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Selective broadband absorption by mode splitting for radiative cooling

Kishin Matsumori, Ryushi Fujimura, and Markus Retsch

Open Access Open Access

Abstract

A plasmonic-photonic structure based on colloidal lithography was designed for a scalable radiative cooling system and its absorption properties were theoretically investigated. The structure comprises a SiO2 core, which is on top of an Au reflector and partially covered by an indium tin oxide (ITO) shell. This simple and scalable structure possesses a strong selective absorption in the primary atmospheric transparency window (8–13 µm). The strong selective absorption is attributed to a mode splitting of the localized surface plasmon (LSP) of the ITO shell. To understand the mechanisms of the mode splitting, a quantitative analysis was conducted using a coupled-oscillator model and a coupled-dipole method. The analysis revealed that the mode splitting is induced by a strong coupling between the LSP of the ITO shell and a magnetic dipole Mie resonance of the SiO2 core.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Daytime radiative cooling is an optics-based technology that can decrease the temperature of an object below the ambient temperature without requiring any energy input [1]. This radiative cooling technology works by completely reflecting/scattering incoming sunlight to minimize heat absorption and emitting thermal radiation in the mid-infrared (MIR) range. In the MIR range, there are two atmospheric transparency windows (ATW); the primary ATW (8–13 µm) and the secondary ATW (16–28 µm). Therefore, the optical properties of the involved materials should have a strong emissivity in those wavelength ranges for a considerable cooling performance. There are two indicators to characterize the cooling performance of such a system [1,2]. One is cooling power, which is the net radiative heat ejection at a certain temperature of the system. A high cooling power can be achieved, if the materials demonstrated a black body emissivity in the MIR range [3–6]. Another indicator is the temperature reduction, which is determined by the difference between the ambient temperature and the temperature of the system at an equilibrium state. At this equilibrium state the energy of radiative heat loss of the system equals the heat load by external sources, such as the thermal radiation of the atmosphere. For a higher temperature reduction, more precise optical engineering is required since the system should have strong selective emissivity only in the primary ATW.

Optical tailoring for radiative cooling can be realized by plasmonic absorbers, which have been extensively investigated because of their variety of energy applications, such as energy harvesting [7] and solar thermophotovoltaics [8–10]. One of the most representative plasmonic absorbers is a metal-insulator-metal (MIM) structure whose top metal layer is structured [11–16]. A strong absorption is induced by localized surface plasmons (LSPs) of the top structure, therefore, a selective broadband absorption can be obtained by mixing different size and shape of structures [8–10,17,18]. However, if a diversity of the structure is increased for a wider absorption bandwidth, a density of each structure decreases, resulting in reduced absorption at all wavelengths [9,17,18]. An alternative approach is designing a multilayered structure [19–21]. A nearly perfect absorption can be obtained in the primary ATW because a single structure has a selective broadband absorption [21]. Broadband absorbers based on 2D photonics structures have also been designed for radiate cooling, and they showed a selective absorption only in the ATWs by phonon-polariton resonances [22–24]. Those plasmonic and photonic structures have almost ideal optical properties for radiative cooling, however, their fabrication heavily relies on top-down processes. Therefore, they are expensive and mostly not suited to produce a large-area radiative cooling system.

Difficulties in the fabrication for a scalable device can be overcome by different approaches. One is by mixing plasmonic or photonic particles in a polymer matrix [2,25–27]. This composite film has a selective absorption in the ATWs by either broadband LSP or superposition of different order modes of Mie resonances, depending on which type of particle was used. Another approach is synthesizing polymer films so that they have intrinsic selective absorption by their molecular vibrations [28,29]. Those two approaches do not require any complicated structures and all fabrication processes are based on bottom-up techniques, paving the way to a scalable radiative cooling system. However, these systems typically have an additional strong absorption outside or a significant absorption dip inside the primary ATW.

For a finer control of the absorption bandwidth, one of the best strategies is still designing a plasmonic or photonic structure such that a single structural motif exhibits the required strong absorption in a specific wavelength range, like multilayered structures do [21]. Therefore, in this contribution, we outline a way to satisfy a high control of absorption bandwidth and a scalable sample processing at the same time. A semishell structure [30–33] composed of a dielectric core and a metal shell is known as a plasmonic nanostructure, which can be fabricated by large-area processes using colloidal lithography [34] and metal deposition. A semishell structure has been utilized for broadband absorbers. MIM structures whose top is composed of a semishell structure possess a broadband absorption in the visible to near-infrared regions [35,36]. A close-packed monolayer of a semishell structure combined with a metal reflector strongly absorbs radiation in a wavelength range from 3 to 6 µm [37]. These structures demonstrated a broadband absorption; however, their absorption bandwidth is not selective.

To realize a selective absorption by a semishell-based absorber, a structure has to be designed by a different approach. In the wavelength regions shorter than the primary ATW, many dielectrics have nearly constant low refractive index. In this case, the optical properties of the core are not important, and the core can be used just as a support of the metal shell. However, in the primary ATW, some dielectrics have a high dispersion, which enable the core to support both electric and magnetic Mie resonances. When the LSP of the metal shell interacts with the Mie resonance of the dielectric core at their resonance condition, the interaction may cause a mode splitting that can be seen in strong coupling, such as electromagnetic induced transparency [38–42] and Rabi splitting [43–46]. This mode splitting has the potential to give additional control on the absorption bandwidth and enable us to realize a selective absorption in the primary ATW. Based on these ideas, we discuss our theoretical investigation to design a plasmonic-photonic semishell structure combined with an Au reflector for radiative cooling applications. Indium tin oxide (ITO) and SiO2 were chosen for the shell and core, respectively. ITO is known as an alternative plasmonic material, which works in the IR range [47,48]. SiO2 has a high dispersion in the primary ATW, which guarantees the excitement of Mie resonances [26]. We discuss the absorption properties of our ITO@SiO2 semishell absorber based on Mie theory and the finite element method (FEM). To understand the absorption mechanisms in detail, we provide a quantitative analysis based on a coupled-oscillator (CO) model and the coupled-dipole method (CDM).

2. Optical properties of the SiO2 core

We first investigated the optical properties of the SiO2 core using Mie theory to determine a suitable size range for the ITO@SiO2 semishell absorber. The extinction (Cext), scattering (Csca), and absorption (Cabs) cross-sections can be calculated as [49];

$$\begin{aligned} {C_{\textrm{ext}}} &= \frac{{2\pi }}{{{k^2}}}({2n + 1} )\textrm{Re}[{{a_n} + {b_n}} ]\\ {C_{\textrm{sca}}} &= \frac{{2\pi }}{{{k^2}}}({2n + 1} )({{{|{{a_n}} |}^2} + {{|{{b_n}} |}^2}})\\ {C_{\textrm{abs}}} &= {C_{\textrm{ext}}} - {C_{\textrm{sca}}}\end{aligned}$$
where k is the angular wavenumber, an and bn are Mie coefficients for n-th order of electric and magnetic modes, respectively. n = 1 and 2 correspond to dipole and quadrupole resonance, respectively. Extinction (Qext), scattering (Qsca), and absorption (Qabs) efficiency are cross-sections normalized by the geometrical cross-section of the core.

Figure 1 shows the absorption efficiency of the electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ) for the radius of 1 µm, 1.5 µm, and 2 µm. The yellow shaded area indicates the Reststrahlen band of SiO2, which is the wavelength range of 8–9.3 µm [50]. In this band, the SiO2 core supports a localized surface phonon polariton (LSPhP) [51]. ED and EQ for smaller radius have an absorption peak at a wavelength that satisfies the Fröhlich condition. Their absorption bandwidth is extremely narrow because of low loss of LSPhP. With increase in radius, both ED and EQ redshift and become broader because of an increase in radiative damping (see Fig. S1(A)-(C), Supplement 1) and an optical resonance induced outside of the Reststrahlen band. Magnetic modes cannot be excited inside the Reststrahlen band, therefore, the absorption peaks of MD and MQ are always outside of this band. Magnetic modes are excited when the dimension of the core is comparable to a wavelength inside the core so that magnetic modes redshift with increasing radius [52]. All modes of the Mie resonance have an absorption at a wavelength of around 12.5 µm, which is attributed to an intrinsic absorption of the transverse optical (TO) mode vibration of the Si-O bond [50]. The absorption at this wavelength increases with an increasing radius because the Mie resonances redshift and enhance electromagnetic fields inside the SiO2 core. Further increase of the radius makes quadrupole modes stronger and enables the core to support higher order modes. This results in a wider absorption bandwidth (see Fig. S1(D)-(F), Supplement 1) and should be avoided for a selective absorption in the primary ATW. In addition, higher order modes cause additional interactions with LSP of the ITO shell, which might degrade a selective absorption of the ITO@SiO2 semishell absorber. Therefore, the core size should be small enough such that dipole modes dominate the total absorption of the SiO2 core.

 figure: Fig. 1.

Fig. 1. Absorption efficiency of the SiO2 sphere with a radius of (A) 1 µm, (B) 1.5 µm, and (C) 2 µm. The red and blue solid curves are the electric and magnetic dipole modes, respectively. The red and blue dashed curves are the electric and magnetic quadrupole modes, respectively.

