Optimization of highly circularly polarized thermal
radiation in α-MoO$_3$/β-Ga$_2$O$_3$ twisted layers

Marco Centini, Chiyu Yang, Maria Cristina Larciprete, Mauro Antezza,
Zhuoming Zhang

► To cite this version:

Marco Centini, Chiyu Yang, Maria Cristina Larciprete, Mauro Antezza, Zhuoming Zhang. Optimization of highly circularly polarized thermal radiation in α-MoO$_3$/β-Ga$_2$O$_3$ twisted layers. Journal of Quantitative Spectroscopy and Radiative Transfer, 2024, 323, pp.109051. 10.1016/j.jqsrt.2024.109051. hal-04608450

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Submitted on 11 Jun 2024

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# Optimization of highly circularly polarized thermal radiation in α-MoO$_3$/β-Ga$_2$O$_3$ twisted layers

Marco Centini, Chiyu Yang, Maria Cristina Larciprete, Mauro Antezza, Zhuoming Zhang

► To cite this version:

Marco Centini, Chiyu Yang, Maria Cristina Larciprete, Mauro Antezza, Zhuoming Zhang. Optimization of highly circularly polarized thermal radiation in α-MoO$_3$/β-Ga$_2$O$_3$ twisted layers. Journal of Quantitative Spectroscopy and Radiative Transfer, 2024, 323, pp.109051. 10.1016/j.jqsrt.2024.109051. hal-04608450

HAL Id: hal-04608450

https://hal.science/hal-04608450v1

Submitted on 11 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
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# Optimization of highly circularly polarized thermal radiation in α-MoO<sub>3</sub>/β-Ga<sub>2</sub>O<sub>3</sub> twisted layers

Marco Centini<sup>a, *</sup>, Chiyu Yang<sup>b</sup>, Maria Cristina Larciprete<sup>a</sup>, Mauro Antezza<sup>c, d</sup>, Zhuomin M. Zhang<sup>b</sup>

## ARTICLE INFO

### Keywords:

Thermal radiation

Circular polarization

Infrared photonics

Optical anisotropic thin films

## ABSTRACT

We investigate a bi-layer scheme for circularly polarized infrared thermal radiation. Our approach takes advantage of the strong anisotropy of low-symmetry materials such as β-Ga₂O₃ and α-MoO₃. We numerically report narrow-band, high degree of circular polarization (over 0.85), thermal radiation at two typical emission frequencies related to the excitation of β-Ga₂O₃ optical phonons. Optimization of the degree of circular polarization is achieved by a proper relative tilt of the crystal axes between the two layers. Our simple but effective scheme could set the basis for a new class of lithography-free thermal sources for IR bio-sensing.

## 1. Introduction

The development of miniaturized and integrable devices operating in the mid-infrared (mid-IR) relies on the availability of narrowband, highly directional, and polarization tunable sources. Indeed, several applications ranging from IR sensing [1,2] to biomedical diagnostics and target detecting [3,4] as well as for the realization of integrated IR photonics [5,6] require highly efficient, narrow-band, and polarization tunable mid-IR sources. An efficient but expensive option is represented by quantum cascade laser (QCL) IR sources. However, with the aim to achieve mass production of low-cost devices, a lot of research effort has been recently spent on the study of optimized and controllable thermal radiation sources. Thanks to the development of micro/nano technologies [7], several approaches to control and tailor the typically incoherent and broadband thermal radiation from heated bodies have been proposed and realized. The mechanism at the base of this tailored emission is the excitation of resonant states i.e. surface plasmon-(SP) or phonon-(SPh) polaritons, Bloch surface waves, magnetic polaritons, to name a few. Gratings [8], metamaterials [9,10], metasurfaces [11] and nanoantennae [12,13] have been successfully proposed to overcome the limits and enhance the performances of natural bulk materials. However, all these approaches rely on the use of lithographic techniques to artificially create absorption/emission resonances as well as a strong

birefringent effective response. In particular, birefringence is necessary for polarization-sensitive applications.

An extreme form of birefringent behavior is called hyperbolicity and it is achieved when one diagonal element of the material dielectric tensor is negative, whilst the others are positive. Hyperbolic metamaterials obtained by periodic sub-wavelength patterned layered media have been extensively studied for photonics application [14-17] as well as for enhanced thermal radiation and nanoscale heat transfer [18,19]. However, it has been recently shown that hyperbolicity can be observed in natural materials, more specifically in van der Waals (vdW) materials IR [20]. Among vdW materials, hexagonal boron nitride hBN is one of the most investigated [21]. The hBN exhibits two different Reststrahlen bands (negative values of the real part of the permittivity) along either the in-plane or out-of-plane directions. The main limitation of hBN is the lack of in-plane anisotropy. In-plane anisotropy is required to distinguish and manipulate orthogonal polarization states of the electromagnetic field at normal incidence.

Among the vdW polar materials displaying in-plane anisotropy, molybdenum trioxide (α-MoO₃) has been the object of a conspicuous number of recent studies [22,23]. Its strong anisotropy, allowing in-plane and off-plane hyperbolicity, has been used to obtain mid-IR waveplates [24] and tunable [25] and broadband [26] absorption in the far-field. Concerning the excitation of surface waves, hyperbolic

<footnote><sup>a</sup> Sapienza University of Rome Department of Basic and Applied Sciences for Engineering, Via A. Scarpa 14, I-00161 Rome, Italy</footnote>

<footnote><sup>b</sup> George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA</footnote>

<footnote><sup>c</sup> Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F- 34095, Montpellier, France</footnote>

<footnote><sup>d</sup> Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex 05, France</footnote>

<footnote>* Corresponding author.
E-mail address: marco.centini@uniromal.it (M. Centini).</footnote>
<figure><img src="image_2.png" /></figure>

# Optimization of highly circularly polarized thermal radiation in α-MoO<sub>3</sub>/β-Ga<sub>2</sub>O<sub>3</sub> twisted layers

Marco Centini<sup>a, *</sup>, Chiyu Yang<sup>b</sup>, Maria Cristina Larciprete<sup>a</sup>, Mauro Antezza<sup>c, d</sup>, Zhuomin M. Zhang<sup>b</sup>

## ARTICLE INFO

### Keywords:

Thermal radiation

Circular polarization

Infrared photonics

Optical anisotropic thin films

## ABSTRACT

We investigate a bi-layer scheme for circularly polarized infrared thermal radiation. Our approach takes advantage of the strong anisotropy of low-symmetry materials such as β-Ga₂O₃ and α-MoO₃. We numerically report narrow-band, high degree of circular polarization (over 0.85), thermal radiation at two typical emission frequencies related to the excitation of β-Ga₂O₃ optical phonons. Optimization of the degree of circular polarization is achieved by a proper relative tilt of the crystal axes between the two layers. Our simple but effective scheme could set the basis for a new class of lithography-free thermal sources for IR bio-sensing.

