New spin-resolved thermal radiation laws for nonreciprocal bianisotropic media

We consider an extended finite-thickness multilayered planar slab much larger than the thermal wavelength at thermal equilibrium with the blackbody radiation at temperature T. As shown in figure 1(a), the power emitted per unit surface area (dA) of the body at an angle θ to the surface normal within the solid angle dΩ per unit frequency interval $\left(\mathrm{d}\omega \right)$ is given by,

$$\begin{equation}{P}_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(\omega ,\theta ,\phi ,\hat{\mathbf{e}}\right)=\eta \left(\omega ,\theta ,\phi ,\hat{\mathbf{e}}\right)\frac{{I}_{\mathrm{b}}\left(\omega ,T\right)}{2}\enspace \mathrm{cos}\enspace \theta \enspace \mathrm{d}\omega \enspace \mathrm{d}{\Omega}\enspace \mathrm{d}A,\end{equation} \tag{ 1 }$$

where η is the dimensionless emissivity dependent on frequency (ω), emission direction (θ, ϕ) in spherical coordinates and orthogonal polarization states ($\hat{\mathbf{e}}$). The angle ϕ ∈ [0, 2π] while the angle θ ∈ [0, π] such that $\theta \in \left[\right.0,\pi /2\left.\right)$ denotes the top hemisphere and θ ∈ [π/2, π] denotes the bottom hemisphere. For a semi-infinite half-space, θ ∈ [0, π/2]. The blackbody radiance at temperature T given by I_b(ω, T) = ω²Θ(ω, T)/(4π³ c²) with Θ(ω, T) = ℏω/[exp(ℏω/k_B T) − 1] being the Planck's function, is divided by two to account for two polarization states separately. Similarly, the power absorbed per unit surface area dA due to the blackbody radiance incident at an angle θ to the surface normal within a solid angle dΩ per unit frequency interval dω is,

$$\begin{equation}{P}_{\text{abs}}\left(\omega ,\theta ,\phi ,\hat{\mathbf{e}}\right)=\alpha \left(\omega ,\theta ,\phi ,\hat{\mathbf{e}}\right)\frac{{I}_{\mathrm{b}}\left(\omega ,T\right)}{2}\enspace \mathrm{cos}\enspace \theta \enspace \mathrm{d}\omega \enspace \mathrm{d}{\Omega}\enspace \mathrm{d}A.\end{equation} \tag{ 2 }$$

We focus on polarization-dependent properties of thermal emission. Instead of the usual s, p-polarization basis, we consider RCP (${\hat{\mathbf{e}}}_{+}$) and LCP (${\hat{\mathbf{e}}}_{-}$) polarization states. The circular polarization basis states in the basis of s, p polarization states (${\hat{\mathbf{e}}}_{\mathrm{s}},{\hat{\mathbf{e}}}_{\mathrm{p}}$) are ${\hat{\mathbf{e}}}_{{\pm}}=\left({\hat{\mathbf{e}}}_{\mathrm{s}}{\pm}\mathrm{i}{\hat{\mathbf{e}}}_{\mathrm{p}}\right)/\sqrt{2}$.

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We derive (as explained in detail in the discussion section) and analyze the spin-resolved emissivities ${\eta }_{\left({\pm}\right)}=\eta \left(\omega ,\theta ,\phi ,{\hat{\mathbf{e}}}_{{\pm}}\right)$ and absorptivities ${\alpha }_{\left({\pm}\right)}=\alpha \left(\omega ,\theta ,\phi ,{\hat{\mathbf{e}}}_{{\pm}}\right)$ defined above for planar configurations of several material classes. As a superset of materials, we consider a linear, time-invariant generic bianisotropic medium whose optical properties are described using the following constitutive relations in the frequency domain assuming local material response:

$$\begin{equation}\mathbf{D}=\bar{\bar{\varepsilon }}{\varepsilon }_{0}\mathbf{E}+\bar{\bar{\xi }}\frac{1}{c}\mathbf{H},\qquad \mathbf{B}=\bar{\bar{\zeta }}\frac{1}{c}\mathbf{E}+\bar{\bar{\mu }}{\mu }_{0}\mathbf{H},\end{equation} \tag{ 3 }$$

where $\bar{\bar{\varepsilon }},\bar{\bar{\mu }}$ are dimensionless permittivity and permeability tensors and $\bar{\bar{\xi }},\bar{\bar{\zeta }}$ are magneto-electric coupling tensors. Such a material is reciprocal if these parameters satisfy the conditions $\bar{\bar{\varepsilon }}={\bar{\bar{\varepsilon }}}^{\mathrm{T}},\enspace \bar{\bar{\mu }}={\bar{\bar{\mu }}}^{\mathrm{T}},\enspace \bar{\bar{\xi }}=-{\bar{\bar{\zeta }}}^{\mathrm{T}}$ where (...)^T denotes the matrix transpose. It is nonreciprocal if any one of these conditions is violated. We consider a list of several material classes as shown in table 1, which includes isotropic, uni/biaxial anisotropic, gyroelectric, gyromagnetic and magneto-electric materials. We discover that for multilayered or composite planar slabs of most of these material classes, the spin-resolved emissivities (η_(±)) and absorptivities (α_(±)) satisfy specific relations described below and summarized in figure 1. They are called as spin-resolved Kirchhoff's laws (SKLs).

Table 1. Reciprocal and nonreciprocal material classes.