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In the Reststrahlen band, the SiO2 core acts like a metallic particle, meaning that the LSP of the inner surface of the ITO shell cannot be induced. Therefore, the LSP can interact only with the magnetic modes of the SiO2 core. The SiO2 core with 1–2 µm radius has its MD at around 10 µm, close to the middle of the primary ATW. This guarantees that an interaction between the LSP and MD inside the primary ATW. From these results, the proper size of the SiO2 sphere is estimated to be 1–2 µm in radius.

3. ITO@SiO2 semishell absorber

3.1 Design of the ITO@SiO2 semishell absorber and FEM simulation

The simulation model of the ITO@SiO2 semishell absorber is shown in Fig. 2(A). The ITO shell with thickness tshell is on top of a SiO2 core with radius rcore. The SiO2 core is immobilized on a thin layer of Au. The Au layer not only reflects incoming radiation but also increases the absorption of the ITO@SiO2 semishell structure. The ITO shell can be produced by ITO deposition on an SiO2 nanoparticle adsorbed on an Au surface based on colloidal lithography. Therefore, we also take a perforated ITO film with thickness of tshell on top of the Au reflector into account. The LSP of the perforated ITO film is not induced because of the electric conduction between the ITO film and the Au reflector. The thickness of the Au reflector is fixed as 200 nm, which is thick enough compared to the skin depth in the MIR range. The unit cell of the model has the periodicity Pcell.

 figure: Fig. 2.

Fig. 2. (A) A schematic illustration of the ITO@SiO2 semishell structure. (B) rcore and (C) tshell dependence of the absorptivity of the ITO@SiO2 semishell structure. The blue shaded area indicates the primary ATW. (D) Transmittance of the atmosphere [61]. The blue and orange shaded areas indicate the actual primary ATW (8–9.3 µm and 10–13 µm) and the Ozone absorption band (9.3–10 µm), respectively. (E-F) Average absorption for Fig. 2(B) and 2(C). The black, blue, and red lines are average absorption in the primary ATW, the actual primary ATW, and the Ozone absorption band, respectively.

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We investigate the optical properties of the ITO@SiO2 semishell absorber using COMSOL multiphysics, which is a commercial software package based on the finite element method (FEM). It is known that the exact optical properties of ITO depend on the deposition method [53–55]. For our investigation, the refractive index of ITO was obtained using Drudés model with parameters in Ref. [53]. Refractive indices of SiO2 and Au were obtained from Ref. [56,57] and Ref. [58], respectively. The surrounding of the structure is air. According to Kirchhoff´s law, emissivity is equal to absorptivity, therefore, absorptivity of the structure was calculated to evaluate the thermal emission properties. The electromagnetic plane wave propagates along – z direction with x polarization. The amplitude of the electric field of the incident wave is E0 = 1 V/m. The Bloch-Floquet periodic boundary conditions were applied in x and y direction. Perfectly matched layers are applied on the top and bottom of the simulation model (in z-axis). The absorptivity A(λ) was calculated by taking a volume integral of dissipated energy density over the structure and dividing the dissipated energy by the energy of the incident wave per area of the unit cell. This method gives the same result as another method, which calculates absorptivity by obtaining reflectance and transmittance of the structure (see Fig. S3(A), Supplement 1). For the evaluation, the average absorption in a wavelength range from λ1 to λ2 was calculated by ${A_{\textrm{ave}}} = \frac{1}{{{\lambda _2} - {\lambda _1}}}\mathop \smallint \nolimits_{{\lambda _1}}^{{\lambda _2}} A(\lambda )\textrm{d}\lambda $.

3.2 Absorption properties of the ITO@SiO2 semishell absorber

We, next, turn our focus to the core-semishell structure and outline the relevance of rcore and tshell on the absorptivity. For all calculations, Pcell is fixed as 7 µm. This periodicity was determined based on two considerations: 1) The distance between the semishell structures needs to be large enough to weaken near-field interactions, which would lead to a broadening of the absorption bandwidth [11]. 2) However, Pcell should not be close to and exceed 8 µm (the shortest wavelength of the primary ATW). This is because the lattice resonance originated from the periodicity can interact with the LSP and the Mie resonances [59,60], and the interaction degrades the absorption properties of the ITO@SiO2 semishell absorber. Therefore, we selected 7 µm as the optimum Pcell even though the near-field interactions between the semishell structures are not negligible with this periodicity (see Fig. S2(A), Supplement 1). For rcore dependence (see Fig. 2(B)), tshell was fixed at 200 nm. The absorption bandwidth increases with an increase in rcore because larger SiO2 core can support higher order modes of Mie resonance. The bandwidth is also broadened by the decrease of face-to-face distance between semishells, resulting in stronger lateral near-field interactions. In addition, the absorption of the semishell redshifts with increasing rcore so that the absorption increases especially in the longer wavelength range. For tshell dependence (see Fig. 2(C)), rcore was fixed at 1.4 µm. tshell changes the spectral shape inside the primary ATW, but the absorption bandwidth does not change a lot for different values of tshell. It is known that the absorption peak of the semishell can be controlled in a wider wavelength range by changing the radius, and in a narrower wavelength range by changing the thickness of the metal shell. Therefore, from rcore and tshell dependences, we can find that rcore can be used to adjust the absorption bandwidth, and further optimizations for the selective absorption can be made by tuning tshell.

The average absorption was calculated for the spectra shown in Fig. 2(B) and (C) at different wavelength ranges; the primary ATW (8–13 µm), Ozone absorption band (9.3–10 µm) [24], and the actual primary ATW (8–9.3 µm and 10–13 µm) (see Fig. 2(D)). Figure 2(E) shows the average absorption for rcore dependence. The absorption is maximized for all wavelength ranges when rcore is around 1.4–1.6µm. The averaged absorption for rcore =1.4 µm and 1.6 µm in the primary ATW is 87% and 88%, respectively. This result shows that rcore = 1.6 µm has the highest average absorption, however, the full-width at half-maximum (FWHM) for rcore = 1.4 µm and 1.6 µm is about 7.7 µm and 9.4 µm, respectively. This indicates that the structure with rcore = 1.4 µm has a better selective absorption. Therefore, the best rcore is at about 1.4 µm. For tshell dependence (see Fig. 2(F)), FWHM is about 7.7 µm for all tshell. The average absorption in the primary ATW increases with increase in tshell, however, this doesńt mean that a thicker tshell has better absorption properties. With increase in tshell, the absorption in the Ozone absorption range increases, which decreases the cooling power. In the actual primary ATW, the averaged absorption is maximized when tshell is about 250 nm, and the value is 89%. When we check for tshell = 200 nm, the average absorption is 88% in the actual primary ATW, which is 1% lower than that for tshell = 250 nm. However, tshell = 200 nm can have a lower absorption in the Ozone absorption range. Also, the ITO film has an intrinsic absorption and thinner ITO film absorbs less electromagnetic waves outside of the primary ATW (see Fig. S2(G), Supplement 1). Therefore, by considering all of those aspects, we decided on 200 nm as the best shell thickness.

To understand the absorption properties of the ITO@SiO2 semishell absorber, the absorption of each component (ITO shell, SiO2 core, perforated ITO film, Au reflector) was calculated for the structure with rcore = 1.4 µm and tshell = 200 nm (see Fig. 3(A)). We find that the total absorption is dominated by the ITO shell and the SiO2 core absorption. The absorption of the ITO shell has three peaks (at 7.78 µm, 11.27 µm, and 13.5 µm) which are separated from each other. The absorption of the SiO2 core has three peaks; two of them are close to each other (8.93 µm and 9.67 µm) and one of them (12.4 µm) is apart from those two peaks. A sharp peak at 7 µm is attributed to the lattice resonance. In Fig. 3(B), the absorption spectra of the structure with only SiO2 core or only ITO shell are shown. Comparing to Fig. 3(A), those spectra are narrower and weaker than the total absorption of the ITO@SiO2 semishell absorber. Therefore, it is found that a selective broadband absorption cannot be obtained from either only the ITO shell or from only the SiO2 core. The SiO2 core with and without the ITO shell has a similar absorption. However, the ITO shell has multiple peaks when it is combined with the SiO2 core. The absorption of the ITO shell is split into two peaks and has a dip at the center of the absorption spectrum of the SiO2 core. This indicates that the interaction between the LSP of the ITO shell and the Mie resonance of the SiO2 core induce a mode splitting in the LSP. When we compare the absorption of the ITO@SiO2 semishell absorber and the absorption of the SiO2 core calculated using Mie theory, we find that the absorption dip of the ITO shell appears at around the absorption peak of the MD of the SiO2 core (see Fig. S3(B), Supplement 1). This suggests that the mode splitting occurs because the LSP of the ITO shell interacts with the MD of the SiO2 core. To support this statement further, we checked the absorption spectra in Fig. 2(B) in detail. With increase in the core size, the absorption dip of the ITO shell redshifts. This may be because the MD of the SiO2 core redshifts (see Fig. S2(B)-(D), Supplement 1). Even if the shell thickness changes, the dip does not shift because the MD does not shift (see Fig. S2(E)-(F), Supplement 1). The ITO shell has one more peak at 13.5 µm. This is also attributed to the mode splitting but not to the interaction between LSP and Mie resonance. This is because of the interaction between the LSP and the TO mode of SiO2 [62].

 figure: Fig. 3.

