Structured coherent thermal emission from non-Hermitian metasurfaces

Domain: biology
Published: 2026-03-16
Editorial outcome: major_revision

Abstract
Metasurfaces can reshape thermal radiation with tailored directionality, spectral selectivity and polarization control, yet coherent thermal emission exhibiting vectorial polarization or structured light property has been rarely demonstrated. Here, we introduce a non-Hermitian thermal meta-emitter that experimentally generates highly directional, rainbow-free, vectorial-polarized thermal emission. Through perturbation-induced Brillouin-zone folding, we couple high-radiative-loss Fabry–Pérot modes to symmetry-protected bound states in the continuum and exploit non-Hermitian dynamics near exceptional points to achieve simultaneous dispersion and loss engineering. This yields a sharp variation of the radiative Q-factor in k-space, flexibly triggering vectorial-polarized emissions only at selected output angles. Experimental studies reveal doughnut-shaped narrowband thermal emission in the 3–5 μm atmospheric transparency window, exhibiting high spectral purity, extreme directionality and distinct vectorial polarizations. Our design establishes a powerful non-Hermitian platform to sculpt incoherent thermal fluctuations into coherent vectorial-polarized beams at a specific frequency, overcoming trade-offs in coherence, spectral selectivity and polarization control.

Introduction
As a ubiquitous electromagnetic radiation phenomenon in nature, the physical essence of thermal radiation stems from the random thermal fluctuations of charges within any object whose temperature exceeds absolute zero. Within the framework of classical thermodynamics, thermal radiation typically exhibits broadband, omnidirectional, and incoherent, and unpolarized characteristics, a traditional understanding that has been perfectly encapsulated in the laws of blackbody radiation and the fluctuation-dissipation theorem. However, recent groundbreaking advancements in nanophotonic and metamaterial technologies are reshaping this fundamental physical landscape1–6. Through the meticulous design of artificial micro- and nanostructures, researchers have successfully achieved precise control over multiple dimensions of thermal emissions (ThE), including spectral, spatial, and polarization, manifesting extraordinary properties such as narrow linewidths, high directionality and controllable polarization states7–12.

This revolutionary concept of “taming thermal fluctuations” not only deepens the foundational understanding of non-equilibrium thermodynamics but also heralds transformative technological opportunities in critical fields such as infrared sensing13,14, thermophotovoltaics15,16, and radiative cooling17,18.

The mid-infrared (MIR) band is widely recognized as the molecular fingerprint region, making ThE with temporal/spatial coherence19,20 and polarization control as a light source particularly appealing. By coupling thermal fluctuations to optical structures that support high-quality (Q) factors, selective spectral components of thermal radiation can be directionally guided into free space, thereby achieving highly coherent ThE. This process is akin to transforming chaotic thermal fluctuations into a harmonious symphony, where each photon is in phase with its neighbors. Recent advances have demonstrated the use of nonlocal metasurfaces21, which exhibits substantial interactions between adjacent meta-atoms to produce high-Q responses, in the flexible reshaping of ThE with high coherence19,22. In terms of polarization control, optical modes such as surface phonon polaritons3, localized surface plasmon resonances23,24, and Mie resonances25 in dielectric structures have been extensively utilized for linearly polarized ThE26–29. Recent breakthroughs in circularly polarized ThE have also provided unique prospects for applications such as circular dichroism infrared sensing and chiral molecule detection9,30–37. Meanwhile, a vectorial-polarized beam, with its rich and unique polarization state distributions (including radial and azimuthal polarizations), has demonstrated great potential in fields such as focused imaging, particle trapping, and laser processing38–41. However, to date, these exotic polarization states have predominantly been achieved by relying on the synergistic effects of external laser sources and bulky optical components, such as spatial light modulators. In contrast, the control of vectorial-polarized ThE using thermal emitters remains a significantly underexplored area of research. Although metal-insulator-metal absorbing metasurfaces can endow individual meta-atoms with tailored polarization states to yield vectorial-polarized ThE, the inherent plasmonic losses severely degrade temporal coherence. To the best of our knowledge, vectorial-polarized or any other form of structured ThE with high temporal coherence (Q > 50) has been scarcely demonstrated. While Zhang et al. recently reported the realization of polarization vortices in ThE42, their system maintains near-critical coupling over a broad range of wavevectors and wavelengths. The persistence of this critical coupling condition along the dispersive BIC band (Fig. 1a) inevitably leads to broad emission angles and the characteristic rainbow effect. In contrast, our non-Hermitian approach enables us to confine the critical coupling state to a specific, narrow angular and spectral window (Fig. 1b). This ensures highly directional, rainbow-free, doughnut-shaped vectorial-polarized ThE, which is essential for meeting the demands of practical applications.

Furthermore, as open systems, thermal emitters can benefit from the rich physical phenomena supported by non-Hermitian descriptions43,44. To date, most studies on their intriguing results have focused on systems that rely on material loss and gain. These systems exhibit parity-time (PT) symmetry and demonstrate optical phase transitions at exceptional points (EPs) between weak and strong coupling regimes45,46. Near an EP, the Hilbert space of the system becomes severely “skewed”, rendering the system’s dynamical behavior highly sensitive to minute perturbations. This sensitivity provides an effective approach for designing and controlling thermal emitters with extreme spatial and spectral properties.

In this work, by leveraging the non-Hermitian engineering in nonlocal metasurfaces, we have designed and implemented a four-hole unit cell structure, through which we successfully realize two types of highly directional, rainbow-free, and doughnut-shaped coherent ThE featuring vectorial polarizations. Specifically, topological charge engineering induced by Brillouin zone folding (BZF) enables flexible triggering of vectorial polarization states at the Γ point. By simultaneously tuning the distances among the four holes, a four-level coupled-mode system is constructed. In this system, high-radiative-loss Fabry–Pérot (FP) modes are strategically coupled to the symmetry-protected BIC, and the complete coupling evolution is demonstrated. During this process, we observe the formation, annihilation, and reformation of EPs and EP-induced band loops, without requiring any artificial optical gain. A pair of BIC-based mini-flatbands emerges as intermediate states during this dynamic process. Furthermore, the non-Hermitian interaction near EPs induces sharp variations in radiative loss with wavevector k, which enables critical coupling between radiation and material absorption losses only at specific k-values—thereby maximizing emissivity with high directionality and suppressing the rainbow effect that plagues conventional dispersive systems whose Q factor depends weakly on k. Away from these k-points, a mismatch between these two loss rates leads to a sharp drop in emissivity, a characteristic inherited by the double-BIC mini-flatband design. Leveraging these unique non-Hermitian physical mechanisms, we theoretically and experimentally demonstrate doughnut-shaped vectorial-polarized ThE with simultaneously high spectral purity and extreme directionality. These remarkable and unconventional results open up new possibilities for on-chip laser-like thermal sources in applications such as infrared photonic manipulation and high-fidelity thermal imaging.

Results
Figure 1a, b schematically compare the ThE characteristics between a traditional dispersive system and a mini-flatband counterpart engineered via non-Hermitian strategies. Both systems support symmetry-protected BICs, which are characterized by infinite Q-factors and integer topological charges in momentum space47–49. However, traditional BIC systems typically exhibit dispersive behavior, and the radiation Q factor (Qrad) usually exhibits two distinct dependencies on k as it deviates from the BIC point50. Initially, Qrad is highly sensitive to changes in k, but it becomes less sensitive as k moves away (see also Fig. S2c). According to the temporal coupled-mode theory (TCMT), the emissivity of the entire system can be determined by the contrast between the absorption and radiation losses7. Perfect absorption (PA) or emission (PE) is achieved under the critical coupling condition Qrad ≈ Qabs, where Qabs is the absorption Q factor. Therefore, for a given intrinsic absorption loss, the control of radiative loss provides additional flexibility in tuning emissivity. The weak dependence of Qrad on k where k > kBIC50, combined with the moderate value of the absorption Q factor (Qabs) in lossy ThE systems, suggests that near-critical coupling conditions can be sustained in the second region over a broad wavevector range. These features result in high emissivity distributed over wide angular spans at frequencies determined by the dispersion band42,51, which is known as the rainbow effect, as illustrated in the bottom panel of Fig. 1a. Consequently, although conventional architectures based on dispersive BIC modes can facilitate the generation of vectorial polarizations in ThE42, they typically exhibit broadband critical coupling behavior. In such systems, high emissivity persists across a wide range of angles as the resonance wavelength shifts, necessitating the use of auxiliary ultra-narrowband filters with omnidirectional responses to achieve precise spectral selection. However, this approach is complex and significantly complicates integration into practical platforms, such as on-chip thermal light sources. In contrast, the mini-flatband system, designed through band and dissipation engineering—specifically by tailoring radiative loss to induce a phase-transition-like critical coupling behavior at small wavevectors—addresses the above challenges. Within this mini-flatband, Qrad remains at moderate values, which are necessary to fulfill the critical coupling, while keeping its high sensitivity to k (see Fig. S2c). As a result, it supports high ThE at a single central frequency across a narrow angular range (see Fig. 1b). Since the mini-flatbands are close to the Γ point (where a BIC with surrounding vortex features in polarizations is located), the system enables highly directional, rainbow-free, doughnut-shaped ThE directly in real space, eliminating the need for additional spectral/spatial filtering. Figure 1c illustrates the working principle of the final thermal emitter designed in this work. The yellow bottom layer represents a sufficiently thick gold film. When the whole structure is heated by applying an electric current across the gold film or placed in a high-temperature environment, thermal fluctuations will be provided that couple into the resonant modes of the upper structure, thereby radiating vectorial-polarized ThE into free space. Here, we present two types of vectorial polarization states: a superposition of radial and azimuthal polarizations carrying a −1 topological charge (left panel of Fig. 1c) and a pure azimuthally polarized state carrying a +1 topological charge (right panel). Besides the vectorial polarizations, the achieved emissions exhibit strong intensity confined to the doughnut ring and rapid suppression at both the center and outer periphery. The flexibility in controlling the vectorial features of ThE holds promise for emerging applications such as high-resolution thermal imaging and target identification.

The above properties stem from the integrated use of non-Hermitian engineering, including topological, band, and dissipation control mechanisms (see Fig. S2 for more details). Figure 2 systematically demonstrates the details of this engineering process. We begin with a meticulously designed partially-etched metasurface structure, whose geometric parameters are shown in Fig. 2a. The top layer is a high-refractive-index Ge film (n = 4.16) with a total thickness of 495 nm, serving to support required guided modes (GMs). A periodic array of circular air holes, with a carefully chosen period of a = 800 nm and radius of r = 195 nm, is partially etched into the Ge layer, leaving an ultrathin slab with a remaining thickness of h = 125 nm. The entire structure, including the patterned layer and the slab, supports both GM and FP modes at the target frequency. A low-index ZnS layer (n = 2.22) with a thickness of t = 865 nm is inserted between the Ge and the Au layer, which mainly helps separate the metasurface from the metal to reduce the impact of the inherent loss of Au52. We numerically calculate the band structures supported by this system using the finite element method (FEM) implemented in the commercial software COMSOL Multiphysics. As shown in Fig. 2b, the structure supports a series of GMs with infinite Q-factors below the light line (yellow region), as well as FP modes (green dashed lines) that couple into free space via the light cone interface. The GMs of interest along the Γ→X and Γ→M directions are labeled GMA, GMC, and GMB in blue, orange, and red, respectively.