No.	Material class	Description	Examples	Kirchhoff's law
1	Reciprocal	$\bar{\bar{\varepsilon }},\bar{\bar{\mu }}$ are scalars, $\bar{\bar{\xi }}=\bar{\bar{\zeta }}=0$	Common dielectric and metallic materials	SKL-1,2,3
	isotropic
2	Reciprocal	Diagonal $\bar{\bar{\varepsilon }}$ with unequal entries,	Uniaxial and biaxial crystals [19]	SKL-1
	anisotropic	scalar $\bar{\bar{\mu }}$, $\bar{\bar{\xi }}=\bar{\bar{\zeta }}=0$.
3	Nonreciprocal	$\bar{\bar{\varepsilon }}\ne {\bar{\bar{\varepsilon }}}^{\mathrm{T}}$, scalar $\bar{\bar{\mu }}$, $\bar{\bar{\xi }}=\bar{\bar{\zeta }}=0$.	Weyl semimetals [26], metals and	SKL-2, 3
	gyroelectric		semiconductors in magnetic field [23–25]
4	Nonreciprocal	$\bar{\bar{\mu }}\ne {\bar{\bar{\mu }}}^{\mathrm{T}}$, scalar $\bar{\bar{\varepsilon }}$, $\bar{\bar{\xi }}=\bar{\bar{\zeta }}=0$.	Ferromagnets, ferrites [28]	SKL-2, 3
	gyromagnetic
5	Reciprocal	Scalar $\bar{\bar{\varepsilon }}$ and $\bar{\bar{\mu }}$, nonzero $\bar{\bar{\xi }}=-{\bar{\bar{\zeta }}}^{\mathrm{T}}$.	Chiral media, metamaterials [21, 22]	SKL-1
	magneto-electric
6	Nonreciprocal	Scalar $\bar{\bar{\varepsilon }}$ and $\bar{\bar{\mu }}$, nonzero $\bar{\bar{\xi }}={\bar{\bar{\zeta }}}^{\mathrm{T}}$.	Topological insulator [30], multi-ferroic	SKL-2, 3
	magneto-electric		media [29] and heterostructures [31]

SKL-1: for multilayered or composite planar slabs of all reciprocal media, the spin-resolved emissivity in (θ, ϕ) direction is equal to the spin-resolved absorptivity in the same direction for each spin (polarization) state.

$$\begin{equation}\begin{aligned}\hfill {\eta }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)\\ \hfill {\eta }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(-\right)}\left(\omega ,\theta ,\phi \right).\end{aligned}\end{equation} \tag{ 4 }$$

SKL-2: for multilayered or composite planar slabs of nonreciprocal media that preserve the rotational symmetry in the plane of the surface such as gyrotropic media with gyrotropy axis perpendicular to the surface and isotropic magneto-electric (Tellegen) media, the spin-resolved emissivity in (θ, ϕ) direction is equal to the spin-resolved absorptivity in the same direction but of opposite spin state.

$$\begin{equation}\begin{aligned}\hfill {\eta }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)\\ \hfill {\eta }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(+\right)}\left(\omega ,\theta ,\phi \right).\end{aligned}\end{equation} \tag{ 5 }$$

SKL-3: for multi-layered or composite planar slabs of nonreciprocal media that do not preserve the rotational symmetry in the plane of the surface such as gyrotropic media with gyrotropy axis parallel to the surface and anisotropic nonreciprocal magneto-electric media that cause cross coupling between perpendicular and parallel components of E − H fields (E_x − H_z , E_z − H_x , E_y − H_z , E_z − H_y ), the spin-resolved emissivity in the direction (θ, ϕ) is equal to the spin-resolved absorptivity for the conjugate direction (θ, ϕ + π) for each spin state.

$$\begin{equation}\begin{aligned}\hfill {\eta }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(+\right)}\left(\omega ,\theta ,\phi +\pi \right)\\ \hfill {\eta }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)={\alpha }_{\left(-\right)}\left(\omega ,\theta ,\phi +\pi \right).\end{aligned}\end{equation} \tag{ 6 }$$

We emphasize that a multilayered or a composite planar slab which combines different material types following the same SKL also follows that same SKL. For instance, a planar slab which combines materials exhibiting gyrotropic and magnetoelectric response simultaneously, will follow SKL-2 or 3 depending on the gyrotropy axis and the isotropic or anisotropic nature of the magnetoelectric response as described above. For conventional isotropic dielectric/metallic materials, all three laws hold since the spin-resolved emissivities and absorptivities are equal [η₍₊₎ = η₍₋₎ = α₍₊₎ = α₍₋₎] in all relevant directions. It follows that a multilayered planar slab which contains layers of such trivial materials (satisfying all three SKLs) along with more exotic bianisotropic media satisfying a specific SKL can be readily described using that specific SKL. However, the laws are not applicable when the nontrivial bianisotropic media following distinct SKLs are combined. One example can be a planar slab containing layers of uniaxial anisotropic media (SKL-1) and gyrotropic media (SKL-2 or 3). Interestingly, we also find that a nonreciprocal magnetoelectric medium that leads to coupling between the parallel components of E − H fields (E_x − H_y or E_y − H_x ) does not satisfy any one of these three SKLs. This special material is well-known in the literature for causing Fiegel effect and related phenomena [37–39]. Its uniqueness is related to the momentum asymmetry of light propagation inside this medium and it will be analyzed in more detail in our future work. Despite the few limiting cases, we note that SKLs are useful for thermal-radiation engineering using multi-layered or composite configurations of most bianisotropic materials, for all frequencies, emission directions, material and geometry parameters. In the following, we propose an experiment to verify these laws using a single material system.