Fig. 3. (A) Absorption spectra of each component of the ITO@SiO2 semishell structure. The red, blue, green, and orange curves are for the ITO shell, SiO2 core, perforated ITO film, and Au reflector, respectively. The black curve is the total absorption of the structure. (B) Color maps of electric and magnetic field distribution at wavelength indicated by arrows in Fig. 3(A). Absolute value of fields is normalized by amplitude of incident wave. The white arrows are electric field vectors.

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The absorption mechanism of the proposed structure was investigated further by focusing on the four highest absorption peaks indicated by arrows in Fig. 3(A). For these resonances, we show the electric and magnetic field distribution color maps (see Fig. 3(C)). Color maps were taken in the x-z plane at the center of the semishell. White arrows in electric maps represent electric field vectors. At 7.78 µm and 11.27 µm, there are electric field hot spots at the edge of the ITO shell and the magnetic field is enhanced inside the shell. This is a characteristic field enhancement of the semishell [32,33]. The LSP interacts with an image pole created in the Au reflector, which enhances the electromagnetic field confinement [12]. At 8.93 µm, where the SiO2 core has a sharp absorption peak, the electric field is not enhanced at the edge of the ITO shell, but it is strongly confined at the bottom of the SiO2 core. The magnetic field is also confined there. Mie theory shows that the SiO2 core has an ED and EQ at 8.93 µm (Fig. S3(B), Supplement 1), therefore, the absorption peak at 8.93 µm is attributed to the mixture of ED and EQ. Mie theory also shows that the second peak of the SiO2 core at 9.67 µm is attributed to the mixture of ED and MD. Since the second peak is closer to the absorption peak of the MD than that of ED, the contribution of the MD is larger (Fig. S3(B), Supplement 1). At this wavelength, the magnetic field is strongly enhanced inside the SiO2 core. There is a rotation of the electric field, and the magnetic field is enhanced at the center of the rotation by the MD of the SiO2 core. The MD also interacts with an image pole in the Au reflector, and the interaction widens the absorption bandwidth of the MD [63,64]. The electric field at the edge of the ITO shell becomes weaker because of the interaction between the LSP of the ITO shell and the MD of the SiO2 core. Therefore, the absorption dip appears in the absorption of the ITO shell at around 9.67 µm. One might think that an interaction between the LSP of the ITO shell and the ED of the SiO2 core also contributes to the mode splitting. However, as mentioned earlier, the absorption of the ED is inside the Reststrahlen band of SiO2, meaning that the LSP and the ED cannot interact with each other. Therefore, we can consider that the mode splitting is attributed to the interaction between the LSP of the shell and the MD of the core.

4. Mode splitting mechanism

4.1 Coupled-oscillator model

We rationalize the mode splitting phenomena using a coupled-oscillator (CO) model, which is frequently used for a phenomenological analysis of plasmonic and photonic systems [39–41]. First, the CO model is built to describe the interaction between the LSP of ITO shell and the MD of the SiO2 core (see Fig. 4(A)). The LSP is considered as the bright oscillator with a higher damping rate and the MD is considered as the dark oscillator with a lower damping rate since the absorption bandwidth of the ITO shell is much wider than that of the MD. The equations of motion can be written as

$$\begin{aligned} &\frac{{{\textrm{d}^2}{x_{\mathrm{\alpha }1}}}}{{\textrm{d}{t^2}}} + {\gamma _1}\frac{{\textrm{d}{x_{\mathrm{\alpha }1}}}}{{\textrm{d}t}} + \omega _1^2{x_{\mathrm{\alpha }1}} - \kappa _{12}^2{x_{\mathrm{\alpha }2}} = {F_1}(t )\\ &\frac{{{\textrm{d}^2}{x_{\mathrm{\alpha }2}}}}{{\textrm{d}{t^2}}} + {\gamma _2}\frac{{\textrm{d}{x_{\mathrm{\alpha }2}}}}{{\textrm{d}t}} + \omega _2^2{x_{\mathrm{\alpha }2}} - \kappa _{12}^2{x_{\mathrm{\alpha }1}} = {F_2}(t ) \end{aligned}$$
where xn is the displacement of mass objects, γn is the damping rate, ωn is the resonance angular frequency determined by the spring constant, and κ1n is the coupling strength. Fn(t) is the external force working on the oscillators. Fn(t) is time harmonic and expressed as ${F_\textrm{n}}(t )= {F_\textrm{n}}\textrm{exp} ({ - i\omega t} )$. Therefore, displacements are also time harmonic and can be expressed as ${x_\textrm{n}} = {c_\textrm{n}}\textrm{exp} ({ - i\omega t} )$. From Eq. (1), xn is written as
$$\begin{aligned} {x_{\mathrm{\alpha }1}} &= \frac{{{\varOmega _2}{F_1} + \kappa _{12}^2{F_2}}}{{{\varOmega _1}{\varOmega _2} - \kappa _{12}^4}}\textrm{exp} ({ - i\omega t} )= {c_{\mathrm{\alpha }1}}\textrm{exp} ({ - i\omega t} )\\ {x_{\mathrm{\alpha }2}} &= \frac{{{\mathrm{\varOmega }_1}{F_2} + \kappa _{12}^2{F_1}}}{{{\varOmega _1}{\varOmega _2} - \kappa _{12}^4}}\textrm{exp} ({ - i\omega t} )= {c_{\mathrm{\alpha }2}}\textrm{exp} ({ - i\omega t} ) \end{aligned}$$
where ${\varOmega _\textrm{n}} = \omega _\textrm{n}^2 - {\omega ^2} - i{\gamma _\textrm{n}}\omega $. Using Eq. (2), the time averaged dissipated power of the whole system is
$$P\left( \omega \right) = \left\langle {\textrm{Re}\left[ {F_1^\textrm{*}\left( t \right) \times \frac{{\textrm{d}{x_1}}}{{\textrm{d}t}}} \right]} \right\rangle + \left\langle {\textrm{Re}\left[ {F_2^\textrm{*}\left( t \right) \times \frac{{\textrm{d}{x_2}}}{{\textrm{d}t}}} \right]} \right\rangle$$

Equation (3) can be divided into two terms using the absolute value of the amplitude of each oscillator (see Eq. (S4), Supplement 1)

$${P_\mathrm{\alpha }}(\omega )= \frac{1}{2}{\gamma _1}{\omega ^2}{|{{c_{\mathrm{\alpha }1}}} |^2} + \frac{1}{2}{\gamma _2}{\omega ^2}{|{{c_{\mathrm{\alpha }2}}} |^2} = {A_{\mathrm{\alpha }1}}(\omega )+ {A_{\mathrm{\alpha }2}}(\omega )$$

 figure: Fig. 4.

Fig. 4. (A) Two-oscillator CO model describing the interaction between the LSP of the ITO shell and the MD of the SiO2 core. (B) Three-oscillator CO model where one additional oscillator is added to the two-oscillator CO model to account for the interaction between the LSP of the ITO shell and the TO mode of SiO2. (C) The CO model is fitted to the absorption of the ITO shell. The red and blue solid curves are absorption spectra of the ITO shell and the SiO2 core in Fig. 3(A), respectively. The black dashed curve is absorption calculated from Eq. (4) and the red, blue, green dashed curves are absorption calculated from Eq. (7).

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In Eq. (4), the first term corresponds to an absorption of the bright oscillator (the LSP of the ITO shell) and the second term is an absorption of the dark oscillator (the MD of the SiO2 core). In addition, the interaction between the LSP and the TO mode of SiO2 induces the mode splitting at around 12 µm, which has to be considered as well to describe the absorption spectrum of the ITO shell. In this case, one more dark oscillator should be added (see Fig. 4(B)). The external force does not work on the third oscillator because the TO mode weakly couples to the incident field. The equations of motion for the three-oscillator CO model are

$$\begin{aligned} \frac{{{\textrm{d}^2}{x_{\mathrm{\beta }1}}}}{{\textrm{d}{t^2}}} + {\gamma _1}\frac{{d{x_{\mathrm{\beta }1}}}}{{dt}} + \omega _1^2{x_{\mathrm{\beta }1}} - \kappa _{12}^2{x_{\mathrm{\beta }2}} - \kappa _{13}^2{x_{\mathrm{\beta }3}} = {F_1}(t )\\ \frac{{{d^2}{x_{\mathrm{\beta }2}}}}{{d{t^2}}} + {\gamma _2}\frac{{d{x_{\mathrm{\beta }2}}}}{{dt}} + \omega _2^2{x_{\mathrm{\beta }2}} - \kappa _{12}^2{x_{\mathrm{\beta }1}} = {F_2}(t )\\ \frac{{{d^2}{x_{\mathrm{\beta }3}}}}{{d{t^2}}} + {\gamma _3}\frac{{d{x_{\mathrm{\beta }3}}}}{{dt}} + \omega _3^2{x_{\mathrm{\beta }3}} - \kappa _{13}^2{x_{\mathrm{\beta }1}} = 0 \end{aligned}$$