## 1. Introduction

The development of miniaturized and integrable devices operating in the mid-infrared (mid-IR) relies on the availability of narrowband, highly directional, and polarization tunable sources. Indeed, several applications ranging from IR sensing [1,2] to biomedical diagnostics and target detecting [3,4] as well as for the realization of integrated IR photonics [5,6] require highly efficient, narrow-band, and polarization tunable mid-IR sources. An efficient but expensive option is represented by quantum cascade laser (QCL) IR sources. However, with the aim to achieve mass production of low-cost devices, a lot of research effort has been recently spent on the study of optimized and controllable thermal radiation sources. Thanks to the development of micro/nano technologies [7], several approaches to control and tailor the typically incoherent and broadband thermal radiation from heated bodies have been proposed and realized. The mechanism at the base of this tailored emission is the excitation of resonant states i.e. surface plasmon-(SP) or phonon-(SPh) polaritons, Bloch surface waves, magnetic polaritons, to name a few. Gratings [8], metamaterials [9,10], metasurfaces [11] and nanoantennae [12,13] have been successfully proposed to overcome the limits and enhance the performances of natural bulk materials. However, all these approaches rely on the use of lithographic techniques to artificially create absorption/emission resonances as well as a strong

birefringent effective response. In particular, birefringence is necessary for polarization-sensitive applications.

An extreme form of birefringent behavior is called hyperbolicity and it is achieved when one diagonal element of the material dielectric tensor is negative, whilst the others are positive. Hyperbolic metamaterials obtained by periodic sub-wavelength patterned layered media have been extensively studied for photonics application [14-17] as well as for enhanced thermal radiation and nanoscale heat transfer [18,19]. However, it has been recently shown that hyperbolicity can be observed in natural materials, more specifically in van der Waals (vdW) materials IR [20]. Among vdW materials, hexagonal boron nitride hBN is one of the most investigated [21]. The hBN exhibits two different Reststrahlen bands (negative values of the real part of the permittivity) along either the in-plane or out-of-plane directions. The main limitation of hBN is the lack of in-plane anisotropy. In-plane anisotropy is required to distinguish and manipulate orthogonal polarization states of the electromagnetic field at normal incidence.

Among the vdW polar materials displaying in-plane anisotropy, molybdenum trioxide (α-MoO₃) has been the object of a conspicuous number of recent studies [22,23]. Its strong anisotropy, allowing in-plane and off-plane hyperbolicity, has been used to obtain mid-IR waveplates [24] and tunable [25] and broadband [26] absorption in the far-field. Concerning the excitation of surface waves, hyperbolic

<footnote><sup>a</sup> Sapienza University of Rome Department of Basic and Applied Sciences for Engineering, Via A. Scarpa 14, I-00161 Rome, Italy</footnote>

<footnote><sup>b</sup> George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA</footnote>

<footnote><sup>c</sup> Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095, Montpellier, France</footnote>

<footnote><sup>d</sup> Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex 05, France</footnote>

<footnote>* Corresponding author.
E-mail address: marco.centini@uniromal.it (M. Centini).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Imaginary part of the in-plane components (xx,xy,yy) of the dielectric permittivity tensor of (a) MoO$_3$, (b) Ga$_2$O$_3$. Real part of the refractive indices along x- (n$_x$) and y- (n$_y$) directions and Δn for (c) MoO$_3$ and Real part of the relative permittivity for (d) Ga$_2$O$_3$.</figcaption></figure>

phonon polaritons have recently been shown in α-MoO$_3$ flakes from 818 cm$^{-1}$ to 974 cm$^{-1}$ [23,27]. Furthermore, the excitation of SPh polaritons waves in twisted flakes of 2D α-MoO$_3$ has been proposed and experimentally verified [28]. Such control of SPh polaritons in twisted flakes has interesting applications for tunable near-field radiative heat transfer [29] by adjusting the tilt angle between the receiver and the emitter. It has been shown that twisted α-MoO$_3$ layers can exhibit circular dichroism [30,31] and can be used to generate spin thermal radiation [32]. However, using twisted structures of identical anisotropic materials to achieve spin thermal radiation has more restrictions on the materials. For this reason, α-MoO$_3$ was combined with an ideal quarter wavelength plate [33]. However, a tunable combination of two real anisotropic materials (i.e. controlling the tilt angle between them) could be adopted for practical applications and integrated thermal sources.

Low-symmetry materials have recently emerged as possible candidates for anisotropic optical applications [30]. Among them, β-Ga$_2$O$_3$ [34,35] has been used to experimentally demonstrate the excitation of shear phonon polaritons in the IR [36,37]. Moreover, this material perfectly matches the usable wavelength range of α-MoO$_3$. More specifically we will show that there are phonon resonances in the β-Ga$_2$O$_3$ which fall in an almost transparent but highly anisotropic band of the α-MoO$_3$. Thus, we propose a combination of two twisted layers obtained from these two materials.

Our calculations, based on a complete Stokes parameters analysis [38] of emitted radiation, show that it is possible to obtain circularly polarized thermal radiation in the mid-IR with a twisted bilayer system having about two microns of total thickness. As a starting point we study the emission properties and the birefringence behavior of single layers of α-MoO$_3$ and β-Ga$_2$O$_3$ on gold substrates. The choice of gold substrate allows for an almost one-sided emissivity as long as we can ignore

thermal emission for the gold back face in the mid-IR. The relative permittivity of Au has been modeled from the data in [39]. After the choice of one of the two materials as the emitter, we add a properly tilted and sized layer of the other, acting as a quarter wave plate to obtain thermally radiated circularly polarized light.