We consider a doped indium antimonide (InSb) slab of thickness t = 1 μm and doping concentration n = 10¹⁷ cm⁻³ (available at vendors like MTI Corp.) on top of a glass substrate of constant permittivity ɛ_s = 2.25. We use the experimentally well-characterized Drude–Lorentz oscillator model [40–42] to calculate the permittivity of InSb with or without applied magnetic field [17]. In the absence of magnetic field, InSb slab has isotropic permittivity ($\bar{\bar{\varepsilon }}$) and permeability ($\bar{\bar{\mu }}$), and $\bar{\bar{\xi }},\bar{\bar{\zeta }}=0$. Therefore, it is reciprocal in nature and follows SKL-1. In the presence of magnetic field, $\bar{\bar{\varepsilon }}$ has nonzero off-diagonal entries and InSb slab acts as a gyroelectric medium with the gyrotropy axis parallel to the applied field. Consequently, when the magnetic field is perpendicular or parallel to the surface, thermal radiation from InSb slab will follow either SKL-2 or SKL-3 respectively. In an experiment, the measurement of spin-resolved emissivities (equation (9)) and absorptivities (equation (10)) can be performed using a specific combination of polarizers and quarter-wave-plate optical components [5] in suitable directions of emission. Here, we calculate them as a function of frequency.

Figure 2 summarizes the three spin-resolved Kirchhoff's laws for the InSb slab (top schematic) and figures 2(a)–(c) demonstrate the calculated spectra of emissivities (left figures) and absorptivities (right figures). For brevity, we focus on the direction (θ = π/4, ϕ = π/4). As shown in figure 2(a), in the absence of magnetic field, the spin-resolved emissivities and absorptivities are equal for both spin states, RCP (red) and LCP (blue). The plots lie on top of each other. Figure 2(b) demonstrates SKL-2, where a magnetic field of strength 1 T is applied perpendicular to the slab. Evidently, η₍₊₎ = α₍₋₎ and η₍₋₎ = α₍₊₎, which verifies SKL-2. Since η₍₊₎ ≠ η₍₋₎ in the given emission direction, the thermal emission from the slab will be partially spin-polarized. In figure 2(c), we consider a magnetic field of strength 2 T applied along the x-axis of the geometry. This example demonstrates that spin-resolved emissivities in (θ, ϕ) direction are equal to spin-resolved absorptivities for the same spin state but in the conjugate (θ, ϕ + π) direction. Thus, SKL-3 can be verified.

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In the context of spin-resolved thermal radiation, it is illuminating to answer two fundamental questions. First, what materials can emit spin-polarized or circularly polarized thermal radiation? And second, what materials can experience nontrivial torque and force upon thermal emission of spin-polarized photons? We answer both these questions with a universal perspective of all material classes listed in the table 1. Note that we consider nonequilibrium situation unlike equilibrium thermal radiation for derivation of SKLs. We focus on thermal emission from the planar slab at temperature T₀ into the surrounding vacuum at T = 0 K. The strength of the circular polarization of thermal emission is characterized using the well-known experimentally accessible circular dichroism parameter or Stokes S₃ parameter defined as:

$$\begin{equation}{S}_{3}\left(\omega ,\theta ,\phi \right)=\frac{{\eta }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)-{\eta }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)}{{\eta }_{\left(+\right)}\left(\omega ,\theta ,\phi \right)+{\eta }_{\left(-\right)}\left(\omega ,\theta ,\phi \right)}.\end{equation} \tag{ 7 }$$

S₃ = +1 denotes pure RCP light while S₃ = −1 denotes pure LCP light in the given emission direction. When the planar slab emits thermal radiation into the environment at lower temperature, it loses linear and angular momentum carried away by the photons, and consequently experiences force and torque respectively. We compute the spectral force per unit area (F_s given by equation (18)) and the spectral torque per unit area ( τ _s given by equation (19)) experienced by the planar slab due to the emission of photons of frequency ω by integrating over all emission directions. The derivation of these quantities is provided in section 3.

Figure 3 summarizes all materials that can emit partially spin-polarized thermal radiation in suitable directions. For illustration, we consider a planar slab of a material characterized by $\bar{\bar{\varepsilon }},\enspace \bar{\bar{\mu }},\enspace \bar{\bar{\xi }},\enspace \bar{\bar{\zeta }}$ at a temperature T₀ emitting into the surrounding vacuum at T = 0 K. Similar to our previous work [17], we provide a universal perspective by analyzing the representative examples of several material classes using reasonable values of the material parameters evaluated at a frequency ω and importantly, satisfying the thermodynamic passivity constraint [43]. While the constitutive relations in equation (3) can, in principle, be used to describe optically active gain media, our radiometry analysis is applicable for passive media based on the detailed balance at thermal equilibrium. Hence, passivity of the medium must be ensured [17]. We assume the thickness of the planar slab to be d = 0.5c/ω. The contour plot for each representative example demonstrates the variation of the Stokes S₃ parameter as a function of the emission direction characterized by (θ, ϕ) in spherical coordinates. The calculated spectral force and torque per unit area provided for each example reveal the zero or nonzero values and the directions of these quantities for all materials belonging to that particular class. While there are many new interesting results in this figure, a detailed discussion involving frequency-dependent response of real materials of each type is beyond the scope of the present work. Nonetheless, this universal perspective is useful because it provides a signature of new interesting possibilities which can then be explored in more detail in the future.