The oscillation state of each oscillator in Eq. (5) can be expressed by the following equations;

$$\begin{aligned} {x_{\mathrm{\beta }1}} &= \frac{{{\varOmega _2}{\varOmega _3}{F_1} + \kappa _{12}^2{\mathrm{\varOmega }_3}{F_2}}}{{{\varOmega _1}{\varOmega _2}{\varOmega _3} - \kappa _{12}^4{\varOmega _3} - \kappa _{13}^4{\varOmega _2}}}\textrm{exp} ({ - i\omega t} )= {c_{\mathrm{\beta }1}}\textrm{exp} ({ - i\omega t} )\\ {x_{\mathrm{\beta }2}} &= \frac{{({{\varOmega _1}{\varOmega _3} - \kappa_{13}^4} ){F_2} + \kappa _{12}^2{\varOmega _3}{F_1}}}{{{\varOmega _1}{\varOmega _2}{\varOmega _3} - \kappa _{12}^4{\varOmega _3} - \kappa _{13}^4{\varOmega _2}}}\textrm{exp} ({ - i\omega t} )= {c_{\mathrm{\beta }2}}\textrm{exp} ({ - i\omega t} )\\ {x_{\mathrm{\beta }3}} &= \frac{{\kappa _{13}^2({{\varOmega _2}{F_1} + \kappa_{12}^2{F_2}} )}}{{{\varOmega _1}{\varOmega _2}{\varOmega _3} - \kappa _{12}^4{\varOmega _3} - \kappa _{13}^4{\varOmega _2}}}\textrm{exp} ({ - i\omega t} )= {c_{\mathrm{\beta }3}}\textrm{exp} ({ - i\omega t} ) \end{aligned}$$

From Eq. (3) and (6), the total dissipated power of the system illustrated in Fig. 4(B) is

$$\begin{aligned} {P_\mathrm{\beta }}(\omega )&= \frac{1}{2}{\gamma _1}{\omega ^2}{|{{c_{\mathrm{\beta }1}}} |^2} + \frac{1}{2}{\gamma _2}{\omega ^2}{|{{c_{\mathrm{\beta }2}}} |^2} + \frac{1}{2}{\gamma _3}{\omega ^2}{|{{c_{\mathrm{\beta }3}}} |^2}\\ &= {A_{\mathrm{\beta }1}}(\omega )+ {A_{\mathrm{\beta }2}}(\omega )+ {A_{\mathrm{\beta }3}}(\omega ) \end{aligned}$$
where the third term describes the absorption of the TO mode of SiO2. First, the dissipated power of a single harmonic oscillator model was fitted to the absorption of the ITO shell in Fig. 3(B) to extract parameters for the bright oscillator (see Fig. S4(A), Supplement 1). From the fitting, the resonance angular frequency and the damping rate are ω1 = 2.15 × 1014 rad/s and γ1 = 1.5 × 1014 rad/s. Next, Aβ1(ω) in Eq. (7) was fitted to the absorption of the ITO shell in Fig. 3(A). The fitting resulted in ω2 = 1.93 × 1014 rad/s, γ2 = 1.4 × 1013 rad/s, κ12 = 1.16 × 1014 rad/s for the MD and ω3 = 1.51 × 1014 rad/s, γ3 = 1.7 × 1013 rad/s, κ13 = 6.6 × 1013 rad/s for the TO mode. In Fig. 4(C), Aβ1(ω), Aβ2(ω) and Aβ3(ω) are shown. Even if the interactions occurring in the ITO@SiO2 semishell absorber are complex, Aβ1(ω) shows good agreement with the absorption spectrum of the ITO shell. The coupling state can be categorized roughly into a strong or weak coupling regime by comparing the total damping rate of the system (γ1γn)1/2 and the coupling strength κ1n [44]. In general, the interaction is considered as a strong coupling when κ1n >> (γ1γn)1/2 because the energy transfer rate between two oscillators can be quicker than their energy dissipation rates. The total damping rate of the LSP-MD is (γ1γ2)1/2 = 4.58 × 1013 rad/s, and that of the LSP-TO mode is (γ1γ3)1/2 = 5.05 × 1013 rad/s. For those interactions, the coupling strengths are larger than the total damping rates. However, the values obtained in our case are comparable, which may be considered as an intermediately strong coupling. In this coupling regime, absorption enhancement takes place in the dark oscillators because the transferred energy is concentrated into the dark oscillators [42–44]. This absorption enhancement can be observed by comparing the absorption of the SiO2 core with and without the ITO shell at around 10–11 µm (see Fig. S4(B), Supplement 1).

It is obvious that the mode splitting by the interaction between the LSP and the MD improves absorption in the primary ATW. To understand how the interaction between the LSP and the TO mode contribute to the absorption of the ITO@SiO2 semishell absorber, Aβ1(ω) is compared with Aα1(ω) in Eq. (4), which is calculated using ω2, γ2, and κ12 from the fitting. Aβ1(ω) has a similar spectral shape to Aα1(ω) in the shorter wavelength range. In the longer wavelength range, Aβ1(ω) is weaker inside and is larger outside of the primary ATW than Aα1(ω). This comparison indicates that the mode splitting by the TO mode weakens the cooling performance, which cannot be eliminated because the TO mode is the intrinsic Si-O bond vibration of SiO2. Overall, an undesired broadening of the absorption occurs outside of the primary ATW by the TO mode, however, absorption inside the primary ATW is improved significantly by the interaction between the LSP and the MD. The absorption of the MD is not strong enough to compensate the absorption dip of the ITO shell. However, there are strong absorptions of ED and EQ of SiO2 core in between two absorption peaks of the ITO shell, which results in a strong broadband absorption in the primary ATW.

4.2 Coupled-dipole method

The mechanisms of the interaction between LSP and a molecular vibration have been extensively investigated not only theoretically but also experimentally, and it is known that the interaction can be described by the CO model [41]. To gain a deeper insight into the mechanism of the LSP-MD interaction, further quantitative analysis was employed using the coupled-dipole method (CDM). The CDM is one of the methods that can describe interactions between electric and magnetic dipoles. In our case, the electric dipole is LSP. Later, this CDM will be compared with the CO model shown in Fig. 4(A). General treatments of the CDM can be found elsewhere [60,63,65]. Here, we treat only one ED and one MD [66]. The schematic of the CDM model for the ED of the ITO shell and the MD of the SiO2 core is shown in Fig. 5(A). The distance between two dipoles is D. The positions of each dipole are r1 = (0, 0, -D/2) for ED and r2 = (0, 0, D/2) for MD. The incident wave propagates along + z direction with x polarization. The electric and magnetic fields are expressed as ${{\textbf E}_{\textrm{in}}} = {E_0}{\hat{{\textbf e}}_\textrm{x}}\textrm{exp} ({\textrm{i}{\textbf k} \cdot {\textbf r}} )$ and ${{\textbf H}_{\textrm{in}}} = {H_0}{\hat{{\textbf e}}_\textrm{y}}\textrm{exp} ({\textrm{i}{\textbf k} \cdot {\textbf r}} )$, where $\hat{{\textbf e}}$ is the unit vector in Cartesian coordinate system and k is the wavevector. Here, the dipole moments are expressed as