To enable the coupling of these GMs to free space, we designed a four-hole unit-cell metasurface to induce bidirectional band folding, as illustrated in Fig. 2d. By introducing a perturbation δ in the positions of four adjacent holes along both the x and y-directions, the first Brillouin zone (FBZ) shrinks because of the length doubling of the unit cell along each direction, which now includes four holes (Fig. 2e). This operation causes the GMs originally below the light cone and at the FBZ boundary to fold into the continuum, forming QGMs50,53,54. When δ = 20 nm, the band structure supported by the metasurface is shown in Fig. 2e, where the color-coded folded bands originate from the corresponding GMs in Fig. 2b. The field distributions in Fig. S3 demonstrate one-to-one correspondence between GMs and QGMs. Importantly, in the unperturbed structure, these GMs lie entirely below the light cone and therefore carry zero topological charge. In contrast, for the bidirectionally band-folded metasurface, the C4v symmetry of the structure leads to the formation of symmetry-protected BICs for the QGMA and QGMB modes at the Γ point (see Fig. S11). Under the constraint of topological charge conservation, these two modes acquire customizable topological charges of −1 and +1, respectively, which give rise to distinct textures of vectorial polarizations in their surrounding momentum space, as shown in Fig. 2f. This topological engineering thus lays a crucial foundation for the realization of tailored vectorial-polarized ThE.

It is noteworthy that this four-hole configuration bears a structural resemblance to second-order topological photonic crystals55–57. However, a fundamental distinction lies in the operational regime: while those topological systems typically function below the light cone to host localized corner or edge states, our design leverages out-of-plane radiation above the light cone. This shift into the radiative regime introduces unique non-Hermitian opportunities; specifically, the interplay between topological lattice symmetries, exceptional points and wavevector-sensitive radiative losses enables the realization of highly directional, rainbow-free ThE. Such dynamics represent a significant extension of the functionality usually explored in the context of second-order topological physics. By investigating the radiative properties of topological metasurfaces, our approach bridges the gap between topological photonics and non-Hermitian thermal photonics, opening new avenues for the development of structured, coherent thermal radiations sources.

In order to elucidate the rich physics arising from near-field coupling between different modes and their influence on radiation losses, we assume zero material absorption loss, modeling an ideal open system. We consider only four interacting modes near the Γ-point: QGMA, QGMB, QGMC, and FP. In this gain-free system, we accurately track the band and dissipation evolution by varying δ. The states of QGMA and QGMB at the Γ point are marked as BICA and BICB, as shown in Fig. 2g. The near-zero imaginary part of the complex eigen frequency and its inverse-quadratic dependence on k confirm their BIC nature (Q factor results, see Fig. S15). In addition, these two modes are coupled with each other near the Γ-point under the introduction of δ. It should be noted that at smaller δ values, the FP mode is far from the high-frequency QGM band. Overall, this results in robust and large imaginary parts of the complex eigenfrequencies, which corresponds to small Qrad. As δ increases, QGMA and QGMB move toward lower frequencies, while the FP mode shifts in the reversed direction. These spectral shifts enable a crossing and interaction between the FP mode and QGMA, as shown in Fig. 2i. Energy exchange occurs in three modes around k ~ 0.035π/a, resulting in a significant increase in the imaginary parts of QGMA and QGMB eigenfrequencies, while the imaginary parts of FP eigen frequencies are significantly suppressed. As δ further increases, the FP mode crosses over QGMB and intercalates between QGMA and QGMB. At δ = 165.6 nm, the interaction between QGMB and FP modes lead to the formation of EPs and significant non-Hermitian EP-induced band loop behavior near the Γ point.

To describe this phenomenon, we employ the TCMT to outline the theoretical framework of the system. Further details of this model are provided in Section 3 of the Supplemental Information (SI). We initially consider the coupling between four modes—QGMA, QGMB, QGMC and FP—using a 4×4 Hamiltonian. As δ increases, we observe a clear phenomenon of EP and EP-induced band loop formation, annihilation, and reformation. When δ approaches certain critical values, the coupling between two of the four modes becomes dominant. Consequently, the Hamiltonian can be simplified to a 2×2 model. At critical coupling conditions, which correspond to the δ values of 165.6 nm and 198 nm, the two eigenvalues of the 2×2 Hamiltonian coalesce into one. This coalesce involves both the resonant frequency and the leakage rates merging simultaneously, signifying the formation of an EP. Furthermore, the real and imaginary parts of the complex eigenfrequencies exhibit a ring-liked distribution in the vicinity of the Γ point (see Fig. 2i and k).

More importantly, during this process, the interplay among the four modes results in distinct topological features. At δ = 173 nm, two mini-flatbands carrying topological charges emerge near the Γ point (the 2D band structure and group velocity are shown in Fig. S4), retaining the unique characteristics induced by the EP effect, such as the steep variation of radiation loss with wavevector, as shown in Fig. 2j. In real devices where one needs to consider the material absorption of the system, there is a fast mismatch between radiation loss and absorption loss as the wavevector deviates from a certain k. The QGMC and FP modes exhibit extremely low Q factors as weakened radiation backgrounds. The radiation loss of two mini-flatbands supporting BIC is significantly enhanced on the EP-induced band loop formed before and after controlling δ and undergoes a phase transition near the EPs. These common behaviors not only ensure the rapid achievement and mismatch of critical coupling but also ensure that it only occurs at a single center output frequency. This is the key to achieve rainbow-free effect in real space and doughnut-shaped ThE with vectorial polarizations.

We conducted numerical simulations of the structure based on the final geometric design, incorporating realistic absorption losses to validate its ThE characteristics. According to Kirchhoff’s law of thermal radiation, the emissivity and absorptivity of an object in thermal equilibrium are equivalent. Consequently, the emissivity can be characterized by the calculated absorptivity of the structure, i.e., E(λ) = A(λ) = 1 − R(λ) − T(λ). Due to the optically thick bottom layer of gold, which prevents light transmission (T(λ) = 0; experimental transmission spectra are shown in Fig. S14), this expression simplifies to E(λ) = A(λ) = 1 − R(λ). For comparison, we first designed a conventional metasurface with C4v point group symmetry that supports BIC modes with a dispersive band, as illustrated in the right panel of Fig. 3a. This metasurface supports an azimuthal far-field polarization distribution near the Γ point (see Fig. S12 for more details). The material composition of this system is identical to that of the metasurface. The geometric parameters are assumed to be a = 1600 nm, h = 485 nm, and t = 500 nm. To simplify calculations, the imaginary parts of the refractive indices of Ge and ZnS are both assumed to be 0.002, closely approximating experimental values for evaporated Ge/ZnS materials in the MIR. The left panel of Fig. 3a presents the calculated total Q (Qtotal), Qrad and Qabs factors as a function of the emission angle for this conventional BIC system with a dispersive band. The angle-resolved ThE spectrum under y-polarization in the Γ→X direction is shown in Fig. 3b. Due to the infinite Qrad of the BIC, the emissivity at the Γ point is nearly zero. As the emission angle increases, the spectrum exhibits significant dispersion, and the emissivity gradually increases, reaching its maximum at approximately θ ~ 13°. However, because Qrad and Qabs remain nearly critically coupled over a broad angular range, the system maintains a relatively high emissivity over an angular range as wide as 20° (emissivity > 0.75), which aligns well with the theoretical explanation in Fig. 1a. Figure 3c, d show the calculated Qtotal, Qrad, and Qabs as a function of the emission angle for the structure with δ = 173 nm, corresponding to the two mini-flatband systems. The slight variation of Qabs with wavevector arises from the electric field mode distribution changes induced by the EP. Notably, the sharp variation in Qrad leads to a rapid mismatch with Qabs, resulting in higher contrast. Figure 3e illustrates the angle-resolved ThE spectrum in the Γ→X direction for y-polarized light under these geometric parameters. The two mini-flatbands located at 4.797 μm and 4.691 μm correspond to QGMB and QGMA in Fig. 3c, d, respectively. The near-zero emissivity at the Γ point is supported by the vortex characteristics of BICB and BICA on the mini-flatbands. It is noteworthy that due to the rapid variation of Qrad on the two mini-flatbands of BICB and BICA, critical coupling with Qabs occurs at emission angles as small as 1.2° and 1.42°, respectively, attaining near-unity emissivity. It is important to note that both systems exhibit Qrad with a k−2 decay sharpness, as shown in Fig. S7. However, our method is able to reduce the wavevector distance between the critical coupling position and the Γ point by approximately one order of magnitude. This represents an approximately one order of magnitude reduction in the emission angle where critical coupling occurs. As the emission angle further increases, the rapid decrease of Qrad and the resulting deviation from the critical coupling condition lead to a sharp decline in emissivity, thereby effectively suppressing ThE at large angles and at other wavelengths (for detailed comparative results over a wider angular range, see Fig. S6). For comparison, we also designed a merged BIC58 thermal emitter that supports a k−6 sharpness, as shown in Fig. S8. Clearly, this system exhibits an infinitely large Qrad factor over a broader wavevector range, approximately three orders of magnitude higher than in our work, resulting in the critical coupling position occurring at larger angles. These results highlight the enhanced versatility of our approach in achieving small divergence angles, high directionality, and high-purity doughnut-shaped coherent vectorial-polarized ThE, while avoiding the broad angular dispersion in emission output.

We employ the topological charge of polarization singularities48 to extract more information on the BICs within the mini-flatbands. Calculations of the topological charge can be found in Section 4 of the SI. By calculating the far-field polarization vector fields of QGMB and QGMA near the Γ point, we observe in Fig. 3f, h that BICB, carrying a topological charge of +1, exhibits an azimuthally polarized far-field distribution (detailed numerical results for a broader wavevector range are provided in Fig. S9). In contrast, BICA, carrying a topological charge of −1, displays an azimuthal polarization along the kx and ky directions but radial polarization along a 45°-rotated axis. The momentum-space ThE patterns of the two modes are presented in Fig. 3g, i. These results confirm that the polarization vortex distribution induced by topological charge-carrying BICs, in conjunction with mini-flatband formation and dissipation engineering, lead to rainbow-free ThE exhibiting distinct doughnut-shaped beams with vectorial polarizations—fully consistent with the expected description in Fig. 1.

Based on the above design, we fabricated such a thermal emitter with δ = 173 nm using standard nanofabrication techniques (see Fig. S1 for more details of the fabrication processes). Figure 4a, b presents the top-view and cross-sectional pseudo-color scanning electron microscope (SEM) images of the fabricated structure, demonstrating high processing accuracy. Figure 4c illustrates a schematic of the custom-built setup for the characterization of ThE. The sample is mounted on a 3D-controllable rotating stage equipped with a heating system, allowing precise multidimensional adjustments to measure ThE at different azimuthal and output angles. Two coaxial apertures (JCOPTIX, China), placed between the sample and a rotatable linear polarizer (JCOPTIX, China), significantly reduce background emission and enhance the accuracy of angle-resolved ThE measurements. The collected ThE is analyzed using a Fourier-transform infrared (FTIR) spectrometer equipped with a mercury cadmium telluride (MCT) detector.