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Circularly polarized thermal emission: first, we note a surprising result that there are many materials whose planar slabs can emit partially spin-polarized thermal radiation. This is evident from the nonzero values of S₃ parameters demonstrated using contour plots in figure 3 for several material classes. This includes all nonreciprocal planar media and also reciprocal media like uni/bi-axial anisotropic materials and reciprocal magnetoelectric (chiral) media. For uni-axial anisotropic media, we assume anisotropy to be in the plane of the slab surface (xy-plane). Note that all materials belonging to these material classes except artificial metamaterials do not require any surface patterning or geometric chirality which is essential for observing circularly polarized thermal radiation in previous experiments [4, 5]. Another result is the subtle difference between spin-resolved emission from gyrotropic media and that from nonreciprocal magnetoelectric media. Particularly, gyrotropic media with gyrotropy axis perpendicular to surface (examples 3, 4, 7) and nonreciprocal isotropic magnetoelectric (Tellegen) media (example 13) both satisfy SKL-2 as discussed above. However, for magnetoelectric media, the spin-resolved emission is symmetric with respect to top (θ ∈ [0, π/2)) and bottom hemispheres (θ ∈ [π/2, π]) whereas it is asymmetric for gyrotropic media. The asymmetry can be predicted based on the microscopic picture of underlying cyclotron motion. If the thermally or statistically averaged cyclotron motion of electrons is anti-clockwise in the xy-plane (perpendicular gyrotropy axis) as viewed from +z-direction, the resulting emission should be RCP along +z-direction and LCP along −z-direction.

We also point out that the material classes lead to unique dependence of S₃ on the emission directions which is evident from the contour plots. This can be informative from the perspective of classifying or identifying materials using infrared imaging polarimetry in an actual experiment.

Nonequilibrium torque: we find that all gyrotropic planar slabs experience a net nonzero torque along the direction of the gyrotropy axis (examples 3–8) because of the overall loss of angular momentum via emission of spin-polarized photons. For isotropic magnetoelectric (Tellegen and Pasteur) media (examples 9, 13), the total spectral torque per unit area is zero which follows from the symmetry in top and bottom hemispheres discussed above. For other materials, the net torque is zero because of the cancellation upon integration over the angle ϕ. Note that we have focused on the planar slabs of individual material classes for illustration.

For multilayered configurations combining different material classes, a net nonzero torque can be obtained along a specific direction even though it is absent for a single-layered slab. For instance, if the isotropic magnetoelectric medium (examples 9, 13) is deposited on top of another thin film of common dielectric or metallic material, a net nonzero torque along z-axis can be obtained because of the asymmetric overall emission of spin-polarized photons in top and bottom hemispheres. We provide these results for multilayered slabs in the supplement. Based on a universal perspective, we identify material classes required for engineering perpendicular and parallel components of the nonequilibrium torque. We find that only materials satisfying SKL-2 (examples 3, 4, 7, 13) and isotropic reciprocal magnetoelectric (Pasteur) media (example 9) can experience perpendicular nonequilibrium torque in a multilayered configuration. Only materials satisfying SKL-3 (examples 5, 6, 8, 15, 16) and anisotropic reciprocal magnetoelectric media (examples 11, 12) can experience parallel nonequilibrium torque in a multilayered configuration.

Nonequilibrium force: figure 3 reveals that for a single-layered geometry, only a reciprocal magnetoelectric planar slab that causes cross coupling between E_x − H_y and E_y − H_x fields (example 10), can lead to a nonzero nonequilibrium force along z-axis. For other materials, the symmetry of overall thermal emission (summed over both spin states) in top and bottom hemispheres leads to cancellation of the resulting force along z-axis. The figure also reveals that only planar slabs of anisotropic nonreciprocal magnetoelectric materials (examples 15, 16) can experience nonequilibrium force parallel to the surface whose direction is determined by the nature of the magneto-electric coupling. If the magnetoelectric tensors ($\bar{\bar{\xi }},\enspace \bar{\bar{\zeta }}$) indicate (E_x − H_z , E_z − H_x )-type cross coupling, the net force is along y-axis. If they indicate (E_y − H_z , E_z − H_y )-type cross coupling, this force is along x-axis. The multilayered slabs of other material classes can also lead to nonequilibrium force as discussed in the supplement. Based on a universal perspective, we find that all material classes can experience a perpendicular nonequilibrium force in a multilayered configuration and only materials satisfying SKL-3 can experience a parallel nonequilibrium force in a multilayered configuration. These results indicate the possibility of engineering thermal-nonequilibrium forces and torques in planar geometries and identifying or classifying materials based on experimentally measurable characteristics.

First, we derive the spin-resolved emissivities and absorptivities for a semi-infinite half-space because of its simplicity and connection with the fluctuational electrodynamic theory as explained in the supplement. We then extend the derivation for a finite-thickness planar slab. Since we assume linear material response and regular reflection from the surface, the frequency ω and the angle θ are not changed upon reflection. Hence, we can apply the principle of total energy conservation for energy-exchange channels characterized by ω, θ separately and focus on the polarization or spin-dependent properties. The flux rates are given by equations (1) and (2).

Figure 4(b) depicts the spin-resolved energy flux rates (RCP and LCP radiation) at frequency ω in the direction (θ, ϕ) which contain emitted, reflected and incident radiation. The fluxes are described at thermal equilibrium within the radiometry paradigm [44]. As shown in the rightmost figure, the incident radiation contains both RCP (I₍₊₎) and LCP (I₍₋₎) radiation. Since the incident blackbody radiation is unpolarized, it follows that I₍₊₎ = I₍₋₎ = I_b/2. The emitted radiation is described using the spin-resolved emissivities as η₍₊₎ I₍₊₎ (RCP) and η₍₋₎ I₍₋₎ (LCP). The reflected radiation arises from the radiation incident along the conjugate direction (θ, ϕ + π) and is described using the polarization interconversion reflectances. For instance, as shown in the second figure in figure 4(b), the incident RCP radiation (I₍₊₎) gets reflected as R₍₊₊₎(θ, ϕ + π)I₍₊₎ (RCP) and R₍₋₊₎(θ, ϕ + π)I₍₊₎ (LCP) along (θ, ϕ) direction. Note that we have characterized directions based on the emitted radiation (indicated by arrows in (a)) and not the incident radiation. This avoids the ambiguity in characterizing directions for a thin film (finite-thickness slab described below) where transmitted radiation is also taken into account. The interconversion reflectances can be measured separately in an experiment or can be calculated for the planar slab using the well-known Fresnel reflection coefficients (r_ss, r_sp, r_ps, r_pp) using the following expressions (omitting the dependence on θ, ϕ for brevity):

