space wave to |*E _{end}*| is calculated by FDTD method in MDM and EMDM structures [not resonator, see Fig. 4(a)], the values of |

*E*| are normalized by the same light source. It clearly indicates the coupling efficiency of EMDM structure is much larger when compared with that of MDM counterpart at 10 <

_{end}*λ*< 15 μm, corresponding to expansion of peak width in Fig. 2(d). Interestingly, in contrast to almost frequency-independent |

_{0}*E*| of MDM structure, the coupling efficiency displays a frequency-dependent periodic enhancement in EMDM structure, which is directly connected to thin film interference of dielectric embedded layer (controlled by optical thickness).

_{end}Finally, in order to evaluate absorptive characteristics of an individual resonator, the absorption cross section (ACS) of MDM and EMDM resonators are calculated. The ACS is defined as follows:

where*σ*,

*P*, and

*I*denote absorption cross-section (unit: μm

^{2}), total absorption power in an individual resonator (unit: W), and intensity of illumination (unit: W/μm

^{2}), respectively. In electromagnetism, absorbed power is calculated as follows:

*ε*(

*x*,

*z*)],

*ω*, and

*A*denote the imaginary part of permittivity, the angular frequency, and the area of cross section, respectively. Hence, a broad absorption peak (the first order peak) in the EMDM resonator can be predicted due to the spectra of |

*E*| [see Fig. 2(d)].

_{end}As shown in Fig. 3(b), ACS in MDM (magenta line) and EMDM (black line) resonators under top illumination are compared. Evidently, the EMDM structure exhibits a significantly wider peak width relative to that of the MDM structure, and an FWHM that is 1.9 times that of the MDM structure. Additionally, a relatively larger value of the peak in EMDM structure is also achieved due to coupling from free space wave [see Fig. 3(a)]. From above discussion, the ACS can be analyzed by F-P model and frequency-dependent coupling from free space wave. F-P model is very simple model without fitting parameters and provide accurate prediction of peak position and linewidth of resonator itself; the coupling from free space wave enhance the ACS at resonance. The expansion of peak width of ACS is benefit from resonator itself and coupling from free space waves in an individual EMDM resonator. A detailed analysis is performed by mode matching method in next section. Consequently, the EMDM resonator provides more excellent absorptive characteristics when compared to its MDM counterpart in an individual case.

## 3. Mode matching analysis of EMDM structures

#### 3.1 Mode characteristic of dielectric-loaded metal waveguide

From the proposed analytical framework in last section, some pertinent questions may be asked, such as the underlying physics of unusual change of *|r|* and *ϕ* in Fig. 2(b); the mode matching between EMDM and other possible modes supporting by embedded layer and free space. To answer these questions, it is necessary to analyze primarily mode characteristics in embedded layer. As shown in region II of Fig. 4(a), the embedded layer and metal substrate are termed as dielectric-loaded metal waveguide (DLMW) working in mid-infrared frequency. The visible or near infrared frequency counterpart of DLMW is well-known dielectric-loaded surface plasmon polaritons waveguide (DLSPPW) [33]. The dispersion relation of the first order TM mode in DLMW (abbreviated as DLMW mode in Sec.3.1) is described by equations [34] as follows:

*β*=

_{0}*k*is the propagation constant along

_{0}n_{eff}*z*-direction.

*k*,

_{d}*k*and

_{f}*k*are

_{m}*x*-component of wave vectors in dielectric (Ge), free space and metal (Cu), respectively.

*ε*,

_{d}*ε*and

_{f}*ε*are the permittivity of corresponding areas.