By heating the sample to 200°C, we conducted precise angle-resolved ThE measurements at different azimuthal and polarization angles. As shown in the inset of Fig. 4c, the emission spectrum was recorded at different azimuthal angles (β) by rotating the stage with respect to the surface normal direction. Angle-resolved emission spectra were then obtained by rotating the stage within a fixed plane containing the z-axis, thus controlling the emission angle θ. The polarization property was characterized by rotating the polarizer at fixed values of θ and β. Due to the structure’s high C4v point group symmetry, we measured the emissions spectra at different azimuthal angles (β = 0°, 22.5°, 45°, 180°, 202.5°, and 225°) and the results are presented in Fig. 5a–c (the corresponding numerical results are presented in Fig. S5). The color amplitude represents the intensity of ThE. All the results in Fig. 5a correspond to the case of β = 0 (see the left panel of Fig. 5a). The middle panel of Fig. 5a displays the angle-resolved ThE spectrum within the xz plane under y-polarization (α = 90°), where the emissivity center wavelengths are fitted by dashed lines corresponding to the theoretical predictions in Fig. 2j. Notably, the two mini-flatbands with output wavelengths of 4.84 and 4.705 μm exhibit near-zero emissivity in the normal direction. As the emission angle increases, the intensity first rises rapidly and then decreases sharply, which is consistent with theoretical predictions. In Fig. S13, we present the measured emission spectrum at the output angle of θ = 2°. The fitting results show that the two modes exhibit Q-factors of 97 and 73, corresponding to bandwidths of 50 nm and 64 nm, respectively. These values indicate relatively high temporal coherence. It should be noted that the experimentally observed critical coupling angle is slightly larger than the simulation results, and the measured Q-factors are slightly lower. This discrepancy is attributed to unavoidable multi-angle collection during the measurement process and additional scattering losses induced by sidewall roughness. The right panel of Fig. 5a presents the angle-resolved ThE spectrum under x-polarization (α = 0°) within the xz plane, where the weak emission feature near 4.63 μm corresponds to an x-polarized FP mode. The presence of mini-flatband and high-intensity ThE for y-polarization and the absence of these features for x-polarization confirm that these modes exhibit azimuthal polarization at β = 0. Considering the C4v symmetry of the structure, the same behavior repeatedly occurs at β = 90, 180 and 270°. The measurement results presented in Fig. 5b, obtained at β = 22.5°, show that the QGMB and QGMA modes reach maximum emission intensities at the polarization angles of α = 112.5° and 67.5°, respectively. This indicates that QGMB maintains azimuthal polarization, while QGMA gradually transitions toward radial polarization. The middle panel of Fig. 5c presents the results obtained at β = 45°, revealing that QGMB achieves maximum emission at α = 135°. This confirms that the QGMB mode maintains azimuthal polarization throughout the entire radiation space, consistent with the C4v symmetry of the structure. In contrast, the right panel of Fig. 5c shows that the QGMA mode achieves peak emission at α = 45°, indicating that its emission alternates between azimuthal and radial polarization at 45° intervals in radiation space. These experimental results are in excellent agreement with the simulations in Fig. 3 and Fig. S5. It should be noted that the slight asymmetry observed in the angle-resolved ThE measurements in Fig. 5 can be attributed to a combination of experimental factors. Primarily, minor optical misalignment—specifically, a slight tilt of the sample relative to the optical axis during the angle-resolved capture—induces the observed intensity deviations between symmetric wavevectors. Additionally, fabrication imperfections, such as sidewall roughness and minor defects in the metasurface structure, may introduce non-uniform scattering losses.

To further visualize the spatial ThE patterns of the two modes, we measured and mapped the ThE intensity in momentum space at the two mini-flatband wavelengths. Figure 5d illustrates the measured azimuthal polarization dependence for QGMB, which exhibits peak emissivity at 4.84 μm and clearly shows a characteristic doughnut-shaped azimuthal polarization distribution in momentum space. The polarization ratio of the x and y components of the emissivity, with respect to the azimuthal angle β, follows the relation (sin β, −cos β). This phenomenon is easily associated with real space scenarios, where the ThE associated with QGMB resonance exhibits azimuthally polarized response, with rainbow-free and highly directional behaviors. The doughnut radius varies with the observation distance, consistent with the schematic in Fig. 1c. Figure 5e presents the emissivity associated with QGMA resonance in momentum space, demonstrating a polarization ratio given by (sin β, cos β). Unlike QGMB, QGMA exhibits azimuthal polarization along the x and y-axes but also radial polarization along the 45° direction. These experimental results, summarized in the right panels of Fig. 5d and e, show excellent agreement with the numerical predictions in Fig. 3g, i, and Fig. S10a, c. We also calculated the angular emission distribution for different polarization angles α, as shown in Fig. S10b, d, which further validates our experimental findings. Importantly, a single thermal emitter can simultaneously generate vectorial-polarized ThE at two distinct wavelengths with different characteristics, offering significant potential for diverse on-chip light source applications.

Methods
Numerical simulations
All numerical simulations were performed based on the FEM using the commercial software of COMSOL Multiphysics. Floquet periodic boundary conditions were implemented in the lateral directions of a unit cell to define the wavevectors, while PML was employed in the z direction to mimic a non-reflecting infinite domain.

Fabrication
First, a 5-nm-thick adhesion layer of Ti, a 100-nm-thick Au film and another 3-nm-thick Ti were successively deposited onto a Si substrate using electron beam evaporation. After that, two layers of 865 nm ZnS and 495 nm Ge were deposited onto the sample using the same equipment. Then, the 180-nm-thick PMMA resist layer was spin coated onto the sample and the reverse-designed structure pattern was exposed onto the PMMA layer by an electron beam lithography system. After developing, 20 nm Al was evaporated onto the sample, and the lift-off process was used to transfer the pattern from PMMA to the Al layer. The Al pattern worked as a mask to obtain the Ge structure in a subsequent inductively coupled plasma-enhanced reaction ion etching process. Finally, the residual Al was completely removed by soaking the sample in NaOH solution.