$$\begin{equation}\begin{aligned}\hfill {R}_{\left(++/--\right)}=\vert \left({r}_{\mathrm{s}\mathrm{s}}+{r}_{\mathrm{p}\mathrm{p}}\right){\pm}\mathrm{i}\left({r}_{\mathrm{s}\mathrm{p}}-{r}_{\mathrm{p}\mathrm{s}}\right){\vert }^{2}/4\\ \hfill {R}_{\left(-+/+-\right)}=\vert \left({r}_{\mathrm{s}\mathrm{s}}-{r}_{\mathrm{p}\mathrm{p}}\right){\pm}\mathrm{i}\left({r}_{\mathrm{s}\mathrm{p}}+{r}_{\mathrm{p}\mathrm{s}}\right){\vert }^{2}/4.\end{aligned}\end{equation} \tag{ 8 }$$

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The Fresnel reflection coefficient r_jk for j, k = [s, p] denotes the amplitude of j-polarized reflected wave due to unit amplitude k-polarized incident wave. Since the overall radiation in the far-field is isotropic and unpolarized at thermal equilibrium, the spin-resolved energy flux rates in opposite directions are equal. This argument is justified by our previous work [17] which shows that the equilibrium spin angular momentum density and the Poynting flux of thermal radiation are always zero in the far-field although they can be nonzero in the near-field of certain nonreciprocal media. By equating the incoming and outgoing spin-resolved radiation flux rates in the far-field and using I₍₊₎ = I₍₋₎ = I_b/2, we obtain the following spin-resolved emissivities in terms of reflectances:

$$\begin{equation}\begin{aligned}\hfill {\eta }_{\left(+\right)}\left(\theta ,\phi \right)=1-{R}_{\left(++\right)}\left(\theta ,\phi +\pi \right)-{R}_{\left(+-\right)}\left(\theta ,\phi +\pi \right)\\ \hfill {\eta }_{\left(-\right)}\left(\theta ,\phi \right)=1-{R}_{\left(--\right)}\left(\theta ,\phi +\pi \right)-{R}_{\left(-+\right)}\left(\theta ,\phi +\pi \right).\end{aligned}\end{equation} \tag{ 9 }$$

Similarly, it is straightforward to obtain the spin-resolved absorptivities by considering the reflection of incident polarized radiation I_(±) separately. By subtracting the reflected flux from the incident radiation flux, we obtain the following spin-resolved absorptivities:

$$\begin{equation}\begin{aligned}\hfill {\alpha }_{\left(+\right)}\left(\theta ,\phi \right)=1-{R}_{\left(++\right)}\left(\theta ,\phi \right)-{R}_{\left(-+\right)}\left(\theta ,\phi \right)\\ \hfill {\alpha }_{\left(-\right)}\left(\theta ,\phi \right)=1-{R}_{\left(--\right)}\left(\theta ,\phi \right)-{R}_{\left(+-\right)}\left(\theta ,\phi \right).\end{aligned}\end{equation} \tag{ 10 }$$

We show in the supplement that the emissivities (equation (9)) can be obtained within the scattering matrix formulation of fluctuational electrodynamics theory. We further validate this derivation by proving the thermodynamic consistency condition of net zero exchange of linear and angular momentum of the planar slab with the environment at thermal equilibrium. But before that, we extend this analysis to obtain the spin-resolved emissivities and absorptivities for a planar slab of finite thickness.

For a finite-thickness slab surrounded by vacuum on both sides, the incident radiation, in addition to getting reflected, also gets transmitted to the other side of the slab. It is straightforward to obtain the emissivities in terms of reflectances and transmittances using similar energy-balance considerations. In particular, as shown in figure 5, the radiation in the direction (θ, ϕ) contains emitted, transmitted, reflected and incident photons. The transmitted radiation in (θ, ϕ) direction arises from the radiation incident in the direction (π − θ, ϕ + π) on the other side as depicted in figure 5(b). The directions are consistently characterized based on the emission directions and not the directions of the incidence or the associated wavevectors. The transmittances are calculated from the associated Fresnel transmission coefficients (t_ss, t_sp, t_ps, t_pp) using the expressions (omitting the dependence on θ, ϕ for brevity):

$$\begin{equation}\begin{aligned}\hfill {T}_{\left(++/--\right)}=\frac{1}{4}\vert \left({t}_{\mathrm{s}\mathrm{s}}+{t}_{\mathrm{p}\mathrm{p}}\right){\pm}\mathrm{i}\left({t}_{\mathrm{s}\mathrm{p}}-{t}_{\mathrm{p}\mathrm{s}}\right){\vert }^{2}\\ \hfill {T}_{\left(-+/+-\right)}=\frac{1}{4}\vert \left({t}_{\mathrm{s}\mathrm{s}}-{t}_{\mathrm{p}\mathrm{p}}\right){\pm}\mathrm{i}\left({t}_{\mathrm{s}\mathrm{p}}+{t}_{\mathrm{p}\mathrm{s}}\right){\vert }^{2}.\end{aligned}\end{equation} \tag{ 11 }$$