## 2. Numerical methods

In order to quantitatively study the degree of polarization (DoP) and the degree of circular polarization (DoCP) of the thermal radiation polarization properties can be characterized by Stokes parameters. For reciprocal materials, the Stokes parameters can be expressed in terms of the polarized emissivities as [38,40]:

$$ \begin{bmatrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{bmatrix} = \frac{S_{0,bb}(\omega, k_{\parallel})}{2} \begin{bmatrix} \epsilon_p + \epsilon_s \\ \epsilon_p - \epsilon_s \\ \epsilon_{45^{\circ}} - \epsilon_{135^{\circ}} \\ \epsilon_R - \epsilon_L \end{bmatrix} \quad (1) $$

where $k_{\parallel}$ is the wave vector projection onto the surface (x-y) plane, $S_{0,bb}$ is the emissivity of the black body, $\epsilon_{p,s}$ are the p- and s- polarized relative emissivities, $\epsilon_{45^{\circ},135^{\circ}}$ are the 45°- and 135°- polarized relative emissivities and $\epsilon_{R,L}$ are the right- and left- circular polarized relative emissivities respectively. Limiting our discussion to reciprocal media at thermal equilibrium, polarized emissivities can be evaluated by calculating the polarized absorptivities $\alpha$, according to Kirchhoff's law:

$$ \epsilon_{p,s,45^{\circ},135^{\circ},R,L} = \alpha_{p,s,45^{\circ},135^{\circ},R,L} \quad (2) $$

Polarized absorptivities have been retrieved by evaluating $1 - R - T$ where R is reflectance and T is transmittance evaluated for the different polarized states of the incident light with a 4 × 4 transfer matrix method
<figure><img src="image_3.png" /><figcaption>Fig. 1. Imaginary part of the in-plane components (xx,xy,yy) of the dielectric permittivity tensor of (a) MoO$_3$, (b) Ga$_2$O$_3$. Real part of the refractive indices along x- (n$_x$) and y- (n$_y$) directions and Δn for (c) MoO$_3$ and Real part of the relative permittivity for (d) Ga$_2$O$_3$.</figcaption></figure>

phonon polaritons have recently been shown in α-MoO$_3$ flakes from 818 cm$^{-1}$ to 974 cm$^{-1}$ [23,27]. Furthermore, the excitation of SPh polaritons waves in twisted flakes of 2D α-MoO$_3$ has been proposed and experimentally verified [28]. Such control of SPh polaritons in twisted flakes has interesting applications for tunable near-field radiative heat transfer [29] by adjusting the tilt angle between the receiver and the emitter. It has been shown that twisted α-MoO$_3$ layers can exhibit circular dichroism [30,31] and can be used to generate spin thermal radiation [32]. However, using twisted structures of identical anisotropic materials to achieve spin thermal radiation has more restrictions on the materials. For this reason, α-MoO$_3$ was combined with an ideal quarter wavelength plate [33]. However, a tunable combination of two real anisotropic materials (i.e. controlling the tilt angle between them) could be adopted for practical applications and integrated thermal sources.

Low-symmetry materials have recently emerged as possible candidates for anisotropic optical applications [30]. Among them, β-Ga$_2$O$_3$ [34,35] has been used to experimentally demonstrate the excitation of shear phonon polaritons in the IR [36,37]. Moreover, this material perfectly matches the usable wavelength range of α-MoO$_3$. More specifically we will show that there are phonon resonances in the β-Ga$_2$O$_3$ which fall in an almost transparent but highly anisotropic band of the α-MoO$_3$. Thus, we propose a combination of two twisted layers obtained from these two materials.

Our calculations, based on a complete Stokes parameters analysis [38] of emitted radiation, show that it is possible to obtain circularly polarized thermal radiation in the mid-IR with a twisted bilayer system having about two microns of total thickness. As a starting point we study the emission properties and the birefringence behavior of single layers of α-MoO$_3$ and β-Ga$_2$O$_3$ on gold substrates. The choice of gold substrate allows for an almost one-sided emissivity as long as we can ignore

thermal emission for the gold back face in the mid-IR. The relative permittivity of Au has been modeled from the data in [39]. After the choice of one of the two materials as the emitter, we add a properly tilted and sized layer of the other, acting as a quarter wave plate to obtain thermally radiated circularly polarized light.

## 2. Numerical methods

In order to quantitatively study the degree of polarization (DoP) and the degree of circular polarization (DoCP) of the thermal radiation polarization properties can be characterized by Stokes parameters. For reciprocal materials, the Stokes parameters can be expressed in terms of the polarized emissivities as [38,40]:

$$ \begin{bmatrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{bmatrix} = \frac{S_{0,bb}(\omega, k_{\parallel})}{2} \begin{bmatrix} \epsilon_p + \epsilon_s \\ \epsilon_p - \epsilon_s \\ \epsilon_{45^{\circ}} - \epsilon_{135^{\circ}} \\ \epsilon_R - \epsilon_L \end{bmatrix} \quad (1) $$

where $k_{\parallel}$ is the wave vector projection onto the surface (x-y) plane, $S_{0,bb}$ is the emissivity of the black body, $\epsilon_{p,s}$ are the p- and s- polarized relative emissivities, $\epsilon_{45^{\circ},135^{\circ}}$ are the 45°- and 135°- polarized relative emissivities and $\epsilon_{R,L}$ are the right- and left- circular polarized relative emissivities respectively. Limiting our discussion to reciprocal media at thermal equilibrium, polarized emissivities can be evaluated by calculating the polarized absorptivities $\alpha$, according to Kirchhoff's law:

$$ \epsilon_{p,s,45^{\circ},135^{\circ},R,L} = \alpha_{p,s,45^{\circ},135^{\circ},R,L} \quad (2) $$

Polarized absorptivities have been retrieved by evaluating $1 - R - T$ where R is reflectance and T is transmittance evaluated for the different polarized states of the incident light with a 4 × 4 transfer matrix method
<figure><img src="image_4.png" /><figcaption>Fig. 2. (a) Schematic of a single β-Ga₂O₃ layer exhibiting optimized DoP at ν=691 cm⁻¹; (b) DoP and DoCP as a function of the frequency for the system sketched in (a); (c) Schematic of a single β-Ga₂O₃ layer exhibiting optimized DoP at ν=737 cm⁻¹; (d) DoP and DoCP as a function of the frequency for the system sketched in (c).</figcaption></figure>

[41]. The DoP and the DoCP are then defined as:

$$ \text{DoP} = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0} \qquad (3) $$

$$ \text{DoCP} = \frac{|S_3|}{S_0} \qquad (4) $$

Being both DoP and DoCP limited in the range [0,1] and DoCP≤DoP. DoP=1 stands for perfectly polarized light, thus DoCP=1 corresponds to perfectly circularly polarized light.