_{m}The spectra of *n _{eff}* is shown in Fig. 4(b), same parameters as Sec. 2.2 are chosen (

*t*= 0.65 μm,

_{d}*n*= 4, air-Ge-Cu structure). The real part of

_{d}*n*[denoted as Re(

_{eff}*n*), red line] decreases gradually with respect to increasing wavelength. With short wavelength range around

_{eff}*λ*= 6 μm, the DLMW mode approximates to dielectric guided mode due to Re(

_{0}*n*) approaching to

_{eff}*n*. In contrast, smaller Re(

_{d}*n*) can be found with long wavelength range of

_{eff}*λ*> 13 μm, the DLMW mode approximates to surface mode of air-Cu interface, resulting from extremely small ratio of

_{0}*t*/

_{d}*λ*. Therefore, the transition feature of DLMW mode is displayed in target wavelength range from 8 to 13 μm. Importantly, it is interesting to find a maximum value of the imaginary part of

_{0}*n*[Im(

_{eff}*n*)

_{eff}_{,}blue line in Fig. 4(b)] around 10.2 μm, which overlaps the same wavelength as |

*r*|

_{min}in Fig. 2(b). From the perspective of mode matching analysis, |

*r*| is determined by degree of mode matching. In contrast to the EMDM mode, the DLMW mode is a propagating mode with weak field confinement. Normally, it is very limited degree of matching between weak and strong confinement modes. However, Im(

*n*)

_{eff}_{max}means the highest confinement (with high loss), which implies the rapidly increased degree of mode matching between EMDM and DLMW modes around Im(

*n*)

_{eff}_{max}.

#### 3.2 Mode matching analysis of EMDM structure

In previous discussion, it points out the Im(*n _{eff}*) of the first order DLMW mode may determine the |

*r*| and

*ϕ*at the terminated end due to mode matching. Nevertheless, the higher order DLMW mode and free space wave are also possible to affect the mode matching in EMDM structure. In order to quantitatively analyze mode matching at single terminated end of EMDM structure [not resonator, Fig. 4(a)], bidirectional eigenmode expansion (EME) [35, 36] is performed to calculate the fraction of power transmitted into specified modes. Generally, the electromagnetic field at the terminated end [see Fig. 4(a), green dashed line] can be expanded as follows:

*f*and

*b*denote the forward and backward propagating modes with mode number

*n*,

*a*and

_{n}*b*denote the complex transmission coefficients of ${E}_{n}{}^{f}$ and ${E}_{n}{}^{b}$, respectively. The net transmittance of the

_{n}*n-th*mode power (

*T*) can be derived from forward (

_{n}*T*) and backward (

_{f, n}*T*) transmitted mode power as follows:where

_{b, n}*a*and

_{n}*b*can be calculated by mode orthogonality and specified modal profiles (such as

_{n}*E*and

_{n}*H*). The details of calculations can be found in a previous study [35]. In the present study, EME is employed to decompose total net transmitted power (

_{n}*T*) at the terminated end,

_{total}*T*is formulated as follows:where

_{total}*T*denote the net transmitted power of DLMW modes and free space wave (generally, free space wave includes infinite number of modes),

_{DLMW}, T_{free space}*n*is the mode number of the DLMW modes. Generally,

*T*,

_{total}*T*, and

_{DLMW}*T*can be normalized by total power (including reflected and transmitted power) at the terminated end. Therefore,

_{free space}*T*= 1- |

_{total}*r*|

^{2}means the total transmittance at the terminated end, likewise,

*T*, and

_{DLMW}*T*are transmittance of DLMW modes and free space wave.

_{free space}In Fig. 5, the spectrum of *T _{total}* (black line) and the transmittance of the first order DLMW mode (

*T*, red line) are compared at the terminated end (

_{DLMW, 1}*t*= 0.65 μm,

_{d}*n*= 4). Evidently,

_{d}*T*dominates

_{DLMW, 1}*T*(i.e.

_{total}*T*/

_{DLMW, 1}*T*is large) in wavelength range of 5 μm <

_{total}*λ*< 15 μm. This indicates that the first order DLMW mode dominates the mode matching between MDM cavity and DLMW. The maximum

_{0}*T*is obtained at

_{DLMW, 1}*λ*= 9.8 μm with the same position of the maximum

_{0}*T*and takes up nearly 80% of it. This result explains the unusual change of |