$${{\textbf p}_1} = {\varepsilon _0}{\alpha _1}{{\textbf E}_{\textrm{tot}}}({{{\textbf r}_1}} )$$
$${{\textbf m}_2} = {\chi _2}{{\textbf H}_{\textrm{tot}}}({{{\textbf r}_2}} )$$
where ɛ0 is the vacuum permittivity, and α1 and χ2 are the electric polarizability of LSP and the magnetic polarizability of MD, respectively. Etot and Htot are total electric and magnetic fields, which are the sum of the incident field and scattered field from other dipoles. The scattered electric field at r1 created by m2 and the scattered magnetic field at r2 created by p1 are
$${{\textbf E}_\textrm{s}}({{{\textbf r}_1}} )={-} {Z_0}{{\textbf G}_\textrm{M}}({{{\textbf r}_1} - {{\textbf r}_2}} )\cdot {{\textbf m}_2}$$
$${{\textbf H}_\textrm{s}}({{{\textbf r}_2}} )= \frac{1}{{{\varepsilon _0}{Z_0}}}{{\textbf G}_\textrm{M}}({{{\textbf r}_2} - {{\textbf r}_1}} )\cdot {{\textbf p}_1}$$
where Z0 is the impedance of free space. GM is the dyad Greeńs function which is written using ${{\textbf r}_1} - {{\textbf r}_2} ={-} D{\hat{{\textbf e}}_\textrm{z}}$, ${{\textbf r}_2} - {{\textbf r}_1} = D{\hat{{\textbf e}}_\textrm{z}}$, and
$${{\textbf G}_\textrm{M}}({{{\textbf r}_1} - {{\textbf r}_2}} )\cdot {\textbf n} ={-} {{\textbf G}_\textrm{M}}({{{\textbf r}_2} - {{\textbf r}_1}} )\cdot {\textbf n} = ({ - {{\hat{{\textbf e}}}_\textrm{z}} \times {\textbf n}} )C(D )$$
where n is an arbitrary vector. C(D) is written as
$$\begin{aligned} C(D )&= \frac{{\textrm{exp} ({ikD} )}}{{4\pi D}}\left( {{k^2} + \frac{{ik}}{D}} \right) = \frac{{\textrm{exp} ({ikD} )}}{{4\pi D}}\sqrt {{k^4} + \frac{{{k^2}}}{{{D^2}}}} \textrm{exp} \left[ {i{{\tan }^{ - 1}}\left( {\frac{1}{{kD}}} \right)} \right]\\ & = i|C |\textrm{exp} \{{i[{kD - {{\tan }^{ - 1}}({kD} )} ]} \}= i|C |\textrm{exp} ({i\phi } ) \end{aligned}$$
where φ is the phase difference. From Eq. (9), Eq. (8) can be written as
$${{\textbf p}_1} = {\varepsilon _0}{\alpha _1}[{{{\textbf E}_{\textrm{in}}}({{{\textbf r}_1}} )+ {{\textbf E}_\textrm{s}}({{{\textbf r}_1}} )} ]= {\varepsilon _0}{\alpha _1}{{\textbf E}_{\textrm{in}}}({{{\textbf r}_1}} )- {\varepsilon _0}{\alpha _1}{Z_0}{{\textbf G}_\textrm{M}}({{{\textbf r}_1} - {{\textbf r}_2}} )\cdot {{\textbf m}_2}$$
$${{\textbf m}_2} = {\chi _2}[{{{\textbf H}_{\textrm{in}}}({{{\textbf r}_2}} )+ {{\textbf H}_\textrm{s}}({{{\textbf r}_2}} )} ]= {\chi _2}{{\textbf H}_{\textrm{in}}}({{{\textbf r}_2}} )+ \frac{1}{{{\varepsilon _0}{Z_0}}}{\; }{\chi _2}{{\textbf G}_\textrm{M}}({{{\textbf r}_2} - {{\textbf r}_1}} )\cdot {{\textbf p}_1}$$

From Eq. (12), a self-consistent form of p1 and m2 are (see Eq. (S5), Supplement 1)

$${{\textbf p}_1} = {p_{1\textrm{x}}}{\hat{{\textbf e}}_\textrm{x}} = {\varepsilon _0}\frac{{{\alpha _1}\textrm{exp} \left( { - i\frac{{kD}}{2}} \right) - {\alpha _1}{\chi _2}C(D )\textrm{exp} \left( {i\frac{{kD}}{2}} \right)}}{{1 + {\alpha _1}{\chi _2}{C^2}(D )}}{E_0}{\hat{{\textbf e}}_\textrm{x}}$$
$${{\textbf m}_2} = {m_{2\textrm{y}}}{\hat{{\textbf e}}_\textrm{y}} = \frac{{{\chi _2}\textrm{exp} \left( {i\frac{{kD}}{2}} \right) + {\alpha _1}{\chi _2}C(D )\textrm{exp} \left( { - i\frac{{kD}}{2}} \right)}}{{1 + {\alpha _1}{\chi _2}{C^2}(D )}}{H_0}{\hat{{\textbf e}}_\textrm{y}}$$
Z0H0 = E0 was used to derive Eq. (13a) and (13b). To understand the excitation state of those dipole moments, α1 and χ2 are considered by approximation using the Lorentzian oscillator
$${\alpha _1} = \frac{{{F_\textrm{p}}}}{{\omega _\textrm{p}^2 - {\omega ^2} - i{\gamma _\textrm{p}}\omega }} = \frac{{{F_\textrm{p}}}}{{{\varOmega _\textrm{p}}}}$$
$${\chi _2} = \frac{{{F_m}}}{{\omega _\textrm{m}^2 - {\omega ^2} - i{\gamma _\textrm{m}}\omega }} = \frac{{{F_\textrm{m}}}}{{{\varOmega _\textrm{m}}}}$$
where Fp and Fm are oscillator strength, ωp and ωm are resonance frequencies, and γp and γm are damping rates for ED and MD, respectively. Note that the size of the ITO shell and the SiO2 core are not in the quasi-static limit (krcore << 1). In this case, radiative damping and retardation effect have to be taken into account in the damping rate of Eq. (14) [67,68]. However, we are only interested in phenomenological analysis, therefore, those effects were ignored. Substituting Eq. (14) into Eq. (13) and using the other form for C(D), we obtain the electric and magnetic dipole moments as
$${p_{1\textrm{x}}} = {\varepsilon _0}\frac{{{\varOmega _\textrm{m}}{F_\textrm{p}}\textrm{exp} \left( { - i\frac{{kD}}{2}} \right) - i{F_\textrm{p}}{F_\textrm{m}}|C |\textrm{exp} ({i\phi } )\textrm{exp} \left( {i\frac{{kD}}{2}} \right)}}{{{\varOmega _\textrm{p}}{\varOmega _\textrm{m}} - {F_\textrm{p}}{F_\textrm{m}}{{|C |}^2}\textrm{exp} ({i2\phi } )}}{E_0}$$
$${m_{2\textrm{y}}} = \frac{{{\varOmega _\textrm{p}}{F_\textrm{m}}\textrm{exp} \left( {i\frac{{kD}}{2}} \right) + i{F_\textrm{p}}{F_\textrm{m}}|C |\textrm{exp} ({i\phi } )\textrm{exp} \left( { - i\frac{{kD}}{2}} \right)}}{{{\varOmega _\textrm{p}}{\varOmega _\textrm{m}} - {F_\textrm{p}}{F_\textrm{m}}{{|C |}^2}\textrm{exp} ({i2\phi } )}}{H_0}$$

It should be mentioned that Eq. (15) from the CDM cannot explain everything about the LSP-MD interaction occurring in the ITO@SiO2 semishell because the ITO shell is touching the SiO2 core. In this situation, their dipoles are extremely close to each other, which the CDM cannot take into account because |C| diverges at D → 0. Therefore, the near-field should be carefully investigated. However, the CDM model is still useful to estimate the excitation states of interacting dipoles. Equation (15a) and (15b) resemble Eq. (2(a)) and (2b) derived from the CO model. Comparing those equations, we find that the coupling strength squared κ122 is proportional to |C|exp(iφ), indicating that there is a phase retardation in the coupling. This phase retardation can be found in the CO model, which describes electromagnetically induced absorption (EIA) of a plasmonic system: a dipolar antenna is vertically stacked over a quadrupolar antenna [69,70]. Reference [69] shows that the interaction can be destructive and show the mode splitting when the phase is 0°. On the other hand, the interaction can be constructive when the phase is 90°, resulting in EIA. In the LSP-MD interaction, D is required to be relatively large to satisfy φ = 90° (Fig. 5(B)). However, the coupling strength is approximately inversely proportional to D so that it is expected to be difficult to excite a strong EIA by the LSP-MD interaction because the coupling becomes weak when φ = 90°. Since our interest is in the mode splitting by the LSP-MD interaction, it is out of scope of this paper to discuss EIA further. However, it is worth to mention that the LSP-MD interaction may be capable of exciting the EIA-like phenomena if the coupling strength could be maintained strong enough at large D.

To see whether the interaction between ED and MD shows mode splitting, the extinction cross-section of the system shown in Fig. 5(A) was calculated following the equation [60,65]

$${C_{\textrm{ext}}} = \frac{k}{{{\varepsilon _0}{{|{{E_0}} |}^2}}}\textrm{Im}[{{\textbf E}_{\textrm{in}}^\ast ({{{\textbf r}_1}} )\cdot {{\textbf p}_1} + {\mu_0}{\textbf H}_{\textrm{in}}^\ast ({{{\textbf r}_2}} )\cdot {{\textbf m}_2}} ]$$
where µ0 is the vacuum permeability and the asterisk denotes the complex conjugate. For the calculation of Eq. (19), Eq. (13a) and (13b) were used for p1 and m2, and Eq. (14a) and (14b) were used for the electric and magnetic polarizabilities in Eq. (13a) and (13b) (see Eq. (S6), Supplement 1). In Fig. 5(C), the extinction spectra calculated using Eq. (16) are shown for different D. When D = 5 µm, there is almost no interaction between the ED and the MD. The sharp extinction peak of the MD of the SiO2 sphere appears on the broad extinction of the ED of the ITO shell because the extinction is the superposition of contributions from ED and MD. With decrease in D, the extinction spectra start splitting and the splitting becomes larger. This is attributed to the fact that the interaction becomes stronger because the coupling strength is approximately inversely proportional to D. From Fig. 5(B), when D is smaller than 1 µm, there is almost no phase difference (less than about 4°). In the case of the ITO@SiO2 semishell, the distance between ED and MD is expected to be smaller than 1µm. Therefore, we can recognize that the interaction is in a strong coupling state without the phase difference, which coincides the CO model shown in Fig. 4(A).

 figure: Fig. 5.