Our approach is based on the combination of two layers on top of gold substrate. The bottom layer acts as an emitter while the second acts as a quarter-wave plate. Both layers are required to be strongly anisotropic, however, the emitting layer should provide high emission/absorption efficiency for only one polarization component while the upper one should be as transparent as possible with a strong anisotropic real part of the refractive index. A particle swarm optimization is used as the search algorithm for the optimal parameters including the thicknesses of the MoO₃ layer (d₁), the Ga₂O₃ layer (d₂), and the twist angle between the two layers (α). In order to reduce the number of optimization parameters and to avoid local minimum solutions for the two-layer system the particle swarm algorithm is performed in two separate steps. At first, we focus on the single bottom layer on gold. This way we find the layer thickness d₂ and emission wavelength for maximized DOP. Then we optimize the thickness and the relative orientation of the second (upper) layer to obtain maximum DoCP. We focus on the fact that this two-step optimization is performed to reduce the number of simultaneous sweeping parameters but it does not affect the accuracy of numerical results. Indeed, the adopted 4 × 4 transfer matrix model takes into account for multiple reflections. A complete single-step full sweep could

lead to improved performances at a slightly shifted emission line. Nevertheless, we expect that this shift, typically related to multiple reflections, could play a relevant role in multilayer stacks or high-Q cavities, here we only focus on the double-layer scheme.

### 3. Materials optical properties

We study both α-MoO₃ and β-Ga₂O₃ optical properties in the mid-IR with the aim of identifying the frequency ranges where they can be efficiently combined. To begin with, we focus our study on the thermal radiation emitted along the direction normal to the surface. Thus, if we consider the x-y plane as the surface plane, the emission direction is toward the z-axis. We also select the orientation of the crystals so that x- and y- axes coincide with the x'- and y'- crystal axes. In this case, we are interested in the in-plane values of the dielectric permittivities. In Fig. 1 (a,b), the imaginary part of xx, yy and xy components of the relative permittivity tensor of α-MoO₃ and β-Ga₂O₃ are plotted as a function of the frequency [23,42]. The imaginary part of the dielectric function is related to absorption/emission. For α-MoO₃, there are two main peaks at 550 cm⁻¹ and 820 cm⁻¹ due to optical phonons while for β-Ga₂O₃, there are three main bands centered around 570 cm⁻¹, 690 cm⁻¹ and 740 cm⁻¹, respectively. As previously mentioned, in our emitter/waveplate approach we want to avoid overlaps in the emission bands in order to clearly distinguish the role of the emitter from the role of the waveplate. For this reason, we exclude from further investigation the lower frequencies (570 cm⁻¹ and 550 cm⁻¹). In Fig. 1(c), the real part of the refractive index components nₓ and nᵧ α-MoO₃ and their Δn = nₓ - nᵧ are shown. The real part of the dielectric function components of β-Ga₂O₃ are shown in Fig. 1(d). We note that α-MoO₃ has a wide band (from 600 cm⁻¹ to 750 cm⁻¹) of high and almost constant anisotropy which makes
<figure><img src="image_4.png" /><figcaption>Fig. 2. (a) Schematic of a single β-Ga₂O₃ layer exhibiting optimized DoP at ν=691 cm⁻¹; (b) DoP and DoCP as a function of the frequency for the system sketched in (a); (c) Schematic of a single β-Ga₂O₃ layer exhibiting optimized DoP at ν=737 cm⁻¹; (d) DoP and DoCP as a function of the frequency for the system sketched in (c).</figcaption></figure>

[41]. The DoP and the DoCP are then defined as:

$$ \text{DoP} = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0} \qquad (3) $$

$$ \text{DoCP} = \frac{|S_3|}{S_0} \qquad (4) $$

Being both DoP and DoCP limited in the range [0,1] and DoCP≤DoP. DoP=1 stands for perfectly polarized light, thus DoCP=1 corresponds to perfectly circularly polarized light.

Our approach is based on the combination of two layers on top of gold substrate. The bottom layer acts as an emitter while the second acts as a quarter-wave plate. Both layers are required to be strongly anisotropic, however, the emitting layer should provide high emission/absorption efficiency for only one polarization component while the upper one should be as transparent as possible with a strong anisotropic real part of the refractive index. A particle swarm optimization is used as the search algorithm for the optimal parameters including the thicknesses of the MoO₃ layer (d₁), the Ga₂O₃ layer (d₂), and the twist angle between the two layers (α). In order to reduce the number of optimization parameters and to avoid local minimum solutions for the two-layer system the particle swarm algorithm is performed in two separate steps. At first, we focus on the single bottom layer on gold. This way we find the layer thickness d₂ and emission wavelength for maximized DOP. Then we optimize the thickness and the relative orientation of the second (upper) layer to obtain maximum DoCP. We focus on the fact that this two-step optimization is performed to reduce the number of simultaneous sweeping parameters but it does not affect the accuracy of numerical results. Indeed, the adopted 4 × 4 transfer matrix model takes into account for multiple reflections. A complete single-step full sweep could

lead to improved performances at a slightly shifted emission line. Nevertheless, we expect that this shift, typically related to multiple reflections, could play a relevant role in multilayer stacks or high-Q cavities, here we only focus on the double-layer scheme.