The transmission coefficient t_jk for j, k = [s, p] denotes the amplitude of j-polarized transmitted wave due to unit amplitude k-polarized incident wave. Equating outgoing and incoming spin-resolved flux rates in the opposite directions in the far-field, we obtain the following expressions for the spin-resolved emissivities:

$$\begin{equation}\begin{aligned}\hfill {\eta }_{\left(+\right)}\left(\theta ,\phi \right)& =1-{R}_{\left(+-\right)}\left(\theta ,\phi +\pi \right)-{R}_{\left(++\right)}\left(\theta ,\phi +\pi \right)-{T}_{\left(+-\right)}\left(\pi -\theta ,\phi +\pi \right)-{T}_{\left(++\right)}\left(\pi -\theta ,\phi +\pi \right)\hfill \\ \hfill {\eta }_{\left(-\right)}\left(\theta ,\phi \right)& =1-{R}_{\left(-+\right)}\left(\theta ,\phi +\pi \right)-{R}_{\left(--\right)}\left(\theta ,\phi +\pi \right)-{T}_{\left(-+\right)}\left(\pi -\theta ,\phi +\pi \right)-{T}_{\left(--\right)}\left(\pi -\theta ,\phi +\pi \right).\hfill \end{aligned}\end{equation} \tag{ 12 }$$

Similarly, the spin-resolved absorptivities are obtained by subtracting the reflected and transmitted flux rates from the incident polarized flux rates.

$$\begin{equation}\begin{aligned}\hfill {\alpha }_{\left(+\right)}\left(\theta ,\phi \right)& =1-{R}_{\left(-+\right)}\left(\theta ,\phi \right)-{R}_{\left(++\right)}\left(\theta ,\phi \right)-{T}_{\left(-+\right)}\left(\theta ,\phi \right)-{T}_{\left(++\right)}\left(\theta ,\phi \right)\hfill \\ \hfill {\alpha }_{\left(-\right)}\left(\theta ,\phi \right)& =1-{R}_{\left(+-\right)}\left(\theta ,\phi \right)-{R}_{\left(--\right)}\left(\theta ,\phi \right)-{T}_{\left(+-\right)}\left(\theta ,\phi \right)-{T}_{\left(--\right)}\left(\theta ,\phi \right).\hfill \end{aligned}\end{equation} \tag{ 13 }$$

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Detailed balance of linear and angular momentum exchange: we now demonstrate the theoretical consistency of the above derivation by showing detailed balance of linear and angular momentum exchange between the slab and its surroundings which must be maintained at thermal equilibrium. Since angular momentum of thermal radiation [33, 35, 45] remains a largely unexplored topic, we explain it in detail. A circularly polarized plane wave in vacuum normalized to describe a single photon carries linear momentum of ℏω/c and angular momentum of ±ℏ along its direction of propagation. It carries only spin angular momentum since the orbital contributions are zero. Intrinsic orbital angular momentum is zero because of the plane wavefront and extrinsic orbital angular momentum is zero because of the cancellation over its infinite transverse extent [46, 47]. It also follows from the spin angular momentum density expression [∼Im{E*(ω) × E(ω)}] that there are no cross-polarization interaction terms in the net angular momentum when the radiation consists of both RCP (${\hat{\mathbf{e}}}_{\left(+\right)}$) and LCP (${\hat{\mathbf{e}}}_{\left(-\right)}$) photons. Therefore, similar to the energy flux, the angular momentum flux of thermal photons can be considered separately for both polarization states. Based on the definitions of spin angular momentum density [46, 48] and the ${\hat{\mathbf{e}}}_{\mathrm{s}},\enspace {\hat{\mathbf{e}}}_{\mathrm{p}}$-polarization vectors (see equation (22)), it follows that the angular momentum carried by a single RCP (${\hat{\mathbf{e}}}_{\left(+\right)}$) photon is −ℏ and that carried by a single LCP photon (${\hat{\mathbf{e}}}_{\left(-\right)}$) is +ℏ along the propagation direction. Calculating the photon number flux rates using equations (1) and (2), we obtain both linear and angular momentum flux rates for a finite-thickness planar slab. The rates for a semi-infinite half-space are obtained by choosing zero transmittances.

The rate of change of linear momentum dp/dt along +z-direction of the planar slab due to absorption, reflection, and transmission of photons incident along (θ, ϕ) direction and emitted by the slab along the same direction is given below (abbreviating dM = |cos θ|dω dΩ dA in equation (1) for simplicity):

$$\begin{equation}\frac{\mathrm{d}{p}_{z}}{\mathrm{d}t}\left(\theta ,\phi \right)=\left[\text{absorption}{\overbrace{-\left({\alpha }_{\left(+\right)}+{\alpha }_{\left(-\right)}\right){I}_{\mathrm{b}}\left({T}_{\text{env}}\right)}}-\text{emission}{\overbrace{\left({\eta }_{\left(+\right)}+{\eta }_{\left(-\right)}\right){I}_{\mathrm{b}}\left({T}_{0}\right)}}+\sum _{j,k=\left(+,-\right)}\underset{\text{reflection}}{\underbrace{-2{R}_{\left(jk\right)}{I}_{\mathrm{b}}\left({T}_{\text{env}}\right)}}\right]\frac{1}{2c}\enspace \mathrm{cos}\theta dM,\end{equation} \tag{ 14 }$$