_{total}*r|*in Fig. 2(b). Note

*T*can’t be neglected especially at 10 μm <

_{free space}*λ*< 15 μm (the maximum of

_{0}*T*is beyond 0.2),

_{free space}*T*and

_{free space}*T*results in expansion of peak width of |

_{DLMW,1}*E*| [see Fig. 2(c), red line]. It is deserved to mention the coupling efficiency from free space wave to EMDM and DLMW modes can also be calculated analytically by applying F-P model at the top interface of EMDM structures [37]. Additionally, the ratio of

_{end}*T*/

_{DLMW, 1}*T*is dropping with respect to 2 μm <

_{total}*λ*< 5 μm, resulting from the contributions of the higher order DLMW (TM) modes (not shown). Therefore, higher order DLMW modes determine mode matching in short wavelength range. Consequently, the matching of DLMW mode (especially the first order mode) and free space wave play vital roles by controlling

_{0}*|r|*,

*ϕ*and the free space coupling efficiency of EMDM structure, which determines the unique characteristics of EMDM resonator and exhibits the possibility of expansion of absorption band in ultra-thin absorber.

## 4. Periodic EMDM resonator

#### 4.1 Single-sized periodic EMDM resonators

In this section, the discussion is extended from an individual resonator to periodic EMDM resonators. For the purposes of simplicity, this also commences with 2D structures (*y*-invariant), and the schematic geometries are shown in Fig. 6(a). Conventionally, in the design of periodic broadband MDM-based absorbers, a common strategy involves a combination of multi-sized resonators in a single super-cell (period). However, strong near field coupling between multi-sized resonators typically occurs within a sub-wavelength sized super-cell, and this significantly hinders the overall performance of absorbers, such as peak positions and absorptivity. In order to eliminate strong near-field coupling between multi-sized resonators, single-sized periodic EMDM resonators and appropriate size of the period are selected. This provides a new approach to overcome the aforementioned challenges.

As shown in Fig. 6(b), the absorption spectra of periodic EMDM and MDM resonators are compared under broadband normal incident light with *z*-direction polarization (top illumination) in which the parameters are selected as *w _{MDM}* = 1.25 μm, and

*P*3 μm for periodic MDM and

_{MDM =}*w*= 0.8 μm, and

_{EMDM}*P*= 3 μm for EMDM. Here,

_{EMDM}*w*and

_{MDM}*w*denote the widths of Cu strips in MDM and EMDM resonators, and

_{EMDM}*P*and

_{MDM}*P*denote the periods of MDM and EMDM resonators, respectively. Furthermore,

_{EMDM}*t*= 0.6 μm,

*s*= 0.05 μm and

*n*= 4 preserve the same values as those in the individual case. With respect to the absorption spectra, the results indicate that the EMDM can increase the FWHM such that it is 3.8 times broader than that of MDM structures. Periodic MDM resonators maintain nearly the same peak positions and FWHM (red line) when compared to an individual resonator. This implies that the near field coupling is negligible under the appropriate size of the period. In contrast, the absorption spectrum of periodic EMDM resonators (blue line) indicates three resonant peaks labeled as P1, P2 and P3, which are located at 12.29 μm, 9.46 μm, and 6.54 μm, respectively.

_{d}In order to further analyze these peaks, EM field distributions of |*E _{x}*| and |

*H*| are calculated at each wavelength. With respect to P1 at 12.29 μm, the first order MDM modal profiles can be typically found [as denoted by the first row in Figs. 7(a) and 7(b)] where major EM fields are confined and enhanced within a-Si gap layer. It can be found P1 displays almost the same peak position as the individual case (Fig. 3), therefore the analytical framework of Sec 0.2 and 3 can be directly extended to periodic structures. With respect to P2 at 9.46 μm, resonant anti-nodes of |

_{y}*H*| [as denoted by the second row in Fig. 7(b)] appear at

_{y}*z*= 0 and

*z*= $\pm $1.5, and the field distribution can be attributed to the hybrid resonance that originates from propagation modes generated by combination of EMDM and DLMW modes. It should be noted that P2 does not exist in the individual case. Similarly, with respect to P3 at 6.54 μm, the |

*E*| and |

_{x}*H*| field distributions are attributed to the second order hybrid resonance [the third row in Figs. 7(a) and (b)].