Optical measurements
The emissivity of the fabricated samples was measured using a Fourier transform infrared spectrometer (FTIR) (Bruker Vertex 80V) equipped with a liquid-nitrogen-cooled mercury cadmium telluride (MCT) detector. The emissivity measurements were carried out by loading the sample onto an emission adapter (Bruker, A540), which was employed to heat the sample at different temperatures to induce thermal emissions. Steel plates of the same size, blackened by the black soot of candles, were used as the standard blackbody reference. Both the blackbody and the fabricated sample were loaded onto the heating plate. When the blackbody and the sample were heated to 200°C, the emitted power was sent to the FTIR system and detected by the MCT detector. The thicknesses of the different films were measured using a Mueller matrix ellipsometer (ME-L ellipsometer, Wuhan Eoptics Technology Co., Wuhan, China).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this work, the authors experimentally demonstrate vector beams generated through thermal emission using a multi-mode non-Hermitian engineering approach. This experimental work is timely with recent theoretical works on shaping non-Hermitian bands [10.1103/378r-pmcv]. While the paper is well written and clear, I cannot recommend publication in Nature Communications as I have major technical questions that must be addressed before publication. With satisfactory changes to the manuscript, I am happy to recommend publication.\nMajor:\n1. What are the advantages of this method over engineering the sharpness of the BIC fall-off, i.e. going from 1/k^2 to 1/k^6 using approaches demonstrated in [10.1038/s41586-019-1664-7]?\n2. What happens to the polarization charges of the exceptional points at 150 nm and 173 nm? Did the singularities move outside the zone shown in Fig. 3f,h (|k|>0.03)? Do they collide with other charges?\n3. What does Fig. 3e look like from -20 degrees to +20 degrees? It looks like decently strong thermal emission should be observed for QGMa since the Q-factor increases from a minimum of 40 to 200 which is only a factor of 5 off the critical coupling condition. This should be very close to the +20-degree case for \"Traditional BICs.\"\n4. All the experimental emission plots are presented in relative emissivity with arbitrary units. True emissivity needs to be reported so readers can understand expected performance levels for this type of non-Hermitian engineering of thermal emitters.\n5. What is the polarizer angle (alpha) in Fig. 5d and e? Moreover, why is the expected polarization behavior so hard to observe? From Fig. 5d, I do not see the near zero intensity I would expect from Fig. 5a with a polarization angle of 0 degrees. The experimental evidence is not clear proof of the expected polarization distribution and the existence of vector beams. Can the authors explain why the behavior in Fig. 5a,b,c is so different from that in Fig. 5d,e?\n6. In the SI, Fig. S3 uses electric field amplitudes which obscures the characters of the modes. Plotting the appropriate field components for TE and TM like modes would greatly clarify for the reader which modes are used.\nMinor:\n1. Line 381, Eps. should be EPs.", "reviewer_2": "The present manuscript reported the design, theoretical analysis, and experimental demonstration of a non-Hermitian thermal metasurface capable of generating highly directional, rainbow-free, coherent thermal emission with vectorial polarization states. The authors introduced a nonlocal metasurface composed of a patterned Ge membrane atop a ZnS spacer and thin gold film. By applying Brillouin-zone folding and non-Hermitian mode engineering, they hybridized Fabry–Perot modes with quasi-guided modes, leading to the couple of symmetry-protected bound states in the continuum (BICs). Through carefully tuning the perturbation parameter, the system undergoes exceptional point (EP) formation, annihilation, and reformation, leading to the emergence of topological mini-flatbands that feature steep, wavevector-sensitive radiative loss. These bands enable critical coupling at narrowly defined angles while suppressing emission elsewhere. The authors demonstrated experimentally the doughnut-shaped vectorial-polarized thermal emission modes within the 3–5 um window, achieving Q-factors up to ~100 and strong directional confinement.\nThe results are solid in both theory and experiment. The work presents a notable conceptual and technological advance in thermal meta-photonics by integrating non-Hermitian physics, BIC engineering, and dissipation control. generate. The demonstration of highly coherent structured thermal radiation will be one step closer for the potential applications in infrared optical manipulation and micro-scale thermal imaging. I will strongly recommend for reconsideration in Nature Communications, if the authors can address my concerns accordingly.\n(1) The four-hole unit-cell metasurface used in this work could be related to second-order topological photonic crystal [e.g. Phys. Rev. Lett. 122, 233902 (2019), Phys. Rev. Lett. 122, 233903 (2019), Optica 6, 786-789 (2019)], which are operating below light cone, while this work focuses on the out-of-plane intriguing radiation above light cone. I encourage the authors to give detailed discussions on these two topics that will broaden the research interests for nanophotonic community.\n(2) The wording ‘bilayer metasurface’ should be seriously reconsidered. Generally, ‘bilayer metasurface’ is patterned with subwavelength meta-structures in the both two layers.\n(3) The 4*4 effective Hamiltonian of TCMT contains substantial physics effects, and the authors developed a specific case of 2*2 model to discuss EP band. Can the Hamiltonian be developed another case for BIC band or mini flat band?\n(4) I am confused about the role of thick gold film, the yellow bottom layer shown in Fig. 1c. Moreover, the details of COMSOL simulation setup should be added (e.g. the absorption loss of gold film), so that it will make others repeat the simulation results.\n(5) What is the definition of emissivity, and how to evaluate the emissivity value of traditional BIC in Fig. 3a? How about the emissivity value of QGM_B [Fig. 3c] and QGM_A [Fig. 3d]?\n(6) Traditional BIC has a trend that Q_rad is proportional to 1/|k|^2. I’m curious about the Q-k trends of QGM_B [Fig. 3c] and QGM_A [Fig. 3d].\n(7) Some minor issues that confuse me, and may also confuse other readers.\n7.1 In Line 313 of manuscript, what is the meaning of Gamma-X-z plane?\n7.2 In Line 91 of supplementary material, the expression of phi(k) may be misunderstanding whether the term arg[…] should be included in the denominator or numerator.", "reviewer_3": "In this manuscript, Sun et al. report the realization of polarization vortices within a thermal emitter, discussing the application of non-Hermitian engineering to control thermal emission. Leveraging topological concepts for thermal management is a timely pursuit, with potential implications for thermophotovoltaics and radiative cooling. Specifically, the authors focus on Bound States in the Continuum (BIC)-induced polarization vortices, utilizing non-Hermitian physics to manipulate dispersion and emissivity.\nWhile the subject matter is undeniably interesting, I have some reservations regarding the novelty of the reported findings and the sufficiency of the experimental data to support the central claims. Therefore, I cannot recommend publication in its current form. I would be willing to reconsider my decision only after the authors have adequately addressed the major concerns detailed below.\nMajor Comments：\n1. Novelty and relationship to Ref. 46: Ref. 46 has already demonstrated polarization vortices in thermal emission. In the introduction, the authors imply that such vortex emission has not been realized, which is not accurate. The authors should fairly acknowledge the prior work and clearly state what additional physics or capabilities the present work provides beyond Ref. 46. Besides, in my opinion, achieving a high Q factor for thermal emission at longer wavelengths is more challenging; therefore, it is not fair to emphasize the Q factor here, given that Ref. 46 operates at much longer wavelengths (>10 um) compared with the present work (~5 um), and the measured Q factor in this work is also not particularly high.\n2. Evidence for the \"mini-flat band\": From Fig. 2, the band in panel (j) does not appear significantly flatter than that in (g). I suggest providing quantitative evidence (e.g., group velocity) to substantiate this claim. Additionally, a 2D band structure plot would help support the claim better.\n3. Critical coupling and spectral signatures: The numerical results show critical coupling occurring within a narrow bandwidth, but this feature is not clearly visible in the emission spectrum. As perfect absorption is typically used to identify critical coupling, can the authors measure transmission and reflection to confirm this? Also, the dispersion shown in Fig. 2 (including real and imaginary parts) could likely be extracted using the same experimental setup, which would strengthen the manuscript.\n4. Quality of experimental data (Fig. 5): The measured emission appears asymmetric, especially in Fig. 5(b). The authors should explain why this occurs—is it due to fabrication deviations or measurement misalignment? Furthermore, the polarization plots in (d) and (e) seem to be numerical rather than experimental. Since the polarization vortex is a key result of this work, experimental validation is needed.\nMinor Issues：\n1.The phrase “convert incoherent thermal fluctuations into coherent vectorial-polarized beams” is inaccurate. Vectorial-polarized beams generally imply stable phase relationships in different directions, but thermal emission is inherently incoherent with random phase. Unless the authors provide experimental evidence demonstrating directional phase coherence, this statement should be revised.\n2.I think it is better to show the numerically calculated dispersion in Fig. 5(b) and (c) for direct comparison.\n3.In Fig. 2(b), how is the angle-resolved emission spectrum computed? Is it actually calculated from the absorption spectrum? Please clarify.\n4.Colorbars in Fig. S4 of the Supplementary Information are missing."}, "author_response": "Response to Reviewers’ Comments\nIn this document the reviewers' comments are pasted in blue, our response in black and the parts newly added to the manuscript are in red.\nReply to Reviewer #1\nReviewer #1:\nIn this work, the authors experimentally demonstrate vector beams generated through thermal emission\nusing a multi-mode non-Hermitian engineering approach. This experimental work is timely with recent\ntheoretical works on shaping non-Hermitian bands [10.1103/378r-pmcv]. While the paper is well\nwritten and clear, I cannot recommend publication in Nature Communications as I have major technical\nquestions that must be addressed before publication. With satisfactory changes to the manuscript, I am\nhappy to recommend publication.\nOur response: We sincerely thank the reviewer for the insightful and constructive comments on our\nmanuscript, and we are encouraged by the positive feedback regarding the clarity and quality of the\npaper.\nWe also thank the reviewer for providing the important reference [10.103/378r-pmcv], which has\nsignificantly informed our revision. In that theoretical work, the authors introduced an elegant effective\nHamiltonian model to describe the key physical characteristics of exceptional points (EPs) and bulk\nFermi arcs (BFAs), systematically investigating how the position of EPs and the associated BFAs\ndepend on the type and degree of symmetry breaking in the 2D unit cell.\nWhile both studies leverage non-Hermitian engineering to manipulate dispersion and losses, our work\ndiffers in several key aspects and focuses on different physical objectives. Unlike the symmetry-\nbreaking approach employed in the cited work, we first achieve topological BICs with opposite charges\nthrough band folding, while maintaining the C symmetry of the structure. By controlling the coupling\n4V\nbetween multiple modes with a single geometric parameter, we realize the formation, annihilation, and\nre-formation of EPs and EP-induced band loop. During this process, we generate two mini-flatbands\nwith opposite topological charges. Importantly, we demonstrate experimentally how non-Hermitian\nengineering can reshape thermal emission in metasurfaces and show that by directly controlling the\ndispersion and losses of photonic bands through EPs, we can precisely control the system's critical\ncoupling, resulting in highly directional, spectrally pure vectorial-polarized thermal emission.\nWe have cited this reference in the revised manuscript (now Ref. 44) and included a discussion on EPs\nto provide a more comprehensive context.\nThe following paragraph has been added to the Introduction section:\n''…Furthermore, as open systems, thermal emitters can benefit from the rich physical phenomena\nsupported by non-Hermitian descriptions43,44. To date, most studies on their intriguing results have\nfocused on systems that rely on material loss and gain. These systems exhibit parity-time (PT)\nsymmetry\nand demonstrate optical phase transitions at exceptional points (EPs) between weak and strong coupling\nregimes45,46. Near an EP, the Hilbert space of the system becomes severely \"skewed\", rendering the\nsystem's dynamical behavior highly sensitive to minute perturbations. This sensitivity provides a novel\napproach for designing and controlling thermal emitters with extreme spatial and spectral properties…''\nReference\n43. Doiron, C. F. & Naik, G. V. Non‐Hermitian Selective Thermal Emitters using Metal–\nSemiconductor Hybrid Resonators. Adv. Mater. 31, 1904154 (2019).\n44. Frau, L., Zanotti, S., Ferrier, L., Gerace, D. & Nguyen, H. S. Shaping bulk Fermi arcs in the\nmomentum space of photonic crystal slabs. Phys. Rev. B 112, 205423 (2025).\n45. Zhang, F. et al. Non-Hermitian singularities in scattering spectra of Mie resonators. Sci. Adv.\n11, eadr9183 (2025).\n46. Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358\n(2015).\nMajor:\n1. What are the advantages of this method over engineering the sharpness of the BIC fall-off, i.e. going\nfrom 1/k^2 to 1/k^6 using approaches demonstrated in [10.1038/s41586-019-1664-7]?\nOur response:\nCompared to traditional methods focusing on adjusting the sharpness of the BIC fall-off, our approach\nreduces the wavevector distance between the critical coupling point and the Γ point by an order of\nmagnitude. This not only facilitates the efficient achievement of the critical coupling condition, but also\nensures a rapid mismatch when deviating from this point.