where T₀ is the temperature of the slab and T_env is the temperature of the environment. The dependence of various quantities on (ω, θ, ϕ) is assumed and not mentioned above for brevity. As an explanation of various terms, we consider incident RCP(+) radiation in the top hemisphere and find the linear momentum change of the slab along +z-axis. It follows that α₍₊₎ portion gets absorbed imparting momentum $\propto -\mathrm{cos}\enspace \theta {\alpha }_{+}{I}_{\mathrm{b},{T}_{\text{env}}}\hslash \omega /c$, R₍₊₊₎ + R₍₋₊₎ portion gets reflected in the positive z-direction imparting momentum $\propto -2\enspace \mathrm{cos}\enspace \theta \left({R}_{\left(++\right)}+{R}_{\left(-+\right)}\right){I}_{\mathrm{b},{T}_{\text{env}}}\hslash \omega /c$, and T₍₊₊₎ + T₍₋₊₎ portion gets transmitted to the bottom hemisphere without imparting any momentum to the slab. Because of the emission of RCP radiation in the same direction, momentum $\propto -\mathrm{cos}\enspace \theta {\eta }_{\left(+\right)}{I}_{\mathrm{b},{T}_{0}}$ is imparted to the slab. Negative sign corresponds to the loss of the linear momentum by the slab.

Similarly, we obtain the rate of change of angular momentum (dJ/dt) of the planar slab whose z-component is:

$$\begin{align}\hfill \frac{\mathrm{d}{J}_{z}}{\mathrm{d}t}\left(\theta ,\phi \right)=& \left[\text{absorption}{\overbrace{-\left({\alpha }_{\left(-\right)}-{\alpha }_{\left(+\right)}\right){I}_{\mathrm{b},{T}_{\text{env}}}}}-\text{emission}{\overbrace{\left({\eta }_{\left(-\right)}-{\eta }_{\left(+\right)}\right){I}_{\mathrm{b},{T}_{0}}}}\right.\hfill \\ \hfill & \left.-\enspace 2\left(\underset{\text{reflection}}{\underbrace{{R}_{\left(--\right)}-{R}_{\left(++\right)}}}+\underset{\text{transmission}}{\underbrace{{T}_{\left(+-\right)}-{T}_{\left(-+\right)}}}\right){I}_{\mathrm{b},{T}_{\text{env}}}\right]\frac{1}{2\omega }\enspace \mathrm{cos}\enspace \theta \enspace \mathrm{d}M.\hfill \end{align} \tag{ 15 }$$

The x-components of both linear and angular momentum transfer rates are:

$$\begin{align}\hfill \frac{d{p}_{x}}{\mathrm{d}t}\left(\theta ,\phi \right)& =\left[-\left({\alpha }_{\left(+\right)}+{\alpha }_{\left(-\right)}\right){I}_{b,{T}_{\text{env}}}-\left({\eta }_{\left(+\right)}+{\eta }_{\left(-\right)}\right){I}_{b,{T}_{0}}\right]\frac{1}{2c}\enspace \mathrm{sin}\theta \enspace \mathrm{cos}\phi dM\hfill \end{align} \tag{ 16 }$$

$$\begin{align}\hfill \frac{\mathrm{d}{J}_{x}}{\mathrm{d}t}\left(\theta ,\phi \right)& =\left[-\left({\alpha }_{\left(-\right)}-{\alpha }_{\left(+\right)}\right){I}_{\mathrm{b},{T}_{\text{env}}}-\left({\eta }_{\left(-\right)}-{\eta }_{\left(+\right)}\right){I}_{\mathrm{b},{T}_{0}}\right.\hfill \\ \hfill & \left.\quad \enspace -2\left({R}_{\left(+-\right)}-{R}_{\left(-+\right)}+{T}_{\left(+-\right)}-{T}_{\left(-+\right)}\right){I}_{b,{T}_{\text{env}}}\right]\frac{\mathrm{sin}\enspace \theta \enspace \mathrm{cos}\enspace \phi \enspace \mathrm{d}M}{2\omega }.\hfill \end{align} \tag{ 17 }$$

The y-components are obtained by replacing cos ϕ with sin ϕ in above two expressions. It is then straightforward to calculate the spectral force per unit area (F_s) and the spectral torque per unit area ( τ _s) experienced by the planar slab due to the thermal emission of photons of frequency ω:

$$\begin{align}\hfill {\mathbf{F}}_{\mathrm{s}}=\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}\omega \enspace \mathrm{d}A}\left(\omega \right)=\int \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t\enspace \mathrm{d}\omega \enspace \mathrm{d}A}\left(\omega ,\theta ,\phi \right)\end{align} \tag{ 18 }$$

$$\begin{align}\hfill {\boldsymbol{\tau }}_{\mathrm{s}}=\frac{\mathrm{d}{\boldsymbol{\tau }}_{\mathrm{s}}}{\mathrm{d}\omega \enspace \mathrm{d}A}\left(\omega \right)=\int \frac{\mathrm{d}\boldsymbol{J}}{\mathrm{d}t\enspace \mathrm{d}\omega \enspace \mathrm{d}A}\left(\omega ,\theta ,\phi \right).\end{align} \tag{ 19 }$$

Here the integration is over the solid angle dΩ and the terms in the denominator (dω dA) come from dM introduced in equations (14)–(17). The final expressions are integrated over the frequency ω for calculating total force and total torque per unit area.