_{y}Hybrid resonance results from the propagation modes of periodic EMDM structures. It is interesting to discover the modal duality of EMDM mode between peaks P1 and P2/P3. With respect to peak P1, the localized nature can be predicted by analytical framework aforementioned (Sec. 2); in contrast, P2/P3 display the propagating (guided) mode characteristics that combines EMDM and DLMW modes, the so-called hybrid mode. In the perspective of diffraction theory [38], it is a special case of guided mode resonance. At the normal incidence, the resonant condition of hybrid mode is formulated as follows:

where*n*denote the real part of effective refractive index of EMDM and DLMW modes, respectively.

_{EMDM}, n_{DLMW}*m*is the order of the hybrid mode resonance (

*m =*1, 2, 3…), P2, and P3 correspond

*m*= 1, and

*m*= 2, respectively.

As shown in Figs. 8(a) and 8(b), the absorption map of periodic EMDM resonators with varying *w _{EMDM}* and

*P*are calculated. As indicated by these results, it is observed that the peak position of P1 is mainly dependent on

_{EMDM}*w*and insensitive to

_{EMDM}*P*. In contrast, the peak positions of P2 and P3 are mainly dependent on

_{EMDM}*P*and insensitive to

_{EMDM}*w*. Black dashed lines indicate the peak positions of P1 and P2 predicted by F-P model [Eqs. (1) - (5)] and Eq. (12), respectively, an excellent agreement with FDTD calculation results is observed. The deviation of P3 results from phase accumulation of higher hybrid modes (the characteristics of propagation modes). Furthermore, when the resonant frequencies of above two peaks approach each other at approximately

_{EMDM}*w*= 0.65 μm, as shown in Fig. 8(a), they exhibit an unusual anti-crossing behavior that corresponds to the signature of a strong coupling regime.

_{EMDM}Similar phenomena are also found in a metallic photonic crystal slab [32] and plasmonic MDM nanobar structures [39] in visible frequency. As indicated in studies by Christ *et al.* [32,40], the anti-crossing behavior originates from strong coupling between localized plasmonic mode and Bragg diffraction-natured propagating waveguide mode. In the present study, a rigorous propagating waveguide mode does not exist in the periodic EMDM resonators, and instead, hybrid propagation modes are formed by combination of EMDM and DLMW modes. Hence, strong coupling occurs in coupled systems comprising the localized EMDM and propagating hybrid modes. In EMDM structures, the largest Rabi splitting $\hslash {\Omega}_{R}$ of 30.2 meV is observed at approximately *w _{EMDM}* = 0.65 μm and

*P*= 3 μm, which can be predicted accurately in this study, i.e. the coupling strength can be directly associated with mode matching in a simple way. The above Rabi splitting value is significantly smaller when compared to the values previously reported in plasmonic structures at visible frequency. However, this type of smaller Rabi splitting is appropriate for broadband absorption since it avoids a rapid decrease and maintains large absorptivity between P1 and P2.

_{EMDM}The above discussions are totally based on normal incidence. For applications of broadband absorber or thermal emitters, one key figure of merit is the angle resolved absorptivity. As shown in Fig. 9, in contrast to normal incidence, P2 splits into two absorption peaks with distinct shifts in absorption map (*w _{EMDM}* = 0.8 μm,

*P*= 3 μm). The blueshift and redshift peaks are denoted as P2 + and P2- in Fig. 9, likewise, P3- is also a redshift peak. At oblique incidence, assuming the same size of period in

_{EMDM}*x*and

*y*axes (

*P*). Equation (12) is extended and formulated as follows:

_{EMDM}*θ*denotes incident angle in

*x-z*plane and the azimuthal angle is always zero.

*m, n*represent the resonant order of hybrid modes in

*x*and

*y*axes. Evidently,

*n*= 0 is confirmed in two-dimensional structures. The prediction of peak positions by Eq. (13) (black dashed line) is compared with FDTD results, the result show a slight deviation (P2 + , P2-) due to the additional phase induced by mode matching or strong coupling between EMDM and DLMW modes. For higher hybrid modes (P3-,

*m*= −2), this deviation is increased by phase accumulation. It noteworthy peaks P1, P2 + and P2- are preserved at the target wavelength range (8-13 μm), although an absorption dip exists between P2 + and P2-. Furthermore, these peaks can keep large absorptivity even in high angle (

*θ*= 60°), which is helpful to keep large average absorptivity in wide angle range.

