Breaking Kirchhoff's law of thermal radiation yields new opportunities in one-way radiative thermal transport and circuitry. We investigate its consequences in the far-field regime in cylindrical cavities, by employing a specular ray-tracing algorithm. At thermal equilibrium, we show that violation of Kirchhoff's law yields non-vanishing heat rectification coefficients within different sections of the cavity, which can be tuned for perfect rectification and circulation, while internal monochromatic currents vanish due to the intrinsic coupling between emission and absorption at specular surfaces. This constraint is lifted under nonequilibrium conditions, where rotational heat fluxes within the cavity can be precisely controlled by appropriately combining reciprocal and nonreciprocal materials. These findings open new avenues for thermal management and provide design principles for nonreciprocal photonic devices.
The classical form of Kirchhoff's law of thermal radiation [1] states that, at a given frequency ω, monochromatic radiation absorbed from a polar-azimuthal direction (θ, φ) must be re-emitted equally in that same direction: ϵ(ω, θ, φ) = α(ω, θ, φ).
(
Recent works have shown that Kirchhoff’s law can be violated along certain directions [2][3][4][5]. This effect finds practical applications in renewable energy, circuitry and communications [6,7]. Within the context of radiative heat engines, nonreciprocal thermal emission enables dramatic performance enhancement [8][9][10] due to improved light-harvesting. For example, breaking reciprocity between absorption and thermal emission improves performance in photovoltaics [11][12][13], thermophotovoltaics [14][15][16], and radiative cooling [17] via redirecting emitted radiation towards directions and channels where it can be maximally utilized. Beyond light-harvesting, breaking Kirchhoff’s law becomes relevant in thermal circuitry and potentially information technology, via effects such as the the photon thermal Hall effect [18][19][20], gyrotropic heat engines [21], and persistent equilibrium heat currents in many-body systems [22][23][24].
Breaking reciprocity between thermal emission and absorption can also lead to nonreciprocal radiative heat transfer [19,23,25]. Several material systems have been explored, theoretically, as platforms for nonreciprocal radiative heat flow, ranging from magneto-optical materials [26][27][28] to Weyl semimetals [29][30][31], and metasurfacebased designs that leverage multiple diffraction channels and predict near-complete violation of Kirchhoff’s law [32], and recent experimental results [33][34][35] confirm the suitability of these systems for breaking reciprocity in thermal radiation. Measurements have confirmed pronounced differences between emission and absorption, per direction, with several works reporting broadband and high-temperature demonstrations of nonreciprocal thermal emission [36,37]. In order to maximize this nonreciprocal effect, multilayer architectures and optimization-based emitter designs are being considered [38,39].
In parallel to progress in uncovering nonreciprocal thermal effects, considerable effort has been devoted to revisiting the theoretical foundations of Kirchhoff’s law. Its classical formulation is fundamentally grounded in Lorentz reciprocity [40] and the linear constitutive relations governing material response [41]. Recent studies have introduced a symmetry-based classification of nonreciprocal thermal emitters and absorbers [42,43] and revealed that, under certain conditions, directional emission and absorption can be tuned independently [44]. Mathematically, this distinction originates from the structure of the constitutive relations and asymmetries of the dielectric and magnetic response tensors [44][45][46]. Materials that allow fully independent control of directional emission and absorption must support multiple reflection channels, typically through diffraction [32]. In contrast, bodies displaying specular reflection, whether reciprocal or not, obey a generalized form of Kirchhoff’s law: emissivity in a given direction is equal to absorptivity in the opposite one [32,47] ϵ(ω, θ, φ) = α(ω, θ, φ + π).
(
When the permittivity and magnetic permeability tensors are symmetric, specifically, when the material is invariant under the adjoint transformation [44], the emissivity and absorptivity profiles become symmetric with respect to the normal direction. In this case, the generalized law reduces to the classical form of Kirchhoff’s law in Eq. (1).
So far, studies investigating nonreciprocal radiative heat transfer and radiative thermal conductance remain mostly limited to planar and near-field geometries [22,23,25], with the exception of the recent work in [48], which considers a two-dimensional, triangular geometry. The implications of nonreciprocal radiative heat flow in nonplanar and more complex configurations remain less understood.
In this work, we investigate how nonreciprocity manifests in far-field radiative heat transfer in a hollow, infinitely long cylindrical cavity. This geometry is chosen as a simple and suitable platform for hosting persistent thermal currents, however it resists analytical treatment. To make the problem tractable, we discretize the cavity walls into vertically aligned, identical elements composed of either reciprocal or nonreciprocal materials. Using a custom-made ray-tracing algorithm tailored to this geometry, we demonstrate that nonreciprocity yields nonvanishing heat rectification coefficients, even at thermal equilibrium. Despite these nonvanishing heat rectification coefficients, owning to the generalized form of Kirchhoff’s law in Eq. ( 2), persistent thermal currents do not arise within the cavity. Under nonequilibrium conditions, by contrast, nonreciprocity enables controllable, rotational heat fluxes, providing pathways for directional therma
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