### 3. Materials optical properties

We study both α-MoO₃ and β-Ga₂O₃ optical properties in the mid-IR with the aim of identifying the frequency ranges where they can be efficiently combined. To begin with, we focus our study on the thermal radiation emitted along the direction normal to the surface. Thus, if we consider the x-y plane as the surface plane, the emission direction is toward the z-axis. We also select the orientation of the crystals so that x- and y- axes coincide with the x'- and y'- crystal axes. In this case, we are interested in the in-plane values of the dielectric permittivities. In Fig. 1 (a,b), the imaginary part of xx, yy and xy components of the relative permittivity tensor of α-MoO₃ and β-Ga₂O₃ are plotted as a function of the frequency [23,42]. The imaginary part of the dielectric function is related to absorption/emission. For α-MoO₃, there are two main peaks at 550 cm⁻¹ and 820 cm⁻¹ due to optical phonons while for β-Ga₂O₃, there are three main bands centered around 570 cm⁻¹, 690 cm⁻¹ and 740 cm⁻¹, respectively. As previously mentioned, in our emitter/waveplate approach we want to avoid overlaps in the emission bands in order to clearly distinguish the role of the emitter from the role of the waveplate. For this reason, we exclude from further investigation the lower frequencies (570 cm⁻¹ and 550 cm⁻¹). In Fig. 1(c), the real part of the refractive index components nₓ and nᵧ α-MoO₃ and their Δn = nₓ - nᵧ are shown. The real part of the dielectric function components of β-Ga₂O₃ are shown in Fig. 1(d). We note that α-MoO₃ has a wide band (from 600 cm⁻¹ to 750 cm⁻¹) of high and almost constant anisotropy which makes
<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Schematic of twisted bi-layer structure designed for ν=691 cm⁻¹, obtained by adding a α-MoO₃ layer on top of the β-Ga₂O₃ layer; (b) circularly polarized emissivity and (c) degree of polarization of the twisted bi-layer structure; (d) S₃/S₀ as a function of the twist angle and of the frequency. We note that by proper tuning of the twist angle, it is possible to switch from LCP to RCP emission at ν=691 cm⁻¹.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Schematic of twisted bi-layer structure designed for ν=737 cm⁻¹, obtained by adding a α-MoO₃ layer on top of the β-Ga₂O₃ layer; (b) circularly polarized emissivity and (c) degree of polarization of the twisted bi-layer structure; (d) S₃/S₀ as a function of the twist angle and of the frequency. We note that by proper tuning of the twist angle it is possible to switch from LCP to RCP emission at ν=737 cm⁻¹.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Schematic of twisted bi-layer structure designed for ν=691 cm⁻¹, obtained by adding a α-MoO₃ layer on top of the β-Ga₂O₃ layer; (b) circularly polarized emissivity and (c) degree of polarization of the twisted bi-layer structure; (d) S₃/S₀ as a function of the twist angle and of the frequency. We note that by proper tuning of the twist angle, it is possible to switch from LCP to RCP emission at ν=691 cm⁻¹.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Schematic of twisted bi-layer structure designed for ν=737 cm⁻¹, obtained by adding a α-MoO₃ layer on top of the β-Ga₂O₃ layer; (b) circularly polarized emissivity and (c) degree of polarization of the twisted bi-layer structure; (d) S₃/S₀ as a function of the twist angle and of the frequency. We note that by proper tuning of the twist angle it is possible to switch from LCP to RCP emission at ν=737 cm⁻¹.</figcaption></figure>
it very suitable to be used as a waveplate. Thus we focus our attention on the design of an efficient β-Ga₂O₃ emitting layer around 690 cm⁻¹ and 740 cm⁻¹ with a high DoP combined with a properly tilted and sized α-MoO₃ layer acting as waveplate to obtain a tailored thermal source with optimized DoCP.

## 4. Results and discussion

We employed particle swarm optimization (PSO), a search algorithm that finds optimal solutions by iteratively trying to improve candidate solutions, and finally obtained structures with the highest circular emission at specific frequencies. The independent parameters needed according to the proposed two-layer scheme are the thicknesses of the MoO₃ layer ($d₁$), of the Ga₂O₃ layer ($d₂$), and the twist angle between the two layers ($α$). These parameters are grouped into a vector, denoted as $r$, which serves as a particle. An initial population of N random particles is generated, and the loss function of each particle is calculated. Subsequently, the particles undergo iterative exploration of the parameter space to minimize the loss function until the predetermined iteration number is reached. More details of the particle swarm optimization algorithm can be found in the reference [43]. However, in order to reduce the number of optimization parameters and to avoid local minimum solutions for the two-layer system the particle swarm algorithm is performed in two separate steps. The first optimization is performed on a single β-Ga₂O₃ layer on gold substrate looking for maximum DOP. We set as the loss function $y(r) = 1 - \text{DoP}$ (thus $r = [d₂]$). We found two schemes, named scheme A and scheme B, corresponding to maximized DoP at two different frequencies: 691 cm⁻¹ and 737 cm⁻¹ respectively. In the first scheme, sketched in Fig. 2a, we note that a $d₂A = 606$ nm thick β-Ga₂O₃ layer has a maximum relative emissivity for a p-polarized field at 691 cm⁻¹ corresponding to a DoP of about 0.97 and zero DoCP (Fig. 2b). On the other hand, the second proposed scheme, sketched in Fig. 2c is based on a thicker Ga₂O₃ layer ($d₂B = 1909$ nm) showing a maximum DoP=0.93 at $ν=737$ cm⁻¹. The main difference with respect to the previous case is that the initial DoCP is now about 0.1 (Fig. 2d). Thus the emitted radiation from the single Ga₂O₃ layer is elliptical.