*e.g.*the average absorptivity at

*θ*= 60° (0.54) is even larger than normal incident case (0.52).

Consequently, the physical origin of resonant peaks, namely P1, P2, and P3, is clarified. The resonance of the localized EMDM mode leads to peak P1. In contrast, hybrid modes resonance result in P2 and P3, where P2 is considered as the first order peak, and P3 is the second order peak. More importantly, strong coupling and Fano resonance (not shown) phenomenon can also be observed in EMDM structures due to the drastic interaction between the localized and hybrid modes. In this study, the periodic EMDM-based absorbers benefit from a strong coupling effect and avoid the destruction of broadband absorption caused by larger $\hslash {\Omega}_{R}$. Additionally, EMDM structures can preserve and even increase the average absorptivity in atmospheric window at wide-angle range.

#### 4.2 Periodic 3D EMDM resonators

In this section, EMDM structures are extended to the 3D (cube) case as shown in Fig. 10(a) in which a Cu strip and a-Si gap layer preserve the same values of *w _{EMDM}* in both

*y*and

*z*axes.

Figure 10(b) shows the calculation result of 3D EMDM resonators with *t _{d}* = 0.65 μm,

*t*= 0.6 μm,

*s*= 0.05 μm,

*w*= 0.9 μm, and

_{EMDM}*P*= 3 μm. The peak positions of EMDM and hybrid modes are still located in the range of 8–13 μm. Although there are slight changes in the absorptivity and line-shape (two peaks within the atmospheric window) when compared to that of the 2D EMDM with

_{EMDM}*w*= 0.8 μm, the broad absorption band is preserved by strong modes coupling. Additionally, two new fine peaks [i.e. (1,1) and (0,2)] are observed in the shorter wavelength region, which are attributed to additional hybrid resonances in

*y*axis. This indicates the potential to further expand the absorption bandwidth.

Figure 11 shows incident angle resolved absorption map ranging from 10°-60° (*w _{EMDM}* = 0.9 μm,

*P*= 3 μm). Similar to 2D EMDM structures, blueshift and redshift peaks (P2 + , P2-) are found and labeled. Interestingly, the absorption dip between P2 + (1,0) and P2- (−1,0) is gradually indistinct (also the dip between P1 and P2 + ), which leads to a broad absorption band within the atmospheric window. Additionally, modes of (1,1) and (0,2) also split at oblique incidence, which further contribute to average absorptivity. Therefore, 3D periodic EMDM resonators preserve (even better at high angle) broadband absorption in wide angle range.

_{EMDM}#### 4.3 Lossy dielectric based periodic 3D EMDM resonators

In Sec. 4.2, though ultrabroadband absorption is achieved by a single-sized 3D periodic EMDM structure, the average absorptivity in the range of 8–13 μm is not sufficient for high performance radiative cooling devices. In order to promote the absorptivity of the EMDM structure in atmospheric window, lossy dielectric materials are employed. It is necessary to point out the absorptivity is not always benefited from lossy materials [*e.g.* hybrid modes (1,1) in Figs. 10 (a) and 12 (a)]. Note the aforementioned strategy by employing lossy dielectric materials is not general law for every type of resonators and optical constant of lossy dielectric should be chosen carefully (high lossy materials may destroy the resonance). As shown in Fig. 12(a), lossy materials contribute to improve P1 and P2 resulting from a complicated balance between imaginary part of refractive index and resonance induced absorptivity. For cavity mode P1 and hybrid mode P2, the absorptivity is improved by increasing absorption area of MDM cavity, resulting from resonance-enhanced absorption in lossy gap (Si) and embedded layer (Ge), furthermore, the absorptivity of P2 also benefits from coupling of P1. In contrast, only metal parts (Cu) contribute to absorptivity in lossless dielectric EMDM structures, generally, small absorption area is limited to interface between metal and dielectric.