Fig. 5. (A) A schematic illustration of the CDM for the interaction between the LSP of the ITO shell and the MD of the SiO2 core. (B) The phase φ calculated using Eq. (11). (C) Extinction spectra of interacting ED and MD.

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5. Conclusion

The absorption properties of an ITO@SiO2 semishell absorber were investigated using Mie theory and FEM for radiative cooling. We demonstrated that this simple plasmonic-photonic composite structure can possess a selective absorption by optimizing the thickness of the ITO shell and the radius of the SiO2 core. The optimized structure has an average absorption of 87% in the primary ATW. When either the shell or the core is absent in the structure, this selective broadband absorption cannot be obtained. By investigating the absorption of each component of the proposed structure, we found that the absorption of the shell splits into two peaks and the absorption dip of the shell appears at the wavelength where the MD is excited in the SiO2 core. Therefore, we attributed the mode splitting to the interaction between the LSP of the shell and the MD of the core. To understand the mode splitting mechanisms, the interaction was quantitatively investigated using a CO model and the CDM. The quantitative analyses proved that the strong interaction between LSP and MD can cause the mode splitting in the LSP of the shell. In addition to this finding, these quantitative models suggested two following things. Firstly, the absorption of the MD may be enhanced by the LSP-MD interaction because the transferred energy is concentrated on the MD. Secondly, the LSP-MD interaction may be possible to induce an EIA-like phenomena by controlling the distance between the electric and magnetic dipole moments. Further investigation is necessary to completely prove them, however, those fundamental understandings of how LSP interacts with MD provide an approach to design a new type of plasmonic and photonic structures not only for radiative cooling systems but also other applications such as chemical sensors.

For high cooling performance, incident angle insensitivity of the absorption is required. However, in this paper, a periodic structure was considered, and the periodicity is close to the primary ATW. In this case, the lattice resonance redshifts with an increase in the incident angle, which result in a degradation of absorption properties (see Eq. (S7), Supplement 1). This undesired lattice resonance can be eliminated by randomly distributing semishells on the reflector [14,15]. The strong selective absorption is attributed to the absorption properties of a single particle of the semishell. Therefore, it is expected that the absorption properties shown in this paper will be preserved for the structure with random distribution of the semishells [14,15]. The proposed structure can be fabricated by relatively simple processes using colloidal lithography and ITO deposition, and the colloidal lithography can produce random distribution of particles on a large area [16,36]. Therefore, we expect that it is possible to produce a scalable radiative cooling system with the proposed structure.

We must mention a drawback of the proposed structure for daytime radiative cooling. In the visible to near-infrared regions, the proposed structure may have a strong absorption because of the intrinsic ITO film and Au reflector absorption properties (see Eq. (S8), Supplement 1). Therefore, we expect that the radiative cooling ability of the proposed structure is attenuated under sunlight exposure. To overcome this problem, improvements in the proposed structure are required by replacing the ITO film, by using a different material for the reflector, or by combining our structure with a solar reflector/scatter [71,72].

Funding

H2020 European Research Council (714968).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

1. B. Zhao, M. Hu, X. Ao, N. Chen, and G. Pei, “Radiative cooling: A review of fundamentals, materials, applications, and prospects,” Appl. Energy 236, 489–513 (2019). [CrossRef]  

2. Z. Huang and X. Ruan, “Nanoparticle embedded double-layer coating for daytime radiative cooling,” Int. J. Heat Mass Transfer 104, 890–896 (2017). [CrossRef]  

3. L. Zhu, A. P. Raman, and S. Fan, “Radiative cooling of solar absorbers using a visibly transparent photonic crystal thermal blackbody,” Proc Natl Acad Sci USA 112(40), 12282–12287 (2015). [CrossRef]  

4. J.-l. Kou, Z. Jurado, Z. Chen, S. Fan, and A. J. Minnich, “Daytime Radiative Cooling Using Near-Black Infrared Emitters,” ACS Photonics 4(3), 626–630 (2017). [CrossRef]  

5. S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective Radiative Cooling by Paint-Format Microsphere-Based Photonic Random Media,” ACS Photonics 5(4), 1181–1187 (2018). [CrossRef]  

6. S. Zeng, S. Pian, M. Su, Z. Wang, M. Wu, X. Liu, M. Chen, Y. Xiang, J. Wu, M. Zhang, Q. Cen, Y. Tang, X. Zhou, Z. Huang, R. Wang, A. Tunuhe, X. Sun, Z. Xia, M. Tian, M. Chen, X. Ma, L. Yang, J. Zhou, H. Zhou, Q. Yang, X. Li, Y. Ma, and G. Tao, “Hierarchical-morphology metafabric for scalable passive daytime radiative cooling,” Science 373(6555), 692–696 (2021). [CrossRef]  

7. H. Chalabi, D. Schoen, and M. L. Brongersma, “Hot-electron photodetection with a plasmonic nanostripe antenna,” Nano Lett. 14(3), 1374–1380 (2014). [CrossRef]  

8. A. K. Azad, W. J. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. Dalvit, and H. T. Chen, “Metasurface Broadband Solar Absorber,” Sci. Rep. 6(1), 20347 (2016). [CrossRef]  

9. H. Wang and L. Wang, “Perfect selective metamaterial solar absorbers,” Opt. Express 21(S6), A1078–1093 (2013). [CrossRef]  

10. X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107(4), 045901 (2011). [CrossRef]  

11. S. D. Rezaei, J. Ho, R. J. H. Ng, S. Ramakrishna, and J. K. W. Yang, “On the correlation of absorption cross-section with plasmonic color generation,” Opt. Express 25(22), 27652–27664 (2017). [CrossRef]  

12. C. Wu, B. Neuner, G. Shvets, J. John, A. Milder, B. Zollars, and S. Savoy, “Large-area wide-angle spectrally selective plasmonic absorber,” Phys. Rev. B 84(7), 075102 (2011). [CrossRef]  

13. J. M. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B 83(16), 165107 (2011). [CrossRef]  

14. A. Tittl, M. G. Harats, R. Walter, X. Yin, M. Schaferling, N. Liu, R. Rapaport, and H. Giessen, “Quantitative angle-resolved small-spot reflectance measurements on plasmonic perfect absorbers: impedance matching and disorder effects,” ACS Nano 8(10), 10885–10892 (2014). [CrossRef]  

15. P. Chevalier, P. Bouchon, J. Jaeck, D. Lauwick, N. Bardou, A. Kattnig, F. Pardo, and R. Haïdar, “Absorbing metasurface created by diffractionless disordered arrays of nanoantennas,” Appl. Phys. Lett. 107(25), 251108 (2015). [CrossRef]  

16. R. Walter, A. Tittl, A. Berrier, F. Sterl, T. Weiss, and H. Giessen, “Large-Area Low-Cost Tunable Plasmonic Perfect Absorber in the Near Infrared by Colloidal Etching Lithography,” Adv. Optical Mater. 3(3), 398–403 (2015). [CrossRef]  

17. C.-W. Cheng, “Wide-angle polarization independent infrared broadband absorbers based on metallic multisized disk arrays,” Opt. Express 20(9), 10376–10381 (2012). [CrossRef]  

18. N. W. Pech-May, L. Tobias, and M. Retsch, “Design of Multimodal Absorption in the Mid-IR: A Metal Dielectric Metal Approach,” ACS Appl. Mater. Interfaces 13(1), 1921–1929 (2021). [CrossRef]  

19. Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012). [CrossRef]  

20. D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015). [CrossRef]  

21. M. M. Hossain, B. Jia, and M. Gu, “A Metamaterial Emitter for Highly Efficient Radiative Cooling,” Adv. Optical Mater. 3(8), 1047–1051 (2015). [CrossRef]  

22. E. Rephaeli, A. Raman, and S. Fan, “Ultrabroadband photonic structures to achieve high-performance daytime radiative cooling,” Nano Lett. 13(4), 1457–1461 (2013). [CrossRef]  

23. A. P. Raman, M. A. Anoma, L. Zhu, E. Rephaeli, and S. Fan, “Passive radiative cooling below ambient air temperature under direct sunlight,” Nature 515(7528), 540–544 (2014). [CrossRef]  

24. Y. Shi, W. Li, A. Raman, and S. Fan, “Optimization of Multilayer Optical Films with a Memetic Algorithm and Mixed Integer Programming,” ACS Photonics 5(3), 684–691 (2018). [CrossRef]  

25. A. R. Gentle and G. B. Smith, “Radiative heat pumping from the Earth using surface phonon resonant nanoparticles,” Nano Lett. 10(2), 373–379 (2010). [CrossRef]  

26. Y. Zhai, Y. Ma, S. N. David, D. Zhao, R. Lou, G. Tan, R. Yang, and X. Yin, “Scalable-manufactured randomized glass-polymer hybrid metamaterial for daytime radiative cooling,” Science 355(6329), 1062–1066 (2017). [CrossRef]  