\nFrom the perspective of perturbation theory, the radiative Q-factor can be expressed as Q = Ck -n. The\nmerging-BIC approach effectively increases the exponent n from 2 to 6, whereas our method primarily\nreduces the prefactor C while maintaining n = 2. This distinction is crucial, as reducing C enables critical\ncoupling to occur at significantly smaller wavevectors without introducing an extended region of ultra-\nhigh Q . While ultra-high-Q resonances are highly beneficial for nonlinear optics or sensing\nrad\napplications, such uncontrollably high Q-factors are detrimental for thermal emission, where critical\ncoupling between radiative and absorptive losses is required. While the approach demonstrated in the\nwork mentioned by the reviewer (Ref. [58]) sharpens the decay of the Q factor from k-2 to k-6 by\nrad\nmerging multiple BICs, such a system leads to an infinitely large Q factor across a broad wavevector\nrad\n(angle) range near the Γ point, where the emissivity remains nearly zero. Additionally, the QBIC\nresonance caused by wavevector perturbations only corresponds to a limited wavevector range. Further\nfrom the BIC, the system transitions into traditional guided-mode resonances, where the Q-factor no\nlonger follows the k-2 or k-6 decay rates, but instead remains stable at a level comparable to the Q\nabs\nfactor, as shown in Fig. 3(a) of Ref [50].\nIn contrast, our method leverages non-Hermitian engineering to control dispersion and radiation loss,\nallowing us to match the Q factor with the Q factor at significantly smaller wavevectors (emission\nrad abs\nangles), while achieving a high-contrast mismatch upon deviation from critical coupling. This not only\nensures thermal emission with a smaller divergence angle but also ensures that perfect emissivity (PE)\noccurs at a single angle, enabling a high-purity, extreme-directional vectorial-polarized thermal meta-\nemitter. We fitted the sharpness of the Q decay in different systems, as shown in Figure S7, to directly\nrad\ncompare the differences between various systems. Clearly, our method reduces the distance between\nthe critical coupling point and the Γ point by an order of magnitude. Additionally, after deviating from\nthe critical coupling point, the Q and Q factors maintain a high contrast, ensuring that high\nrad abs\nemissivity only occurs at the critical coupling point. In contrast, traditional BIC systems cover a wide\nwavevector (emission angle) range near critical coupling, resulting in the \"rainbow effect\" of high\nemissivity over a broad range.\nFurthermore, we also designed a Merging BIC thermal emitter that supports a k-6 decay rate, as shown\nin Figure S8. Clearly, this system exhibits an infinitely large Q factor over a broader wavevector\nrad\nrange—approximately three orders of magnitude higher than our approach—leading to the critical\ncoupling occurring at larger angles, as shown in Figure S8b and c. However, the k-6 decay rate occurs\nonly over a limited wavevector range; farther from the BIC, the Q factor is no longer dominated by\nrad\nthe BIC but by guided-mode resonances with more robust Q factors. This results in a \"rainbow effect\"\nrad\nof high emissivity over a wide wavevector (output angle) range, similar to traditional BIC systems. In\nFigure S8d, we show the variation trends of the Q factor for the three different systems, where the\nrad\nyellow markers near critical coupling highlight that our work offers greater versatility in achieving small\ndivergence angles, high-directionality, and high-purity doughnut-shaped coherent vectorial-polarized\nthermal emission, while avoiding broad-angle thermal emission outputs.\nFinally, we emphasize that in merging-BIC systems, the emergence of ultra-high Q-factors significantly\nsuppresses non-Hermitian effects, such as exceptional points (EPs). In contrast, the stronger radiative\nlosses in our system naturally enhance non-Hermitian characteristics, enabling the emergence of EP-\nrelated physics that plays a central role in our approach. To make it clearer for the readers, we have\nadded the following content to the main text:\n''…It is noteworthy that due to the rapid variation of Q on the two mini-flatbands of BIC and BIC ,\nrad B A\ncritical coupling with Q occurs at emission angles as small as 1.2° and 1.42°, respectively, attaining\nabs\nnear-unity emissivity. It is important to note that both systems exhibit Q with a k-2 decay sharpness,\nrad\nas shown in Fig. S7. However, our method is able to reduce the wavevector distance between the critical\ncoupling position and the Γ point by approximately one order of magnitude. This represents an\napproximately one order of magnitude reduction in the emission angle where critical coupling occurs.\nAs the emission angle further increases, the rapid decrease of Q and the resulted deviation from\nrad\ncritical coupling condition leads to a sharp decline in emissivity, thereby effectively suppressing ThE at\nlarge angles and at other wavelengths (for detailed comparative results over a wider angular range, see\nFig. S6). For comparison, we also designed a merged BIC58 thermal emitter that supports a k-6 sharpness,\nas shown in Fig. S8. Clearly, this system exhibits an infinitely large Q factor over a broader\nwavevector range, approximately three orders of magnitude higher than in our work, resulting in the\ncritical coupling position occurring at larger angles. These results highlight the enhanced versatility of\nour approach in achieving small divergence angles, high directionality, and high-purity doughnut-\nshaped coherent vectorial-polarized ThE, while avoiding the broad angular dispersion in emission\noutput.'' \nAdditionally, we have included the following discussions in the Supplementary Information:\n''…Fig. S7 illustrates the calculated relationship between the Q factors and the wavevector for various\nsystems, with Q fitted to show a k-2 decay rate. As shown in Fig. S7a-c, the right panel provides a\nrad\nmagnified view near the critical coupling region, where the green line indicates the wavevector\ncorresponding to the critical coupling point. The results reveal that the wavevector for the critical\ncoupling position in traditional BIC systems is one order of magnitude higher than that in our non-\nHermitian-engineered four-hole metasurface system, while exhibiting near-critical coupling over a wide\nangular range. This demonstrates that our method can achieve extremely directional thermal emission.\nFig. S7 | Numerical analysis of the Q factors for various metasurface systems. a, Q factors as a function of\nwavevector k for a traditional BIC system. The blue lines represent the corresponding fitting curves with k-2. b\nand c, Q factors as a function of wavevector k for the four-hole metasurface system. The blue lines represent\nthe corresponding fitting curves with k-2. The right panel provides a magnified view near the critical coupling\npoint, with the green solid line indicating the wavevector at which critical coupling occurs.\nFor comparison, we also designed a thermal emitter based on the Merging BIC system, as shown in Fig.\nS8a, with the following geometric parameters: a = 1400 nm, h = 1300 nm, t = 620 nm, and r = 450 nm.\nFig. S8b illustrates the Q factor as a function of wavevector k, with the blue line representing the k-6 fit\nfor Q . The green line marks the wavevector corresponding to the critical coupling point, which is\nrad\napproximately 27 times larger than that for our non-Hermitian-engineered four-hole metasurface system,\nindicating that for the Merging BIC system, high emissivity is shifted toward significantly larger angles.\nFig. S8c shows the angle-resolved emission spectrum of the Merging BIC system along the ΓX direction,\ncorroborating the analysis in Fig. S8b. In particular, the emissivity is near zero within the range of -10°\nto 10°, while the maximum emissivity occurs around 50°, exhibiting near-critical coupling effects over\na broad wavevector range. This behavior arises because the k-6 decay rate holds only within a specific\nwavevector range; at larger wavevectors, the system emissivity is dominated by traditional guided-\nmode resonance that approaches the Q factor. For a clearer comparison, we plot the Q factors of the\nabs rad\nthree systems as a function of wavevector k in Fig. S8d, where the yellow near-critical coupling region\nhighlights the significant advantage of our work in achieving highly directional, doughnut-shaped\nvectorial-polarized thermal emission.\nFig. S8 | Design and performance of the Merging BIC thermal emitter. a, Schematic of the Merging BIC\nthermal emitter structure. b, Calculated Q , Q , and Q factors as a function of the wavevector k. The blue\ntotal rad abs\nlines represent the corresponding fitting curves with k-6. c, Calculated angle-resolved emission spectra (in color\nscale) for the Merging BIC system in the ΓX direction. d, Comparison of the Q factors as a function of the\nrad\nwavevector k between our work, traditional BIC system, and the merging BIC system. The yellow shading\nindicates the near-critical coupling regime. The inset shows the fitted Q , demonstrating the k-2, k-2, and k-6\nrad\ndecay rates…''\nReferences\n50. Sun, K., Wang, W. & Han, Z. High- Q resonances in periodic photonic structures. Phys. Rev. B\n109, 085426 (2024).\n58. Jin, J. et al. Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering.\nNature 574, 501–504 (2019).\n2. What happens to the polarization charges of the exceptional points at 150 nm and 173 nm? Did the\nsingularities move outside the zone shown in Fig. 3f,h (|k|>0.03)? Do they collide with other charges?\nOur response: We appreciate the reviewer’s insightful question regarding the evolution of polarization\nsingularities. To address the concern, we have calculated the far-field polarization distribution over a\nsubstantially wider wavevector range (k = [-0.1, 0.1]*π/a) to track the behavior of these singularities\nbefore and after the formation of the exceptional points.\nFor δ = 150 nm, Fig. S9a and b illustrate the far-field polarization (top) and the orientation of the\npolarization long axis (bottom) for the QGM and QGM . Both modes exhibit clear topological charges\nB A\nof +1 and -1, respectively. As δ increases to 173 nm, the polarization charge of the QGM mode remains\nB\nunchanged. However, for the QGM mode, at larger wavevectors, multiple polarization charges of 1/2\nA\nappear due to mode coupling, i.e. C-points. However, since our EPs are located at high-symmetric\ndirection of the Brillouin zone, while the C-points are symmetry-forbidden in such directions, the EPs\nthemselves have no polarization changes. Moreover, these C-points are observed only outside the mini\nflatband. We have marked the wavevector corresponding to critical coupling (orange circles) and the\nwavevector range covering the mini flatband (red and blue circles). This analysis further shows that the\nchanges in topological charge at larger wavevectors do not affect the high purity and extreme\ndirectionality of the vectorial-polarized thermal emission, as the large-angle emission is significantly\nsuppressed when δ = 173 nm.\nIn summary, the exceptional points in our system have no topological charges, and the additional C-\npoints only appear at larger wavevectors, further from the critical coupling. This means that these\nchanges do not interfere with the anticipated high purity and directionality of the thermal emission. To\nfurther clarify, we have included the relevant numerical results and discussions in the Supplementary\nInformation.\nUpdated main text:\n''…By calculating the far-field polarization vector fields of QGM and QGM near the Γ point, we\nB A\nobserve in Figs. 3f and h that BIC , carrying a topological charge of +1, exhibits an azimuthally\nB\npolarized far-field distribution (detailed numerical results for a broader wavevector range are provided\nin Fig. S9). In contrast, BIC , carrying a topological charge of -1, displays an azimuthal polarization\nA\nalong the k and k directions but radial polarization along a 45°-rotated axis…''\nx y\nUpdated Supplementary Information:\n''We calculated the far-field polarization distribution over an extended wavevector range to monitor the\nevolution of singularities before and after the formation of the perfect EP. For δ = 150 nm, Fig. S9a and\nb show the far-field polarization (top) and the orientation of the polarization long axis (bottom) for\nQGM and QGM , respectively. For δ = 173 nm, while the topological charge near the Γ point remains\nB A\nunchanged, QGM exhibits multiple polarization singularities carrying topological charges of 1/2 at\nA\nlarger wavevectors due to enhanced mode coupling. We have explicitly marked the wavevector\ncorresponding to critical coupling (orange circles) and the wavevector range covering the mini flatband\n(red and blue circles). This analysis indicates that changes in the topological charge at larger\nwavevectors do not compromise the intended functionality, as large-angle emission is significantly\nsuppressed at δ = 173 nm, ensuring high spectral purity and extreme directionality of the vectorial\npolarization thermal emission,\nFig. S9 | Numerical results of far-field polarization and topological singularity evolution. a and b,\nCalculated far-field polarization (top) and the orientation of the polarization long axis (bottom) for QGM and\nB\nQGM modes at δ = 150 nm, over a wide wavevector range. c and d, Corresponding far-field polarization (top)\nA\nand the orientation of the polarization long axis (bottom) at δ = 173 nm. The orange circles represent the\nwavevector corresponding to the critical coupling position. The red and blue circles indicate the wavevector\nrange of the respective mini-flatbands''\n3. What does Fig. 3e look like from -20 degrees to +20 degrees? It looks like decently strong thermal\nemission should be observed for QGMa since the Q-factor increases from a minimum of 40 to 200\nwhich is only a factor of 5 off the critical coupling condition. This should be very close to the +20-\ndegree case for \"Traditional BICs.\"\nOur response: We thank the reviewer for raising this insightful question regarding the thermal emission\nover a wider angular range. To address this, we have added Fig. S6 in the Supporting information to\nillustrate the emission behavior within the -20° to +20° range.\nWhile the reviewer correctly notes that the Q-factor for QGM is within a factor of 5 of the critical\nA\ncoupling condition at larger angles, our results show that the resulting emissivity remains relatively\nsuppressed compared to the near-unity emission at the Γ point. Specifically, for QGM , high emissivity\nB\nis strictly confined to the mini-flatband near 0°. For QGM , although the emissivity gradually increases\nA\nbeyond the 10° output angle, it remains below 0.6 across the calculated range. In practical applications,\nany residual large-angle emission can be further eliminated using a spatial filter (e.g., a simple aperture),\nensuring a high-purity, small-angle vectorial-polarized output.