Then we added a MoO₃ cap layer to both schemes and applied a second PSO varying the thickness $d₁$ and the relative orientation $α$ with respect to the bottom layer. For the first scheme an initial population of $N = 30$ particles (with $τA = [d₁, d₂A, α]$ and random $d₁$ and $α$) is generated. The loss function of each particle, defined as $y(r) = 1 - \text{DoP}$, is calculated and a predetermined iteration number of 150 is chosen. We found an optimal condition corresponding to a DoCP=0.87 at 691 cm⁻¹ when a $d₁A = 740$ nm α-MoO₃ layer, tilted by an angle $αA = 83°$ in the x-y plane with respect to the x'y' axes of the β-Ga₂O₃ layer is considered (sketch of Fig. 3a). All the results have been obtained with the indirect method, (i.e., calculating the absorptivity and applying Kirchhoff's law) and validated by the direct method based on fluctuation electrodynamics [38]. Fig. 3b shows a high peak of almost left circularly polarized (LCP) emission at $ν=691$ cm⁻¹. The corresponding DoP and DoCP are reported in Fig. 3c. We note that DoP=DoCP=0.87 at nearly the same frequency (see Fig. 3c). Finally, we report in Fig. 3d the value of $S₃/S₀$ as a function of the twist angle between the two layers and of the frequency. Its modulus is the DOCP, however, its sign determines the Right (if positive) or Left (if negative) circular polarization. The reported map clearly shows that at $ν=691$ cm⁻¹ it is possible to switch from almost perfect (DoCP=0.87) RCP to LCP by properly tuning the twist angle.

An analogous approach has been applied for the scheme B. Optimized performances are achieved for the second scheme by adding a $d₁B = 324$ nm thick layer of α-MoO₃ as sketched in Fig. 4a, rotated in the x-y plane by $αB = 114°$ with respect to the x'y' axes of the β-Ga₂O₃ layer. We note that in this case, we obtain a resulting DoP = DoCP = 0.85 (Fig. 4c) and the thermal emitted radiation at 737 cm⁻¹ is mostly left circularly polarized (Fig. 4b). However, right-hand circular polarization at 737 cm⁻¹ can be obtained similarly to the previous scheme, by varying the twist angle of the MoO₃ layer as shown in Fig. 4d.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Convergence plot of the particle swarm optimization process for scheme A (as illustrated in Fig. 3) and scheme B (as illustrated in Fig. 4).</figcaption></figure>

Fig. 5 illustrates the final optimization process of scheme A (Fig. 3) and scheme B (Fig. 4), denoted by the red solid line and black dashed line, respectively. The minimum obtained loss function among $N = 30$ particles is plotted as a function of iteration number. It can be observed that after the 25th iteration, the optimization reaches a converged value. Scheme A has a loss function of 0.13, corresponding to a DoCP of 0.87, while scheme B has a loss function of 0.15, indicating a DoCP of 0.85. We finally explore the robustness of the proposed schemes by varying the emission angle i.e. the azimuth angle ($φ$) and zenith angle ($θ$)) to show that the results are maintained over a large angular range for both schemes (Fig. 6a and 6b respectively).

Further evidence of the robustness of the bi-layer approach is reported in Fig. 6. We performed single parameters sweep (on Ga₂O₃ layer and on MoO₃ layer thickness) around the optimal conditions for scheme A (Fig. 7a,b) and scheme B (Fig. 7c,d). We show that high degree of circular polarization is maintained over a large range of the thickness of the layers corresponding to hundreds of nm. We note that scheme 1 allows for a narrow band emission with a good tunability of 5 cm⁻¹, varying the thickness of the Ga₂O₃ layer and maintaining a DoCP higher than 0.7. On the other hand, scheme B emits on a larger bandwidth and it is extremely robust with respect to tolerances on the layer's thickness.

## 5. Conclusion

We proposed a lithography-free method based on a double-twisted layer scheme to obtain circularly polarized thermal radiation in the mid-IR, at two specific frequencies related to β-Ga₂O₃ optical phonons excitation. The almost linearly polarized emission from the single slab is converted into circular polarization by adding a properly sized and tilted α-MoO₃ layer acting as a quarter-wavelength plate. In both cases, we take advantage of the strong natural anisotropy related to the low symmetry class of the investigated materials without the need for further processing techniques. The achieved degree of circular polarization is higher than 0.85 for the two proposed schemes and it is maintained over a broad range of angles of emission. We believe that this approach could lead to the development of low-cost, integrated thermal sources of circularly polarized light for IR sensing applications.

### Credit authorship contribution statement

**Marco Centini:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing.
**Chiyu Yang:** Conceptualization, Data curation, Formal analysis, Methodology, Software, Writing – original draft, Writing – review & editing.
**Maria Cristina Larciprete:** Formal analysis,
it very suitable to be used as a waveplate. Thus we focus our attention on the design of an efficient β-Ga₂O₃ emitting layer around 690 cm⁻¹ and 740 cm⁻¹ with a high DoP combined with a properly tilted and sized α-MoO₃ layer acting as waveplate to obtain a tailored thermal source with optimized DoCP.

## 4. Results and discussion

We employed particle swarm optimization (PSO), a search algorithm that finds optimal solutions by iteratively trying to improve candidate solutions, and finally obtained structures with the highest circular emission at specific frequencies. The independent parameters needed according to the proposed two-layer scheme are the thicknesses of the MoO₃ layer ($d₁$), of the Ga₂O₃ layer ($d₂$), and the twist angle between the two layers ($α$). These parameters are grouped into a vector, denoted as $r$, which serves as a particle. An initial population of N random particles is generated, and the loss function of each particle is calculated. Subsequently, the particles undergo iterative exploration of the parameter space to minimize the loss function until the predetermined iteration number is reached. More details of the particle swarm optimization algorithm can be found in the reference [43]. However, in order to reduce the number of optimization parameters and to avoid local minimum solutions for the two-layer system the particle swarm algorithm is performed in two separate steps. The first optimization is performed on a single β-Ga₂O₃ layer on gold substrate looking for maximum DOP. We set as the loss function $y(r) = 1 - \text{DoP}$ (thus $r = [d₂]$). We found two schemes, named scheme A and scheme B, corresponding to maximized DoP at two different frequencies: 691 cm⁻¹ and 737 cm⁻¹ respectively. In the first scheme, sketched in Fig. 2a, we note that a $d₂A = 606$ nm thick β-Ga₂O₃ layer has a maximum relative emissivity for a p-polarized field at 691 cm⁻¹ corresponding to a DoP of about 0.97 and zero DoCP (Fig. 2b). On the other hand, the second proposed scheme, sketched in Fig. 2c is based on a thicker Ga₂O₃ layer ($d₂B = 1909$ nm) showing a maximum DoP=0.93 at $ν=737$ cm⁻¹. The main difference with respect to the previous case is that the initial DoCP is now about 0.1 (Fig. 2d). Thus the emitted radiation from the single Ga₂O₃ layer is elliptical.