27. R. Zhu, D. Hu, Z. Chen, X. Xu, Y. Zou, L. Wang, and Y. Gu, “Plasmon-Enhanced Infrared Emission Approaching the Theoretical Limit of Radiative Cooling Ability,” Nano Lett. 20(10), 6974–6980 (2020). [CrossRef]  

28. A. Srinivasan, B. Czapla, J. Mayo, and A. Narayanaswamy, “Infrared dielectric function of polydimethylsiloxane and selective emission behavior,” Appl. Phys. Lett. 109(6), 061905 (2016). [CrossRef]  

29. U. Banik, A. Agrawal, H. Meddeb, O. Sergeev, N. Reininghaus, M. Gotz-Kohler, K. Gehrke, J. Stuhrenberg, M. Vehse, M. Sznajder, and C. Agert, “Efficient Thin Polymer Coating as a Selective Thermal Emitter for Passive Daytime Radiative Cooling,” ACS Appl. Mater. Interfaces 13(20), 24130–24137 (2021). [CrossRef]  

30. A. I. Maaroof, M. B. Cortie, N. Harris, and L. Wieczorek, “Mie and Bragg plasmons in subwavelength silver semi-shells,” Small 4(12), 2292–2299 (2008). [CrossRef]  

31. M. Frederiksen, V. E. Bochenkov, M. B. Cortie, and D. S. Sutherland, “Plasmon Hybridization and Field Confinement in Multilayer Metal–Dielectric Nanocups,” J. Phys. Chem. C 117(30), 15782–15789 (2013). [CrossRef]  

32. M. Cortie and M. Ford, “A plasmon-induced current loop in gold semi-shells,” Nanotechnology 18(23), 235704 (2007). [CrossRef]  

33. N. A. Mirin and N. J. Halas, “Light-Bending Nanoparticles,” Nano Lett. 9(3), 1255–1259 (2009). [CrossRef]  

34. N. Vogel, M. Retsch, C. A. Fustin, A. Del Campo, and U. Jonas, “Advances in colloidal assembly: the design of structure and hierarchy in two and three dimensions,” Chem. Rev. 115(13), 6265–6311 (2015). [CrossRef]  

35. K. Takatori, T. Okamoto, and K. Ishibashi, “Surface-plasmon-induced ultra-broadband light absorber operating in the visible to infrared range,” Opt. Express 26(2), 1342–1350 (2018). [CrossRef]  

36. K. Matsumori and R. Fujimura, “Broadband light absorption of an Al semishell-MIM nanostrucure in the UV to near-infrared regions,” Opt. Lett. 43(12), 2981–2984 (2018). [CrossRef]  

37. W. Yu, Y. Lu, X. Chen, H. Xu, J. Shao, X. Chen, Y. Sun, J. Hao, and N. Dai, “Large – Area, Broadband, Wide – Angle Plasmonic Metasurface Absorber for Midwavelength Infrared Atmospheric Transparency Window,” Adv. Optical Mater. 7(20), 1900841 (2019). [CrossRef]  

38. P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1(2), 131–138 (2012). [CrossRef]  

39. N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]  

40. R. Yahiaoui, J. A. Burrow, S. M. Mekonen, A. Sarangan, J. Mathews, I. Agha, and T. A. Searles, “Electromagnetically induced transparency control in terahertz metasurfaces based on bright-bright mode coupling,” Phys. Rev. B 97(15), 155403 (2018). [CrossRef]  

41. V. Krivenkov, S. Goncharov, I. Nabiev, and Y. P. Rakovich, “Induced Transparency in Plasmon-Exciton Nanostructures for Sensing Applications,” Laser Photonics Rev. 13(1), 1800176 (2019). [CrossRef]  

42. B. Gallinet, T. Siegfried, H. Sigg, P. Nordlander, and O. J. Martin, “Plasmonic radiance: probing structure at the Angstrom scale with visible light,” Nano Lett. 13(2), 497–503 (2013). [CrossRef]  

43. N. Murata, R. Hata, and H. Ishihara, “Crossover between Energy Transparency Resonance and Rabi Splitting in Antenna–Molecule Coupled Systems,” J. Phys. Chem. C 119(45), 25493–25498 (2015). [CrossRef]  

44. T. J. Antosiewicz, S. P. Apell, and T. Shegai, “Plasmon–Exciton Interactions in a Core–Shell Geometry: From Enhanced Absorption to Strong Coupling,” ACS Photonics 1(5), 454–463 (2014). [CrossRef]  

45. J. A. Faucheaux, J. Fu, and P. K. Jain, “Unified Theoretical Framework for Realizing Diverse Regimes of Strong Coupling between Plasmons and Electronic Transitions,” J. Phys. Chem. C 118(5), 2710–2717 (2014). [CrossRef]  

46. A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, “Near-field mediated plexcitonic coupling and giant Rabi splitting in individual metallic dimers,” Nano Lett. 13(7), 3281–3286 (2013). [CrossRef]  

47. S. Shrestha, Y. Wang, A. C. Overvig, M. Lu, A. Stein, L. Dal Negro, and N. F. Yu, “Indium Tin Oxide Broadband Metasurface Absorber,” ACS Photonics 5(9), 3526–3533 (2018). [CrossRef]  

48. M. Abb, Y. Wang, N. Papasimakis, C. H. de Groot, and O. L. Muskens, “Surface-enhanced infrared spectroscopy using metal oxide plasmonic antenna arrays,” Nano Lett. 14(1), 346–352 (2014). [CrossRef]  

49. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

50. Y. Kayaba and T. Kikkawa, “Quantitative Determination of Complex Dielectric Function of Amorphous Silicon Dioxide on Silicon Substrate from Transmission Spectrum,” Jpn. J. Appl. Phys. 48(12), 121406 (2009). [CrossRef]  

51. J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015). [CrossRef]  

52. J. van de Groep and A. Polman, “Designing dielectric resonators on substrates: combining magnetic and electric resonances,” Opt. Express 21(22), 26285–26302 (2013). [CrossRef]  

53. S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid. Films 335(1-2), 1–5 (1998). [CrossRef]  

54. J. W. Cleary, E. M. Smith, K. D. Leedy, G. Grzybowski, and J. Guo, “Optical and electrical properties of ultra-thin indium tin oxide nanofilms on silicon for infrared photonics,” Opt. Mater. Express 8(5), 1231–1245 (2018). [CrossRef]  

55. J. Lian, D. Zhang, R. Hong, P. Qiu, T. Lv, and D. Zhang, “Defect-Induced Tunable Permittivity of Epsilon-Near-Zero in Indium Tin Oxide Thin Films,” Nanomaterials 8(11), 922 (2018). [CrossRef]  

56. I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]  

57. S. Popova, T. Tolstykh, and V. Vorobev, “Optical characteristics of amorphous quartz in the 1400-200 cm−1 region,” Opt. Spectrosc. 46(33), 8118 (2007). [CrossRef]  

58. H. J. Hagemann, W. Gudat, and C. Kunz, “Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3,” J. Opt. Soc. Am. 65(6), 742–744 (1975). [CrossRef]  

59. B. Auguie and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. 101(14), 143902 (2008). [CrossRef]  

60. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]  

61. “IR Transmission Spectra,” https://www.gemini.edu/observing/telescopes-and-sites/sites#Transmission.

62. J. Li, R. Gan, Q. Guo, H. Liu, J. Xu, and F. Yi, “Tailoring optical responses of infrared plasmonic metamaterial absorbers by optical phonons,” Opt. Express 26(13), 16769–16781 (2018). [CrossRef]  

63. A. E. Miroshnichenko, A. B. Evlyukhin, Y. S. Kivshar, and B. N. Chichkov, “Substrate-Induced Resonant Magnetoelectric Effects for Dielectric Nanoparticles,” ACS Photonics 2(10), 1423–1428 (2015). [CrossRef]  

64. I. Sinev, I. Iorsh, A. Bogdanov, D. Permyakov, F. Komissarenko, I. Mukhin, A. Samusev, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, and Y. S. Kivshar, “Polarization control over electric and magnetic dipole resonances of dielectric nanoparticles on metallic films,” Laser Photonics Rev. 10(5), 799–806 (2016). [CrossRef]  

65. O. Merchiers, F. Moreno, F. González, and J. M. Saiz, “Light scattering by an ensemble of interacting dipolar particles with both electric and magnetic polarizabilities,” Phys. Rev. A 76(4), 043834 (2007). [CrossRef]  

66. S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the Interplay of Electric and Magnetic Modes via Fano-like Plasmon Resonances,” Nano Lett. 11(9), 3927–3934 (2011). [CrossRef]  

67. G. C. Schatz, “Theoretical Studies of Surface Enhanced Raman Scattering,” Acc. Chem. Res. 17(10), 370–376 (1984). [CrossRef]  

68. C. Deeb, X. Zhou, J. Plain, G. P. Wiederrecht, R. Bachelot, M. Russell, and P. K. Jain, “Size Dependence of the Plasmonic Near-Field Measured via Single-Nanoparticle Photoimaging,” J. Phys. Chem. C 117(20), 10669–10676 (2013). [CrossRef]  