\nTo provide a more quantitative comparison, we calculated the angle-resolved emission spectra with an\nemissivity greater than 0.6, as shown in Fig. S6. Within the -20° to +20° range, our system exhibits high\nemissivity above 0.6 only near 0°. In stark contrast, traditional BIC systems exhibit high emissivity\nabove 0.6 across a broad range from 5° to 20°, accompanied by a pronounced rainbow effect.\nFurthermore, we demonstrate that using materials with lower absorption losses, thermal meta-emitters\nengineered through non-Hermitian methods could further suppress the emission at large angles. These\nfindings underscore the significant advantage of our approach in achieving high-purity, doughnut-\nshaped coherent vectorial-polarized thermal emission with high directionality.\nUpdated main text:\n''…As the emission angle further increases, the rapid decrease of Q and the resulted deviation from\nrad\ncritical coupling condition leads to a sharp decline in emissivity, thereby effectively suppressing ThE at\nlarge angles and at other wavelengths (for detailed comparative results over a wider angular range, see\nFig. S6)…''\nUpdated Supplementary Information:\n''…Fig. S6 presents the angular-resolved thermal emission spectra for different systems and absorption\nloss levels. Fig. S6a displays the emission spectrum over a wider angular range (-20° to 20°) than Fig.\n3e in the main text. Figs. S6b and d highlight the cases where the emissivity exceeds 0.6 for both systems.\nThe results confirm that the four-hole metasurface achieves high emissivity only near the Γ point, with\nsignificant emission suppression over a wide angular range. In contrast, the traditional BIC system\nmaintains high emissivity over a much broader angular range.\nFor systems with lower material absorption losses (k ~ 0.001), the angular-resolved emission\nspectrum for the four-hole metasurface is calculated and shown in Fig. S6e. Compared to Fig. S6a, the\ntemporal coherence is significantly improved, and the emissivity at larger angles is further suppressed,\nas corroborated by the results in Fig. S6f.\nFig. S6 | Comparative analysis of wide-angle thermal emission spectra. a, Angle-resolved thermal emission\nspectrum of the four-hole metasurface presented over an extended angular range, compared to Fig. 3e in the\nmain text. c, Calculated emission spectrum for a traditional BIC system for comparison. b and d, Extracted\nspectra from a and c, respectively, where the emissivity exceeds a threshold of 0.6. e, Angular-resolved emission\nspectra for the four-hole metasurface assuming reduced intrinsic loss (k~0.001). f, Corresponding spectra where\nthe emissivity exceeds 0.4 in e. ''\n4. All the experimental emission plots are presented in relative emissivity with arbitrary units. True\nemissivity needs to be reported so readers can understand expected performance levels for this type of\nnon-Hermitian engineering of thermal emitters.\nOur response: We thank the reviewer for this constructive feedback regarding the presentation of\nexperimental results. We fully agree that reporting the true emissivity is essential for evaluating the\nperformance of our non-Hermitian thermal meta-emitter.\nIn response to this suggestion, we have normalized all experimental results to the emission of a\nblackbody reference at the same area and temperature. The emissivity was determined using the relation:\nEmissivity = E / E . We have updated all relevant experimental plots in the revised manuscript\nsample blackbody\nand Supplementary Information to reflect these true emissivity values. We believe these revisions will\nprovide a clearer understanding of the system's behavior and its potential for practical applications.\nIn the main text:\n''…As the emission angle increases, the intensity first rises rapidly and then decreases sharply, which is\nconsistent with theoretical predictions. In Fig. S13, we present the measured emission spectrum at the\noutput angle of θ=2°. The fitting results show that the two modes exhibit Q-factors of 97 and 73,\ncorresponding to bandwidths of 50 nm and 64 nm respectively…''\n''…To further visualize the spatial ThE patterns of the two modes, we measured and mapped the ThE\nintensity in momentum space at the two mini-flatband wavelengths. Fig. 5d illustrates the measured\nazimuthal polarization dependence for QGM , which exhibits peak emissivity at 4.84 μm and clearly\nB\nshows a characteristic doughnut-shaped azimuthal polarization distribution in momentum space. The\npolarization ratio of the x and y components of the emissivity, with respect to the azimuthal angle β,\nfollows the relation (sinβ, -cosβ). This phenomenon is easily associated with real space scenarios, where\nthe ThE associated with QGM resonance exhibits azimuthally polarized response, with rainbow-free\nB\nand highly directional behaviours. The doughnut radius varies with the observation distance, consistent\nwith the schematic in Fig. 1c. Fig. 5e presents the emissivity associated with QGM resonance in\nA\nmomentum space, demonstrating a polarization ratio given by (sinβ, cosβ). Unlike QGM , QGM\nB A\nexhibits azimuthal polarization along the x and y axes but also radial polarization along the 45° direction.\nThese experimental results, summarized in the right panels of Figs. 5d and e, show excellent agreement\nwith the numerical predictions in Figs. 3g and i, and Figs. S10a and c. We also calculated the angular\nemission distribution for different polarization angles α, as shown in Figs. S10b and d, which further\nvalidate our experimental findings. Importantly, a single thermal emitter can simultaneously generate\nvectorial-polarized ThE at two distinct wavelengths with different characteristics, offering significant\npotential for diverse on-chip light source applications.\nFig. 5 | Experimental results of structured thermal emission. a-c, angle resolved ThE measurements\nfor different azimuthal (β) and polarization angles (α), where the color amplitude represents the emissivity.\nd and e, the momentum spatial emissivity distribution corresponding to the wavelengths of two mini\nflatbands. The right panels show the dependency of the emissivity on the polarization angle α and\nazimuthal angle β.''\nIn the Supplementary Information:\n''…We present the measured emission spectrum and the corresponding fitting results of the four-hole\nstructure at β = 0°, α = 90°, and θ = 2°. As shown in Fig. S13a, the QGM and QGM modes exhibit\nA B\nhigh Q-factors of 73 and 97, respectively, demonstrating their high temporal coherence. To calibrate\nthe absolute emissivity, a blackbody reference was measured at the same temperature (200 °C) and\neffective area (represented by the black solid line). The normalized emissivity spectrum, shown in Fig.\nS13b, was subsequently obtained by dividing the measured sample emission by the blackbody reference\nspectrum, ensuring an accurate quantitative assessment of the emitter's performance.\nFig. S13 | Experimental thermal emission spectra results. a, Experimental data (yellow circles) combined\nwith the corresponding fitting results (solid lines) of the four-hole metasurface at β = 0°, α = 90°, and θ = 2°.\nThe black line represents the reference blackbody emission at the same area and temperature. b, Normalized\nemissivity spectrum derived from the experimental data presented in a.''\n5. What is the polarizer angle (alpha) in Fig. 5d and e? Moreover, why is the expected polarization\nbehavior so hard to observe? From Fig. 5d, I do not see the near zero intensity I would expect from Fig.\n5a with a polarization angle of 0 degrees. The experimental evidence is not clear proof of the expected\npolarization distribution and the existence of vector beams. Can the authors explain why the behavior\nin Fig. 5a,b,c is so different from that in Fig. 5d,e?\nOur response: We sincerely thank the reviewer for these critical comments, which allow us to clarify\nthe interpretation of our experimental results presented in Fig. 5d and e.\nThe primary reason for the perceived difference between Figs. 5a–c and Figs. 5d–e lies in the\nmeasurement dimensionality. Figs. 5a–c represent angle-resolved spectra at specific, fixed polarization\nangles (α), whereas Figs. 5d and e are 2D momentum-space emissivity maps at resonance wavelengths,\nwhere the local polarization state (black arrows) is reconstructed from multiple measurements.\nTherefore, the polarization angle (α) in these maps is not a single fixed value; instead, the plots illustrate\nhow the local polarization state evolves with the azimuthal angle (β).\nSpecifically, for the QGM mode (Fig. 5d), the x- and y-emissivity components—measured as a\nB\nfunction of the polarization angle α within our angle-resolved setup—follow the (sinβ, -cosβ)\nrelationship relative to β. In contrast, for QGM mode (Fig. 5e), these components conform to a (sinβ,\nA\ncosβ) dependence. The local polarization orientations of the emission output, represented by the black\narrows in both panels, rigorously align with the theoretically predicted topological distributions shown\nin Figs. 3f–i, thereby providing direct experimental evidence of the generated vectorial thermal beams.\nRegarding the \"near-zero intensity\" concern, we would like to highlight that a distinct region of near-\nzero emissivity is clearly observed at the center of the momentum-space map (the Γ point) in Fig. 5d.\nThis central null is the hallmark of the doughnut-shaped emission profile, which arises from the\ntopological singularity of the BIC at the Γ point. This observation is perfectly consistent with the line-\n\ncut data in Fig. 5a, where the emissivity drops to near-zero as the angle approaches 0°.\nTo provide further clarity, we have added the dependency between α and β in the right panels of Figs.\n5d and e. Additionally, we have included the polarization distribution for both modes at fixed α values\n(α = 90°, 0°, 135°, and 45°) in the Supplementary Information, as shown in Figs. S10. These additional\nresults clearly show the expected intensity nulls and rotating nodes, further validating the vectorial\nnature of the emission and ensuring consistency with the behavior observed in Figs. 5d and e.\nRevised main text:\n''…Unlike QGM , QGM exhibits azimuthal polarization along the x and y axes but also radial\nB A\npolarization along the 45° direction. These experimental results, summarized in the right panels of Figs.\n5d and e, show excellent agreement with the numerical predictions in Figs. 3g and i, and Figs. S10a and\nc. We also calculated the angular emission distribution for different polarization angles α, as shown in\nFigs. S10b and d, which further validate our experimental findings.\nFig. 5 | Experimental results. a-c, angle resolved ThE measurements for different azimuthal (β) and\npolarization angles (α), where the color amplitude represents the emissivity. d and e, the momentum spatial\nemissivity distribution corresponding to the wavelengths of two mini flatbands. The right panels show the\ndependency between α and β…''\nIn the Supplementary Information:\n''…We calculated the emission angular distribution for two modes (BIC and BIC ) under different\nA B\npolarizations. In Fig. S10a, the thermal emission output under azimuthal polarization for BIC is shown,\nB\nconsistent with the results in Fig. 5d. We also calculated the polarization-resolved distribution of the\nBIC at α = 90°, 0°, 135°, and 45°, as shown in Fig. S10b, confirming the polarization behavior observed\nB\nin Fig. S10a. In Fig. S10c, the thermal emission polarization-resolved distribution of the BIC mode,\nA\nalternating between azimuthal and radial polarization at 45° intervals, is shown, also in agreement with\nthe results in Fig. 5d. The polarization angle distribution at different α values, shown in Fig. S10d,\nfurther confirms the polarization behavior depicted in Fig. S10c.\nFig. S10 | Numerical results of vectorial polarization characteristics. a, Calculated momentum-space\nemissivity distribution of the BIC mode, desmontrating a pure azimuthal polarization profile. b, Polarization-\nB\nresolved emissvity distribution of the BIC mode at α = 90°, 0°, 135°, and 45°, respectively. c, Momentum-\nspace emissivity distribution of the BIC mode, exhibiting an alternating azimuthal and radial polarization state\nA\nat 45° intervals. d, Polarization-resolved emissvity distribution of the BIC mode at α = 90°, 0°, 135°, and 45°,\nrespectively.''\n6. In the SI, Fig. S3 uses electric field amplitudes which obscures the characters of the modes. Plotting\nthe appropriate field components for TE and TM like modes would greatly clarify for the reader which\nmodes are used.\nOur response: We thank the reviewer for this constructive suggestion. We agree that visualizing\nspecific field components is essential for identifying the TE- and TM-like nature of the resonance modes.\nIn response, we have revised Fig. S3 to replace the same amplitude distributions with vectorial electric\nfield distributions. Specifically, we have superimposed blue arrows to represent the E , E , and E\nx y z\ncomponents for the different modes. This visualization clearly reveals the symmetry and polarization\ncharacteristics of each mode, allowing for a straightforward distinction between TE-like and TM-like\nbehaviors. We have also updated the corresponding text in the supplementary information to provide a\nmore detailed analysis of these field components and their origins.\nRevised Supplementary Information:\n''Figure S3 presents the electric field amplitude and vectorial distributions for the four modes supported\nby the two structures shown in Fig. 2, calculated at various wavevectors. By plotting the specific electric\nfield components (E, E, and E) as vector field, the TE-like and TM-like symmetries of the modes are\nx y z\nclearly identified. The results reveal a one-to-one correspondence between the modes before and after\nBrillouin zone folding, demonstrating that the modes in the four-hole unit-cell metasurface originate\nfrom the folded bands of the original Brillouin zone.\nFig. S3| Vectorial analysis of the electric field distributions. a, Electric field amplitude distributions at\nvarious wavevectors corresponding to the resonance modes in Fig. 2b of the main text. b, Electric field\namplitude distributions at k = 0.05π/a, corresponding to different coloured bands in Fig. 2e of the main text.\nx\nThe blue arrows represent the vectorial distributions of the E, E, and E components. ''\nx y z\nMinor:\n1. Line 381, Eps. should be EPs.\nOur response: We thank the reviewer for the careful reading of the manuscript. In the revised version,\nwe have made the necessary correction. Additionally, we took this opportunity to thoroughly review the\nentire manuscript to prevent similar errors from occurring.