We now substitute the spin-resolved emissivities (equation (12)) and absorptivities (equation (13)) in the above expressions and compute the rates of linear and angular momentum transfer under thermal equilibrium condition (T₀ = T_env). We find that the following vectorial equalities always hold true:

$$\begin{equation}\begin{aligned}\hfill \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\left(\theta ,\phi \right)& +\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\left(\theta ,\phi +\pi \right)+\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\left(\pi -\theta ,\phi \right)+\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\left(\pi -\theta ,\phi +\pi \right)=0\hfill \\ \hfill \frac{\mathrm{d}\boldsymbol{J}}{\mathrm{d}t}\left(\theta ,\phi \right)& +\frac{\mathrm{d}\boldsymbol{J}}{\mathrm{d}t}\left(\theta ,\phi +\pi \right)+\frac{\mathrm{d}\boldsymbol{J}}{\mathrm{d}t}\left(\pi -\theta ,\phi \right)+\frac{\mathrm{d}\boldsymbol{J}}{\mathrm{d}t}\left(\pi -\theta ,\phi +\pi \right)=0.\hfill \end{aligned}\end{equation} \tag{ 20 }$$

It follows upon integration over all emission directions that there is no net force or torque on the planar slab which is at thermal equilibrium with the surrounding environment. Since the spin-resolved emissivities (equation (12)) are derived without invoking the concept of electromagnetic reciprocity, and are new to the best of our knowledge, we note that the above analysis of detailed balance of linear and angular momentum transfer provides a thermodynamic consistency of the derivation. The above derivation is also crucial for computing the fluctuations-induced force and torque on the planar slabs under the nonequilibrium condition when T_env ≠ T₀ as discussed above.

The calculation of spin-resolved emissivities and absorptivities requires the calculation of Fresnel reflection and transmission coefficients. While this is a very well-known topic in the literature, we note that, apart from the trivial isotropic media, the closed-form semi-analytic expressions of reflection coefficients are possible only for semi-infinite half-spaces of very few bianisotropic material classes [49–51]. We separately show this calculation in the supplement for a gyromagnetic medium. However, the semi-analytic derivation for simple cases is inadequate because we are interested in finding the laws which are applicable for generic cases of finite-thickness slabs that can possibly contain multiple layers of various materials. It can also be a composite medium where a single layer can simultaneously exhibit response of multiple material classes ¹ . Since meaningful closed-form expressions are not possible for these complicated cases, we have developed numerical tools to calculate the reflection and transmission coefficients in an exact manner as discussed in the methods section. We have made them open-source [¹] so that the reader can reproduce and verify the results in this work and also use these versatile tools for any other research activities. Our computer-assisted but exact approach for the discovery of Kirchhoff's laws is similar in spirit to that of an increasing use of machine learning, optimization, and inverse design computational techniques in many scientific disciplines for making fundamental discoveries.

Reciprocal materials: for a planar slab of reciprocal media, we find that the reflection coefficients satisfy the relations r_ss,pp(θ, ϕ) = r_ss,pp(θ, ϕ + π) and r_sp(θ, ϕ) = −r_ps(θ, ϕ + π). The transmission coefficients satisfy t_ss,pp(θ, ϕ) = t_ss,pp(π − θ, ϕ + π) and t_sp(θ, ϕ) = −t_ps(π − θ, ϕ + π). These relations can be inferred from the Green's function on the vacuum side of the geometry [52]. Substituting these relations in equations (12) and (13), we find that the spin-resolved emissivity is equal to the spin-resolved absorptivity for reciprocal media. η_±(ω, θ, ϕ) = α_±(ω, θ, ϕ). This is SKL-1 given in equation (4). For a chiral absorber α₍₊₎ ≠ α₍₋₎ from which it follows that η₍₊₎ ≠ η₍₋₎. Therefore, a chiral absorber can emit circularly polarized thermal radiation. But this well-known law is valid only for reciprocal media. Interestingly, for nonreciprocal media, we find other formulations of this spin-resolved Kirchhoff's law.

Nonreciprocal materials: for gyrotropic materials with the gyrotropy axis perpendicular to the planar surface and for nonreciprocal isotropic magneto-electric materials (Tellegen media) having diagonal $\bar{\bar{\xi }},\enspace \bar{\bar{\zeta }}$ tensors, we find that the reflection coefficients satisfy the relations r_ss,pp(θ, ϕ) = r_ss,pp(θ, ϕ + π) and r_sp(θ, ϕ) = r_ps(θ, ϕ + π). The transmission coefficients satisfy the relations t_ss,pp(θ, ϕ) = t_ss,pp(π − θ, ϕ + π) and t_sp(θ, ϕ) = t_ps(π − θ, ϕ + π). Substituting these relations in equations (12) and (13), we obtain η_±(ω, θ, ϕ) = α_∓(ω, θ, ϕ). This is SKL-2 given in equation (5).

Interestingly, multilayered planar slabs of other nonreciprocal gyrotopic and anisotropic magneto-electric materials also exhibit simplifying relations between the Fresnel coefficients. In particular, if the magnetoelectric coupling tensors ($\bar{\bar{\xi }},\enspace \bar{\bar{\zeta }}$) indicate cross coupling between electric and magnetic fields lying perpendicular and parallel to the slab (E_x − H_z , E_z − H_x , E_y − H_z or E_z − H_y ) or if the medium is gyrotropic with the gyrotropy axis parallel to the surface, the reflection coefficients satisfy the condition r_sp(θ, ϕ) = −r_ps(θ, ϕ) and the transmission coefficients satisfy t_ss,pp(π − θ, ϕ + π) = t_ss,pp(θ, ϕ + π) and t_sp(π − θ, ϕ + π) = −t_ps(θ, ϕ + π). Substituting these relations in equations (12) and (13), we obtain η_±(ω, θ, ϕ) = α_±(ω, θ, ϕ + π). This is SKL-3 given in equation (6). We note that the above underlying relations are not obvious and their universal applicability for multilayered or composite configurations of many bianisotropic material classes has not been reported before. The explanation of their specific form based on the underlying structural characteristics or symmetries of material classes is a nontrivial problem that motivates new analogies or methods [53] across a broad set of materials.