Then we added a MoO₃ cap layer to both schemes and applied a second PSO varying the thickness $d₁$ and the relative orientation $α$ with respect to the bottom layer. For the first scheme an initial population of $N = 30$ particles (with $τA = [d₁, d₂A, α]$ and random $d₁$ and $α$) is generated. The loss function of each particle, defined as $y(r) = 1 - \text{DoP}$, is calculated and a predetermined iteration number of 150 is chosen. We found an optimal condition corresponding to a DoCP=0.87 at 691 cm⁻¹ when a $d₁A = 740$ nm α-MoO₃ layer, tilted by an angle $αA = 83°$ in the x-y plane with respect to the x'y' axes of the β-Ga₂O₃ layer is considered (sketch of Fig. 3a). All the results have been obtained with the indirect method, (i.e., calculating the absorptivity and applying Kirchhoff's law) and validated by the direct method based on fluctuation electrodynamics [38]. Fig. 3b shows a high peak of almost left circularly polarized (LCP) emission at $ν=691$ cm⁻¹. The corresponding DoP and DoCP are reported in Fig. 3c. We note that DoP=DoCP=0.87 at nearly the same frequency (see Fig. 3c). Finally, we report in Fig. 3d the value of $S₃/S₀$ as a function of the twist angle between the two layers and of the frequency. Its modulus is the DOCP, however, its sign determines the Right (if positive) or Left (if negative) circular polarization. The reported map clearly shows that at $ν=691$ cm⁻¹ it is possible to switch from almost perfect (DoCP=0.87) RCP to LCP by properly tuning the twist angle.

An analogous approach has been applied for the scheme B. Optimized performances are achieved for the second scheme by adding a $d₁B = 324$ nm thick layer of α-MoO₃ as sketched in Fig. 4a, rotated in the x-y plane by $αB = 114°$ with respect to the x'y' axes of the β-Ga₂O₃ layer. We note that in this case, we obtain a resulting DoP = DoCP = 0.85 (Fig. 4c) and the thermal emitted radiation at 737 cm⁻¹ is mostly left circularly polarized (Fig. 4b). However, right-hand circular polarization at 737 cm⁻¹ can be obtained similarly to the previous scheme, by varying the twist angle of the MoO₃ layer as shown in Fig. 4d.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Convergence plot of the particle swarm optimization process for scheme A (as illustrated in Fig. 3) and scheme B (as illustrated in Fig. 4).</figcaption></figure>

Fig. 5 illustrates the final optimization process of scheme A (Fig. 3) and scheme B (Fig. 4), denoted by the red solid line and black dashed line, respectively. The minimum obtained loss function among $N = 30$ particles is plotted as a function of iteration number. It can be observed that after the 25th iteration, the optimization reaches a converged value. Scheme A has a loss function of 0.13, corresponding to a DoCP of 0.87, while scheme B has a loss function of 0.15, indicating a DoCP of 0.85. We finally explore the robustness of the proposed schemes by varying the emission angle i.e. the azimuth angle ($φ$) and zenith angle ($θ$)) to show that the results are maintained over a large angular range for both schemes (Fig. 6a and 6b respectively).

Further evidence of the robustness of the bi-layer approach is reported in Fig. 6. We performed single parameters sweep (on Ga₂O₃ layer and on MoO₃ layer thickness) around the optimal conditions for scheme A (Fig. 7a,b) and scheme B (Fig. 7c,d). We show that high degree of circular polarization is maintained over a large range of the thickness of the layers corresponding to hundreds of nm. We note that scheme 1 allows for a narrow band emission with a good tunability of 5 cm⁻¹, varying the thickness of the Ga₂O₃ layer and maintaining a DoCP higher than 0.7. On the other hand, scheme B emits on a larger bandwidth and it is extremely robust with respect to tolerances on the layer's thickness.

## 5. Conclusion

We proposed a lithography-free method based on a double-twisted layer scheme to obtain circularly polarized thermal radiation in the mid-IR, at two specific frequencies related to β-Ga₂O₃ optical phonons excitation. The almost linearly polarized emission from the single slab is converted into circular polarization by adding a properly sized and tilted α-MoO₃ layer acting as a quarter-wavelength plate. In both cases, we take advantage of the strong natural anisotropy related to the low symmetry class of the investigated materials without the need for further processing techniques. The achieved degree of circular polarization is higher than 0.85 for the two proposed schemes and it is maintained over a broad range of angles of emission. We believe that this approach could lead to the development of low-cost, integrated thermal sources of circularly polarized light for IR sensing applications.

### Credit authorship contribution statement

**Marco Centini:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing.
**Chiyu Yang:** Conceptualization, Data curation, Formal analysis, Methodology, Software, Writing – original draft, Writing – review & editing.
**Maria Cristina Larciprete:** Formal analysis,
<figure><img src="image_8.png" /><figcaption>Fig. 6. DoCP of thermal radiation emitted (a) at ν=691 cm⁻¹ for the structure sketched in Fig. 3a and (b) at ν=737 cm⁻¹ for the structure sketched in Fig. 4a as a function of angle (φ) and zenith angle (θ).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. DoCP as a function of the frequency and the (a) β-Ga2O3 layer thickness, (b) α-MoO3 layer thickness for scheme A. DoCP as a function of the frequency and the (c) β-Ga2O3 layer thickness, (d) α-MoO3 layer thickness for scheme B.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6. DoCP of thermal radiation emitted (a) at ν=691 cm⁻¹ for the structure sketched in Fig. 3a and (b) at ν=737 cm⁻¹ for the structure sketched in Fig. 4a as a function of angle (φ) and zenith angle (θ).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. DoCP as a function of the frequency and the (a) β-Ga2O3 layer thickness, (b) α-MoO3 layer thickness for scheme A. DoCP as a function of the frequency and the (c) β-Ga2O3 layer thickness, (d) α-MoO3 layer thickness for scheme B.</figcaption></figure>
Investigation, Visualization, Writing – original draft, Writing – review & editing. **Mauro Antezza:** Conceptualization, Supervision, Writing – original draft, Writing – review & editing. **Zhuomin M. Zhang:** Formal analysis, Investigation, Methodology, Supervision, Visualization, Writing – original draft, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgements

M.A. thanks the SAPIENZA University of Rome and the Department of Basic and Applied Sciences for Engineering for hospitality during his stay in Rome under the visiting professor program, where this work has been initiated. M.C, M.C.L, M.A. and Z.M.Z. acknowledge the KITP program ‘Emerging Regimes and Implications of Quantum and Thermal Fluctuational Electrodynamics’ 2022, where part of this work has been done. This research was supported in part by the National Science Foundation under Grant No. PHY-1748958. C.Y. was supported by the National Science Foundation (CBET-2029892).