\nIn the main text:\n''…and have demonstrated the dynamic processes of formation, annihilation, and reformation of EPs.\nand EP-induced band loop…''\nReply to Reviewer #2\nReviewer #2 (Remarks to the Author):\nThe present manuscript reported the design, theoretical analysis, and experimental demonstration of a\nnon-Hermitian thermal metasurface capable of generating highly directional, rainbow-free, coherent\nthermal emission with vectorial polarization states. The authors introduced a nonlocal metasurface\ncomposed of a patterned Ge membrane atop a ZnS spacer and thin gold film. By applying Brillouin-\nzone folding and non-Hermitian mode engineering, they hybridized Fabry–Perot modes with quasi-\nguided modes, leading to the couple of symmetry-protected bound states in the continuum (BICs).\nThrough carefully tuning the perturbation parameter, the system undergoes exceptional point (EP)\nformation, annihilation, and reformation, leading to the emergence of topological mini-flatbands that\nfeature steep, wavevector-sensitive radiative loss. These bands enable critical coupling at narrowly\ndefined angles while suppressing emission elsewhere. The authors demonstrated experimentally the\ndoughnut-shaped vectorial-polarized thermal emission modes within the 3–5 um window, achieving Q-\nfactors up to ~100 and strong directional confinement.\nThe results are solid in both theory and experiment. The work presents a notable conceptual and\ntechnological advance in thermal meta-photonics by integrating non-Hermitian physics, BIC\nengineering, and dissipation control. generate. The demonstration of highly coherent structured thermal\nradiation will be one step closer for the potential applications in infrared optical manipulation and\nmicro-scale thermal imaging. I will strongly recommend for reconsideration in Nature Communications,\nif the authors can address my concerns accordingly.\nOur response: We sincerely thank reviewer #2 for the thoughtful and encouraging comments. We are\nparticularly grateful for the recognition that our results are ''solid in both theory and experiment.'' It\nis also highly encouraging to receive the reviewer's appreciation of our work as a ''notable conceptual\nand technological advance'' in the field of thermal meta-photonics. We concur with the reviewer’s\nperspective that the demonstration of highly coherent structured thermal radiation represents a\nmeaningful step toward practical applications in infrared optical manipulation and micro-scale thermal\nimaging.\nWe have carefully reviewed and addressed all the specific concerns and suggestions raised by reviewer\n#2. The manuscript has been revised correspondingly to further strengthen the clarity and rigor of our\nfindings. We believe these revisions have significantly enhanced the quality of our work.\nPlease find our detailed point-by-point responses to the reviewer's comments below. Once again, we\ngreatly appreciate the reviewer's insightful feedback and the positive recognition of the significance of\nour research.\n1. The four-hole unit-cell metasurface used in this work could be related to second-order topological\nphotonic crystal [e.g. Phys. Rev. Lett. 122, 233902 (2019), Phys. Rev. Lett. 122, 233903 (2019), Optica\n6, 786-789 (2019)], which are operating below light cone, while this work focuses on the out-of-plane\nintriguing radiation above light cone. I encourage the authors to give detailed discussions on these two\ntopics that will broaden the research interests for nanophotonic community.\nOur response: We appreciate the reviewer's insightful observation regarding the potential connection\nbetween our four-hole metasurface and second-order (higher-order) topological photonic crystals. We\nvalue the opportunity to broaden the scientific context of our study by addressing this sophisticated\ntopological framework.\nIndeed, the symmetry-protected nature of our system and the lattice reconfiguration mechanisms—\nspecifically the controlled displacement of holes within the unit cell—echo the strategies employed to\ninduce topological phase transitions in higher-order topological insulators (HOTIs). However, as the\nreviewer astutely noted, a fundamental distinction lies in the operational regime: while traditional\nHOTIs are primarily designed to host localized corner or edge states below the light cone, our work\nexplores the leaky-mode regime above the light cone.\nThis distinction is important because the out-of-plane radiation in our system introduces unique features\nand challenges, particularly in the context of thermal emission and non-Hermitian physics. The interplay\nbetween topological features, exceptional points, and wavevector-sensitive radiative loss in our system\nleads to directional, rainbow-free thermal emission, a topic not typically explored in the context of\nsecond-order topological photonic crystals.\nIn response to the reviewer's suggestion, we have now included a detailed discussion of the similarities\nand differences between second-order topological photonic crystals and our work in the manuscript.\nThis comparison highlights the unique aspects of our system, particularly the out-of-plane radiation\nabove the light cone, and the potential for further research in these two areas. We believe this discussion\nwill broaden the research interests within the nanophotonic community and stimulate future\nexplorations of topological metasurfaces in the context of thermal photonics.\nWe have added this discussion to the revised manuscript to enhance the clarity and depth of our work.\nWe thank the reviewer once again for the valuable suggestion.\nRevised main text:\n''It is noteworthy that this four-hole configuration bears a structural resemblance to second-order\ntopological photonic crystals55–57. However, a fundamental distinction lies in the operational regime:\nwhile those topological systems typically function below the light cone to host localized corner or edge\nstates, our design leverages out-of-plane radiation above the light cone. This shift into the radiative\nregime introduces unique non-Hermitian opportunities; specifically, the interplay between topological\nlattice symmetries, exceptional points and wavevector-sensitive radiative losses enables the realization\nof highly directional, rainbow-free ThE. Such dynamics represent a significant extension of the\nfunctionality usually explored in the context of second-order topological physics. By investigating the\nradiative properties of topological metasurfaces, our approach bridges the gap between topological\nphotonics and non-Hermitian thermal photonics, opening new avenues for the development of\nstructured, coherent thermal radiation sources.''\nReferences\n55. Xie, B.-Y. et al. Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional\nDielectric Photonic Crystals. Phys. Rev. Lett. 122, 233903 (2019).\n56. Chen, X.-D. et al. Direct Observation of Corner States in Second-Order Topological Photonic\nCrystal Slabs. Phys. Rev. Lett. 122, 233902 (2019).\n57. Ota, Y. et al. Photonic crystal nanocavity based on a topological corner state. Optica 6, 786 (2019).\n2. The wording ‘bilayer metasurface’ should be seriously reconsidered. Generally, ‘bilayer metasurface’\nis patterned with subwavelength meta-structures in the both two layers.\nOur response: We sincerely thank the reviewer for the valuable feedback. We agree that the term\n''bilayer metasurface'' is generally associated with metasurfaces patterned with structures in both layers.\nTo avoid any confusion, we have revised the manuscript to refer to it as a ''partially-etched metasurface''\nto more accurately describe the structure. We believe this change will improve the clarity of the\nmanuscript.\nUpdated main text:\n''…The above properties stem from the integrated use of non-Hermitian engineering, including\ntopological, band, and dissipation control mechanisms (see Fig. S2 for more details). Fig. 2\nsystematically demonstrates the details of this engineering process. We begin with a meticulously\ndesign partially-etched metasurface structure, whose geometric parameters are shown in Fig. 2a. The\ntop layer is a high-refractive-index Ge film (n = 4.16) with a total thickness of 495 nm, serving to\nsupport required guided modes (GMs). A periodic array of circular air holes, with a carefully chosen\nperiod of a = 800 nm and radius of r = 195 nm, is partially etched into the Ge layer, leaving an ultrathin\nslab with a remaining thickness of h = 125 nm…''\n2\n''…Fig. 2 | Non-Hermitian engineering in nonlocal metasurfaces. a, cross-sectional and top-view schematics\nof the original partially-etched metasurface. b, calculated photonic band structure supported by the unperturbed\nstructure in (a) along the Γ→X, Y→M, and M→Γ directions. c, top-view schematic of the new metasurface after\nintroducing perturbation δ…''\n3. The 4*4 effective Hamiltonian of TCMT contains substantial physics effects, and the authors\ndeveloped a specific case of 2*2 model to discuss EP band. Can the Hamiltonian be developed another\ncase for BIC band or mini flat band?\nOur response: We thank the reviewer for this profound suggestion regarding the effective Hamiltonian.\nThis allows us to clarify the physical intuition behind our choice of simplified modes.\nBy applying temporal coupled-mode theory (TCMT), our goal is to distill the essential physics of the\nsystem while retaining sufficient complexity to faithfully capture the underlying non-Hermitian\nphenomena. The 2×2 Hamiltonian presented in the manuscript was intentionally derived as a reduced\neffective model to isolate and illustrate the non-Hermitian coupling between two specific modes near\nthe exceptional point (EP). This simplification is highly effective for visualizing the topological\ntransition and the coalescence of eigenvalues.\nIn contrast, the formation of bound states in the continuum (BIC) bands—particularly the symmetry-\nprotected, robust BICs at the Γ point—is subject to much more stringent symmetry requirements\nimposed by the two-dimensional photonic crystal. In general photonic crystal systems, BICs can be\nregarded as polarization singularities in two-dimensional momentum space. Reproducing such BIC\nbands within a TCMT framework therefore requires the effective Hamiltonian to rigorously respect the\nintrinsic lattice symmetries and multi-channel interference conditions. While TCMT can sometimes\nsuccessfully describe BICs in simpler grating-based systems, constructing a reduced-dimensional\neffective Hamiltonian that accurately captures BIC bands in a general two-dimensional photonic crystal\nis nontrivial.\nTherefore, we find that describing the BIC bands and the mini-flatbands requires the more\ncomprehensive framework. The formation of these features is fundamentally rooted in the collective\ninterference and symmetry constraints among all four basis modes (i.e., the hybridization between the\nFabry–Pérot modes and the quasi-guided modes). While a further reduction to another case might be\nmathematically feasible under certain highly specific limits, it would inevitably obscure the multi-path\ninterference and the non-local coupling effects that are essential for the existence and flattening of the\nmini-flatbands.\nConsequently, we believe that the current TCMT Hamiltonian remains the most rigorous, efficient and\nphysically transparent tool for capturing the BICs and mini-flatbands, as it preserves the full symmetry\nand interaction landscape of the system. We hope this clarifies why we utilized the model specifically\nfor the EP physics while maintaining the model for the broader band analysis.\n4. I am confused about the role of thick gold film, the yellow bottom layer shown in Fig. 1c. Moreover,\nthe details of COMSOL simulation setup should be added (e.g. the absorption loss of gold film),\nso that it will make others repeat the simulation results.\nOur response: We thank the reviewer for raising this important point. We agree that a detailed\nclarification of the gold film's functional role and the simulation methodology is essential for the\ncompleteness of the manuscript.\nAs shown in Fig. 1c, the role of the thick gold film is indeed crucial in our design. The gold layer\nprimarily serves to prevent light transmission and acts as a reflective surface. This configuration\ntransforms the metasurface into a one-port system where, according to Kirchhoff's law of thermal\nradiation, the emissivity is maximized by minimizing reflectivity (E=1−R). By providing a reflective\nboundary, the gold layer also facilitates the specific phase-matching conditions within the ZnS spacer\nthat are required to achieve the critical coupling state for high-emissivity resonance.\nTo clarify the specific function of the gold film in the metasurface, we have now included the measured\ntransmission spectra of the fabricated sample in the supplementary information (Fig. S14), which\nconfirms that the gold film is sufficiently thick to block light transmission. Additionally, we have\nprovided more details on the COMSOL simulation setup and parameters in the revised manuscript. We\nhave also included references for the dielectric constant of the gold film. We believe these additions\nsignificantly enhance the technical transparency and reproducibility of our work. Once again, we\nappreciate the reviewer’s constructive feedback on these essential details.\nUpdated main text:\n''…A low-index ZnS layer (n = 2.22) with a thickness of t = 865 nm is inserted between the Ge and the\nAu layer, which mainly helps separate the metasurface from the metal to reduce the impact of the\ninherent loss of Au52. We numerically calculate the band structures supported by this system using the\nfinite element method (FEM) implemented in the commercial software COMSOL Multiphysics…''\n''…Due to the optically thick bottom layer of gold, which prevents light transmission (T(λ) = 0;\nexperimental transmission spectra are shown in Fig. S14), this expression simplifies to E(λ) = A(λ) = 1\n− R(λ)…''\n''…Numerical simulations\nAll numerical simulations were performed based on the FEM using the commercial software of\nCOMSOL Multiphysics. Floquet periodic boundary conditions were implemented in the lateral\ndirections of a unit cell to define the wavevectors, while PML were employed in the z direction to mimic\na non-reflecting infinite domain…''\nReference\n52. Ordal, M. A., Bell, R. J., Alexander, R. W., Long, L. L. & Querry, M. R. Optical properties of\nfourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and\nW. Appl. Opt. 24, 4493 (1985).\nUpdated Supplementary Information:\n''We measured the transmission spectrum of the four-hole metasurface device, as shown in Fig. S14.\nThe overall transmission is near zero across the spectral range, confirming that the gold film is optically\nthick enough. This configuration ensures that the structure emits only upward, where the high emissivity\nis achieved at the critical coupling conditions, as predicted by the temporal coupled mode theory4.