With respect to the choice of materials, the heavily doped semiconductor exhibits increased electric conductivity when compared to an intrinsic semiconductor, and this is equivalent to the increased imaginary part of refractive index (*k*) or absorption loss in optics. In detail, the *k* values of n-type Si and Ge are 0.1~1 at the doping concentration of ~10^{19} cm^{−3} [31,41]. Generally, optical constants (*n* and *k*) of heavily doped semiconductor are frequency dependent, which induces unwanted peak shift with respect to intrinsic materials. In order to circumvent this inconvenience and directly compare with intrinsic case, it is reasonably assumed that doped silicon and germanium with complex refractive indexes corresponding to *n _{Si}* = 3.42 + 0.3i and

*n*= 4 + 0.1i at the wavelength range of 6–15 μm. Doped Si and Ge are employed as gap layers in the MDM cavity and embedded layer, respectively.

_{Ge}Figure 12. (a) shows the optimized absorption band of lossy 3D cube EMDM resonators with *t _{d}* = 0.6 μm,

*t*= 0.55 μm,

*s*= 0.05 μm,

*w*= 0.9 μm, and

_{EMDM}*P*= 3 μm. Near unity absorptivity of 0.98 (9.26 μm) and 0.99 (11.15 μm) is achieved, and the total averaged absorptivity corresponds to 0.95 within the range of 9–11.5 μm. Finally, averaged absorptivity of 0.85 is obtained in the entire atmospheric window (8–13 μm), and this is significantly larger than that in the lossless dielectric case. Hence, it is feasible to improve the performance of radiative cooling devices by utilizing lossy dielectric-based periodic single-sized EMDM resonators. Additionally, a lossy dielectric maintains lower absorptivity (around 0.1) in the non-resonant frequency due to the relatively low value of

_{EMDM}*k*. This also indicates that lossy dielectric-based EMDM resonators preserve an excellent selectivity of the wavelength. Furthermore, it is possible to achieve polarization independence in cubic periodic EMDM resonators due to rotation symmetry.

Figure 12(b) shows angle resolved absorption map of 3D lossy dielectric based EMDM resonators, evidently, the strong absorption broadband overlaps the target atmospheric window in wide angle range. P2- is indistinguishable due to lossy dielectric induced strong absorption between P1 and P2 + . The excellent performance of EMDM resonators are preserved even in high incident angle, *e.g.* average absorptivity as large as 0.82 can be achieved, which preserves almost same value of normal incidence (0.85). The results indicate lossy dielectric based EMDM resonators remarkably improve absorptive performance even in wide angle range. Interestingly, the proposed strategy and EMDM resonators are not only limited in application of radiative cooling, but also light-trapping of thin film solar cell [42], plasmonic color palettes [7] and broadband absorbers at other wavelengths [14].

## 5. Conclusion

In this study, a novel structure noted EMDM structure that is appropriate for ultrabroadband absorption is proposed and systematically analyzed. With respect to an individual resonator, the EMDM resonator exhibits a 1.9 times broader FWHM relative to that of the MDM resonator, and this results from the match between MDM mode and DLMW mode and enhanced scattering intensity. Periodic single-sized resonators have an FWHM that is 3.8 times broader than that of the MDM resonator. In addition, a strong coupling effect is confirmed between localized MDM and hybrid modes. The existence of a hybrid mode enables EMDM resonators to significantly expand the absorption band. In the application of a periodic 3D EMDM resonator an average absorptivity of 0.85 could be achieved in the entire atmospheric window (8–13 μm) by lossy-dielectric single-sized ultra-thin EMDM resonators. The pertinent wide angular responses are also confirmed. The results suggest that these types of periodic 3D EMDM structures are extremely promising candidates for radiative cooling devices and other ultrabroadband absorbers.

## Acknowledgments

The authors thank Dr. Masashi Miyata and Mr. Hideaki Hatada for their assistance with the software. Tianji Liu is supported by the Japanese government's scholarship (Monbukagakusho: MEXT) for research students.

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