References

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[3] Wood MF, Côté D, Vitkin IA. Combined optical intensity and polarization methodology for analyte concentration determination in simulated optically clear and turbid biological media. J Biomed Opt 2008;13(4):044037.

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[5] Chen Y, Francescato Y, Caldwell JD, Giannini V, Maß TWW, Glembocki OJ, Bezares FJ, Taubner T, Kasica R, Hong M, Maier SA. Spectral tuning of localized surface phonon polariton resonators for low-loss mid-IR applications. ACS Photonics 2014;1(8):718-24.

[6] Caldwell JD, et al. Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons. Nanophotonics 2015;4:44-68.

[7] Zhang ZM. Nano/microscale heat transfer. 2nd ed. Springer; 2020.

[8] Greffet J-J, Carminati R, Joulain K, Mulet J-P, Mainguy S, Chen Y. Coherent emission of light by thermal sources. Nature 2002;416(6876):61-4.

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[10] Lochbaum A, Fedoryshyn Y, Dorodnyy A, Koch U, Hafner C, Leuthold J. On-chip narrowband thermal emitter for mid-IR optical gas sensing. ACS Photonics 2017;4 (6):1371-80.

[11] Overvig AC, Mann SA, Alù A. Thermal metasurfaces: complete emission control by combining local and nonlocal light-matter interactions. Phys Rev X 2021;11: 021050.

[12] Centini M, Larciprete MC, Voti RLI, Bertolotti M, Sibilia C, Antezza M. Hybrid thermal Yagi-Uda nanoantennas for directional and narrow band long-wavelength IR radiation sources. Opt Express 2020;28:19334-48.

[13] Li C, Krachmalnicoff V, Bouchon P, Jaeck J, Bardou N, Haïdar R, de Wilde Y. Near-field and far-field thermal emission of an individual patch nanoantenna. Phys Rev Lett 2018;121(24):243901.

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Investigation, Visualization, Writing – original draft, Writing – review & editing. **Mauro Antezza:** Conceptualization, Supervision, Writing – original draft, Writing – review & editing. **Zhuomin M. Zhang:** Formal analysis, Investigation, Methodology, Supervision, Visualization, Writing – original draft, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgements

M.A. thanks the SAPIENZA University of Rome and the Department of Basic and Applied Sciences for Engineering for hospitality during his stay in Rome under the visiting professor program, where this work has been initiated. M.C, M.C.L, M.A. and Z.M.Z. acknowledge the KITP program ‘Emerging Regimes and Implications of Quantum and Thermal Fluctuational Electrodynamics’ 2022, where part of this work has been done. This research was supported in part by the National Science Foundation under Grant No. PHY-1748958. C.Y. was supported by the National Science Foundation (CBET-2029892).

References

[1] Hodgkinson J, Tatam RP. Optical gas sensing: a review. Meas Sci Technol 2013;24 (1):012004.

[2] Furstenberg R, Kendziora CA, Stepnowski J, Stepnowski SV, Rake M, Papantonakis MR, Nguyen V, Hubler GK, McGill RA. Stand-off detection of trace explosives via resonant infrared photothermal imaging. Appl Phys Lett 2008;93 (22):224103.

[3] Wood MF, Côté D, Vitkin IA. Combined optical intensity and polarization methodology for analyte concentration determination in simulated optically clear and turbid biological media. J Biomed Opt 2008;13(4):044037.

[4] F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Har-rington, A.D. Martino, D. Mawet, J. Riedi, and J.S. Tyo, in Polarization: measurement, analysis, and re-mote sensing XI, Vol. 9099, D. B. Chenault and D. H. Goldstein, International society for optics and photonics (SPIE, 2014) p. 90990.

[5] Chen Y, Francescato Y, Caldwell JD, Giannini V, Maß TWW, Glembocki OJ, Bezares FJ, Taubner T, Kasica R, Hong M, Maier SA. Spectral tuning of localized surface phonon polariton resonators for low-loss mid-IR applications. ACS Photonics 2014;1(8):718-24.

[6] Caldwell JD, et al. Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons. Nanophotonics 2015;4:44-68.

[7] Zhang ZM. Nano/microscale heat transfer. 2nd ed. Springer; 2020.

[8] Greffet J-J, Carminati R, Joulain K, Mulet J-P, Mainguy S, Chen Y. Coherent emission of light by thermal sources. Nature 2002;416(6876):61-4.

[9] Larciprete MC, Centini M, Voti RLI, Sibilia C. Selective and tunable thermal emission in metamaterials composed of oriented polar inclusions. J Opt Soc Am B 2017;34(7):1459-64.

[10] Lochbaum A, Fedoryshyn Y, Dorodnyy A, Koch U, Hafner C, Leuthold J. On-chip narrowband thermal emitter for mid-IR optical gas sensing. ACS Photonics 2017;4 (6):1371-80.

[11] Overvig AC, Mann SA, Alù A. Thermal metasurfaces: complete emission control by combining local and nonlocal light-matter interactions. Phys Rev X 2021;11: 021050.

[12] Centini M, Larciprete MC, Voti RLI, Bertolotti M, Sibilia C, Antezza M. Hybrid thermal Yagi-Uda nanoantennas for directional and narrow band long-wavelength IR radiation sources. Opt Express 2020;28:19334-48.

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