69. R. Taubert, M. Hentschel, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption: simulations, experiments, and coupled oscillator analysis,” J. Opt. Soc. Am. B 30(12), 3123–3134 (2013). [CrossRef]  

70. R. Taubert, M. Hentschel, J. Kastel, and H. Giessen, “Classical analog of electromagnetically induced absorption in plasmonics,” Nano Lett. 12(3), 1367–1371 (2012). [CrossRef]  

71. N. W. Pech-May and M. Retsch, “Tunable daytime passive radiative cooling based on a broadband angle selective low-pass filter,” Nanoscale Adv. 2(1), 249–255 (2020). [CrossRef]  

72. A. Leroy, B. Bhatia, C. C. Kelsall, A. Castillejo-Cuberos, H. M. Di Capua, L. Zhao, L. Zhang, A. M. Guzman, and E. N. Wang, “High-performance subambient radiative cooling enabled by optically selective and thermally insulating polyethylene aerogel,” Sci. Adv. 5(10), eaat9480 (2019). [CrossRef]  

Supplementary Material (1)

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Supplement 1       Additional information to support the manuscript

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Absorption efficiency of the SiO2 sphere with a radius of (A) 1 µm, (B) 1.5 µm, and (C) 2 µm. The red and blue solid curves are the electric and magnetic dipole modes, respectively. The red and blue dashed curves are the electric and magnetic quadrupole modes, respectively.

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Fig. 2.
Fig. 2. (A) A schematic illustration of the ITO@SiO2 semishell structure. (B) rcore and (C) tshell dependence of the absorptivity of the ITO@SiO2 semishell structure. The blue shaded area indicates the primary ATW. (D) Transmittance of the atmosphere [61]. The blue and orange shaded areas indicate the actual primary ATW (8–9.3 µm and 10–13 µm) and the Ozone absorption band (9.3–10 µm), respectively. (E-F) Average absorption for Fig. 2(B) and 2(C). The black, blue, and red lines are average absorption in the primary ATW, the actual primary ATW, and the Ozone absorption band, respectively.

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Fig. 3.
Fig. 3. (A) Absorption spectra of each component of the ITO@SiO2 semishell structure. The red, blue, green, and orange curves are for the ITO shell, SiO2 core, perforated ITO film, and Au reflector, respectively. The black curve is the total absorption of the structure. (B) Color maps of electric and magnetic field distribution at wavelength indicated by arrows in Fig. 3(A). Absolute value of fields is normalized by amplitude of incident wave. The white arrows are electric field vectors.

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Fig. 4.
Fig. 4. (A) Two-oscillator CO model describing the interaction between the LSP of the ITO shell and the MD of the SiO2 core. (B) Three-oscillator CO model where one additional oscillator is added to the two-oscillator CO model to account for the interaction between the LSP of the ITO shell and the TO mode of SiO2. (C) The CO model is fitted to the absorption of the ITO shell. The red and blue solid curves are absorption spectra of the ITO shell and the SiO2 core in Fig. 3(A), respectively. The black dashed curve is absorption calculated from Eq. (4) and the red, blue, green dashed curves are absorption calculated from Eq. (7).

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Fig. 5.
Fig. 5. (A) A schematic illustration of the CDM for the interaction between the LSP of the ITO shell and the MD of the SiO2 core. (B) The phase φ calculated using Eq. (11). (C) Extinction spectra of interacting ED and MD.

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Equations (23)

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C ext = 2 π k 2 ( 2 n + 1 ) Re [ a n + b n ] C sca = 2 π k 2 ( 2 n + 1 ) ( | a n | 2 + | b n | 2 ) C abs = C ext − C sca
d 2 x α 1 d t 2 + γ 1 d x α 1 d t + ω 1 2 x α 1 − κ 12 2 x α 2 = F 1 ( t ) d 2 x α 2 d t 2 + γ 2 d x α 2 d t + ω 2 2 x α 2 − κ 12 2 x α 1 = F 2 ( t )
x α 1 = Ω 2 F 1 + κ 12 2 F 2 Ω 1 Ω 2 − κ 12 4 exp ( − i ω t ) = c α 1 exp ( − i ω t ) x α 2 = Ω 1 F 2 + κ 12 2 F 1 Ω 1 Ω 2 − κ 12 4 exp ( − i ω t ) = c α 2 exp ( − i ω t )
P ( ω ) = ⟨ Re [ F 1 * ( t ) × d x 1 d t ] ⟩ + ⟨ Re [ F 2 * ( t ) × d x 2 d t ] ⟩
P α ( ω ) = 1 2 γ 1 ω 2 | c α 1 | 2 + 1 2 γ 2 ω 2 | c α 2 | 2 = A α 1 ( ω ) + A α 2 ( ω )
d 2 x β 1 d t 2 + γ 1 d x β 1 d t + ω 1 2 x β 1 − κ 12 2 x β 2 − κ 13 2 x β 3 = F 1 ( t ) d 2 x β 2 d t 2 + γ 2 d x β 2 d t + ω 2 2 x β 2 − κ 12 2 x β 1 = F 2 ( t ) d 2 x β 3 d t 2 + γ 3 d x β 3 d t + ω 3 2 x β 3 − κ 13 2 x β 1 = 0
x β 1 = Ω 2 Ω 3 F 1 + κ 12 2 Ω 3 F 2 Ω 1 Ω 2 Ω 3 − κ 12 4 Ω 3 − κ 13 4 Ω 2 exp ( − i ω t ) = c β 1 exp ( − i ω t ) x β 2 = ( Ω 1 Ω 3 − κ 13 4 ) F 2 + κ 12 2 Ω 3 F 1 Ω 1 Ω 2 Ω 3 − κ 12 4 Ω 3 − κ 13 4 Ω 2 exp ( − i ω t ) = c β 2 exp ( − i ω t ) x β 3 = κ 13 2 ( Ω 2 F 1 + κ 12 2 F 2 ) Ω 1 Ω 2 Ω 3 − κ 12 4 Ω 3 − κ 13 4 Ω 2 exp ( − i ω t ) = c β 3 exp ( − i ω t )
P β ( ω ) = 1 2 γ 1 ω 2 | c β 1 | 2 + 1 2 γ 2 ω 2 | c β 2 | 2 + 1 2 γ 3 ω 2 | c β 3 | 2 = A β 1 ( ω ) + A β 2 ( ω ) + A β 3 ( ω )
p 1 = ε 0 α 1 E tot ( r 1 )
m 2 = χ 2 H tot ( r 2 )
E s ( r 1 ) = − Z 0 G M ( r 1 − r 2 ) ⋅ m 2
H s ( r 2 ) = 1 ε 0 Z 0 G M ( r 2 − r 1 ) ⋅ p 1
G M ( r 1 − r 2 ) ⋅ n = − G M ( r 2 − r 1 ) ⋅ n = ( − e ^ z × n ) C ( D )
C ( D ) = exp ( i k D ) 4 π D ( k 2 + i k D ) = exp ( i k D ) 4 π D k 4 + k 2 D 2 exp [ i tan − 1 ( 1 k D ) ] = i | C | exp { i [ k D − tan − 1 ( k D ) ] } = i | C | exp ( i ϕ )
p 1 = ε 0 α 1 [ E in ( r 1 ) + E s ( r 1 ) ] = ε 0 α 1 E in ( r 1 ) − ε 0 α 1 Z 0 G M ( r 1 − r 2 ) ⋅ m 2
m 2 = χ 2 [ H in ( r 2 ) + H s ( r 2 ) ] = χ 2 H in ( r 2 ) + 1 ε 0 Z 0 χ 2 G M ( r 2 − r 1 ) ⋅ p 1
p 1 = p 1 x e ^ x = ε 0 α 1 exp ( − i k D 2 ) − α 1 χ 2 C ( D ) exp ( i k D 2 ) 1 + α 1 χ 2 C 2 ( D ) E 0 e ^ x
m 2 = m 2 y e ^ y = χ 2 exp ( i k D 2 ) + α 1 χ 2 C ( D ) exp ( − i k D 2 ) 1 + α 1 χ 2 C 2 ( D ) H 0 e ^ y
α 1 = F p ω p 2 − ω 2 − i γ p ω = F p Ω p
χ 2 = F m ω m 2 − ω 2 − i γ m ω = F m Ω m
p 1 x = ε 0 Ω m F p exp ( − i k D 2 ) − i F p F m | C | exp ( i ϕ ) exp ( i k D 2 ) Ω p Ω m − F p F m | C | 2 exp ( i 2 ϕ ) E 0
m 2 y = Ω p F m exp ( i k D 2 ) + i F p F m | C | exp ( i ϕ ) exp ( − i k D 2 ) Ω p Ω m − F p F m | C | 2 exp ( i 2 ϕ ) H 0
C ext = k ε 0 | E 0 | 2 Im [ E in ∗ ( r 1 ) ⋅ p 1 + μ 0 H in ∗ ( r 2 ) ⋅ m 2 ]
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