\nFig. S14 | Transmission spectra results. a, Measured transmission spectrum of the sample. b,\nLongitudinally magnified view of the transmission spectrum shown in a. ''\n5. What is the definition of emissivity, and how to evaluate the emissivity value of traditional BIC in\nFig. 3a? How about the emissivity value of QGM_B [Fig. 3c] and QGM_A [Fig. 3d]?\nOur response: We thank the reviewer for the insightful question. In our work, the emissivity is\nevaluated based on Kirchhoff's law of thermal radiation, which states that the emissivity and\nabsorptivity of a thermally balanced object are identical. As a result, the emissivity can be expressed as\nE(λ) =A(λ)=1−R(λ)−T(λ), where R(λ) and T(λ) represent the reflectivity and transmissivity, respectively.\nGiven the optically thick gold layer in our structure, which prevents light transmission (i.e., T(λ) = 0),\nthis simplifies to E(λ) = A(λ) = 1−R(λ).\nFor the emissivity of traditional BIC in Fig. 3a, as well as the emissivity values of QGM (Fig. 3c) and\nB\nQGM (Fig. 3d), we used this simplified expression. Specifically, we calculated the absorptivity A(λ)\nA\nfor each configuration. This approach ensures that we accurately capture the emissive behavior of the\nsystem across the relevant wavelength range.\nWe hope this explanation clarifies how the emissivity values were evaluated in our work.\nUpdated main text:\n''…We conducted numerical simulations of the structure based on the final geometric design,\nincorporating realistic absorption losses to validate its ThE characteristics. According to Kirchhoff's law\nof thermal radiation, the emissivity and absorptivity of an object in thermal equilibrium are equivalent.\nConsequently, the emissivity can be characterized by the calculated absorptivity of the structure, i.e.,\nE(λ) = A(λ) = 1 − R(λ) − T(λ). Due to the optically thick bottom layer of gold, which prevents light\ntransmission (T(λ) = 0; experimental transmission spectra are shown in Fig. S14), this expression\nsimplifies to E(λ) = A(λ) = 1 − R(λ). For comparison, we first designed a conventional metasurface with\nC point group symmetry that supports BIC modes with a dispersive band, as illustrated in the right\n4V\npanel of Fig. 3a…''\nUpdated Supplementary Information:\n''We measured the transmission spectrum of the four-hole metasurface device, as shown in Fig. S14.\nThe overall transmission is near zero across the spectral range, confirming that the gold film is optically\nthick enough. This configuration ensures that the structure emits only upward, where the high emissivity\nis achieved at the critical coupling conditions, as predicted by the temporal coupled mode theory4.\nFig. S14 | Transmission spectra results. a, Measured transmission spectrum of the sample. b, Longitudinally\nmagnified view of the transmission spectrum shown in a…''\nReference in the Supplementary Information:\n4. Sun, K., Cai, Y., Huang, L. & Han, Z. Ultra-narrowband and rainbow-free mid-infrared thermal\nemitters enabled by a flat band design in distorted photonic lattices. Nat. Commun. 15, 4019 (2024).\n6. Traditional BIC has a trend that Q_rad is proportional to 1/|k|^2. I’m curious about the Q-k trends of\nQGM_B [Fig. 3c] and QGM_A [Fig. 3d].\nOur response: We sincerely thank the reviewer for this insightful and thoughtful comment. We have\nanalyzed and fitted the Q decay characteristics for different systems, with the results now detailed in\nrad\nFig. S7. As the reviewer pointed out, traditional BICs exhibit a scaling trend where Q is proportional\nrad\nto k-2. Our results confirm that for the QGM and QGM modes after non-Hermitian engineering, Q\nB A rad\nstill follows a similar trend proportional to k-2.\nFrom the perspective of perturbation theory, the radiative Q-factor can be expressed as Q=Ck-n. The\nmerging-BIC approach effectively increases the exponent n from 2 to 6, whereas our method primarily\nreduces the prefactor C while maintaining n=2. This distinction is crucial, as reducing C enables critical\ncoupling to occur at significantly smaller wavevectors without introducing an extended region of ultra-\nhigh Q . While ultra-high-Q resonances are highly beneficial for nonlinear optics or sensing\nrad\napplications, such uncontrollably high Q-factors are detrimental for thermal emission, where critical\ncoupling between radiative and absorptive losses is required.\nHowever, the core advantage of our approach is the ability to shift the critical coupling wavevector\nsignificantly closer to the Γ point, allowing us to achieve critical coupling at exceptionally small angles\n(1.2° and 1.42°). Furthermore, as the wavevector deviates from the critical coupling point, the Q\nrad\nfactor decreases to a much lower level, ensuring a high contrast with the Q factor. This mechanism\nabs\neffectively suppresses off-resonance emission and results in highly directional, rainbow-free, doughnut-\nshaped and vectorial-polarized thermal emission.\nIn contrast, traditional BIC systems exhibit near-critical coupling behavior at much larger angles and\ncovering a wide range of wavevectors (or angles), which lead to the undesirable rainbow effect with\nhigh emissivity. We have also included a comparison with a \"merged\" BIC system (k−6 scaling) in Fig.\nS8 to further highlight the superior angular confinement of our platform.\nWe have added a more detailed discussion of these trends in the revised manuscript, highlighting the\ndifferences compared to traditional BIC behavior.\nUpdated main text:\n''…It is noteworthy that due to the rapid variation of Q on the two mini-flatbands of BIC and BIC ,\nrad B A\ncritical coupling with Q occurs at emission angles as small as 1.2° and 1.42°, respectively, attaining\nabs\nnear-unity emissivity. It is important to note that both systems exhibit Q with a k-2 decay sharpness,\nrad\nas shown in Fig. S7. However, our method is able to reduce the wavevector distance between the critical\ncoupling position and the Γ point by approximately one order of magnitude. This represents an\napproximately one order of magnitude reduction in the emission angle where critical coupling occurs.\nAs the emission angle further increases, the rapid decrease of Q and the resulted deviation from\nrad\ncritical coupling condition leads to a sharp decline in emissivity, thereby effectively suppressing ThE at\nlarge angles and at other wavelengths (for detailed comparative results over a wider angular range, see\nFig. S6). For comparison, we also designed a merged BIC58 thermal emitter that supports a k-6 sharpness,\nas shown in Fig. S8. Clearly, this system exhibits an infinitely large Q factor over a broader\nwavevector range, approximately three orders of magnitude higher than in our work, resulting in the\ncritical coupling position occurring at larger angles. These results highlight the enhanced versatility of\nour approach in achieving small divergence angles, high directionality, and high-purity doughnut-\nshaped coherent vectorial-polarized ThE, while avoiding the broad angular dispersion in emission\noutput.'' \nAdditionally, we have included the following discussions in the Supplementary Information:\n''…Fig. S7 illustrated the calculated relationship between the Q factor and the wavevector for various\nsystems, with Q fitted to show a k-2 decay rate. As shown in Fig. S7a-c, the right panel provides a\nrad\nmagnified view near the critical coupling region, where the green line indicates the wavevector\ncorresponding to the critical coupling point. The results reveal that the wavevector for the critical\ncoupling position for traditional BIC systems is one order of magnitude higher than that in our non-\nHermitian-engineered four-hole metasurface system, while exhibiting near-critical coupling over a wide\nangular range. This demonstrates that our method can achieve extremely directional thermal emission.\nFig. S7 | Numerical analysis of the Q factors for various metasurface systems. a, Q factors as a function of\nwavevector k for a traditional BIC system. The blue lines represent the corresponding fitting curves with k-2. b\nand c, Q factors as a function of wavevector k for the four-hole metasurface system. The blue lines represent\nthe corresponding fitting curves with k-2. The right panel provides a magnified view near the critical coupling\npoint, with the green solid line indicating the wavevector at which critical coupling occurs.\nFor comparison, we also designed a thermal emitter based on the Merging BIC system, as shown in Fig.\nS8a, with the following geometric parameters: a = 1400 nm, h = 1300 nm, t = 620 nm, and r = 450 nm.\nFig. S8b illustrates the Q factor as a function of wavevector k, with the blue line representing the k-6 fit\nfor Q . The green line marks the wavevector corresponding to the critical coupling point, which is\nrad\napproximately 27 times larger than that for the non-Hermitian-engineered four-hole metasurface system,\nindicating that for the Merging BIC system, high emissivity is shifted toward significantly larger angles.\nFig. S8c shows the angle-resolved emission spectrum of the Merging BIC system along the ΓX direction,\ncorroborating the analysis in Fig. S8b. In particular, the emissivity is near zero within the range of -10°\nto 10°, while the maximum emissivity occurs around 50°, exhibiting near-critical coupling effects over\na broad wavevector range. This behavior arises because the k-6 decay rate holds only within a specific\nwavevector range; at larger wavevectors, the system emissivity is dominated by traditional guided-\nmode resonance that approaches the Q factor. For a clearer comparison, we plot the Q factors of\nabs rad\nthe three systems as a function of wavevector k in Fig. S8d, where the yellow near-critical coupling\nregion highlights the significant advantage of our work in achieving highly directional, doughnut-\nshaped vectorial-polarized thermal emission.\nFig. S8 | Design and performance of the Merging BIC thermal emitter. a, Schematic of the Merging BIC\nthermal emitter structure. b, Calculated Q , Q , and Q factors as a function of the wavevector k. The blue\ntotal rad abs\nlines represent the corresponding fitting curves with k-6. c, Calculated angle-resolved emission spectra (in color\nscale) for the Merging BIC system in the ΓX direction. d, Comparison of the Q factors as a function of the\nrad\nwavevector between our work, traditional BIC system, and the merging BIC system. The yellow shading\nindicates the near-critical coupling regime. The inset shows the fitted Q , demonstrating the k-2, k-2, and k-6\nrad\ndecay rates.'' \nReferences\n50. Sun, K., Wang, W. & Han, Z. High- Q resonances in periodic photonic structures. Phys. Rev. B\n109, 085426 (2024).\n58. Jin, J. et al. Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering.\nNature 574, 501–504 (2019).\n7. Some minor issues that confuse me, and may also confuse other readers.\n7.1 In Line 313 of manuscript, what is the meaning of Gamma-X-z plane?\n7.2 In Line 91 of supplementary material, the expression of phi(k) may be misunderstanding whether\nthe term arg[…] should be included in the denominator or numerator.\nOur response: We thank the reviewer for the meticulous review and constructive feedback.\nWe agree that the term \"Γ-X-z plane\" was ambiguous; accordingly, we have revised it to \"xz plane\"\nthroughout the revised manuscript. Regarding the expression for ϕ(k), we have updated it to avoid any\nmisunderstanding:\n1\n''…where 𝜙(𝒌) = arg[S (𝒌)+iS (𝒌)] (-π/2 ≤ ϕ ≤π/2) represents…''\n1 2\ntechnical consistency and eliminate any similar errors. We once again thank the reviewer for these\nconstructive suggestions, which have greatly enhanced the content and clarity of the manuscript.\nReply to Reviewer #3\nReviewer #3 (Remarks to the Author):\nIn this manuscript, Sun et al. report the realization of polarization vortices within a thermal emitter,\ndiscussing the application of non-Hermitian engineering to control thermal emission. Leveraging\ntopological concepts for thermal management is a timely pursuit, with potential implications for\nthermophotovoltaics and radiative cooling. Specifically, the authors focus on Bound States in the\nContinuum (BIC)-induced polarization vortices, utilizing non-Hermitian physics to manipulate\ndispersion and emissivity.\nWhile the subject matter is undeniably interesting, I have some reservations regarding the novelty of\nthe reported findings and the sufficiency of the experimental data to support the central claims.\nTherefore, I cannot recommend publication in its current form. I would be willing to reconsider my\ndecision only after the authors have adequately addressed the major concerns detailed below.\nOur response: We sincerely reviewer #3 for his/her critical reading of our manuscript. We are\nencouraged by the recognition that leveraging topological concepts for thermal photonics is a \"timely\npursuit\" with significant potential for applications in thermophotovoltaics and radiative cooling. We\nalso appreciate the positive remark that our subject matter is \"undeniably interesting.\"\nWe have taken the reviewer’s reservations regarding the novelty and the sufficiency of our experimental\ndata very seriously. In response, we have substantially revised the manuscript to explicitly articulate the\nfundamental physics that distinguish our work from existing studies. Furthermore, we have included\nadditional experimental characterizations and theoretical analyses (as detailed in the responses below\nand the revised Supplementary Information) to provide more robust support for our central claims.\nWe believe these revisions have significantly strengthened the manuscript, addressing the concerns\nraised while enhancing the overall clarity and impact of the work. Detailed point-by-point responses to\neach of the reviewer's comments follow below. We hope the revised version now meets the high\nstandards of Nature Communications.\nMajor Comments：\n1. Novelty and relationship to Ref. 46: Ref. 46 has already demonstrated polarization vortices in thermal\nemission. In the introduction, the authors imply that such vortex emission has not been realized, which\nis not accurate. The authors should fairly acknowledge the prior work and clearly state what additional\nphysics or capabilities the present work provides beyond Ref. 46. Besides, in my opinion, achieving a\nhigh Q factor for thermal emission at longer wavelengths is more challenging; therefore, it is not fair to\nemphasize the Q factor here, given that Ref. 46 operates at much longer wavelengths (>10 um) compared\nwith the present work (~5 um), and the measured Q factor in this work is also not particularly high.\nOur response:\nWe agree with Reviewer #3 about his/her comments on the pioneering work of Ref. 42 (Previously Ref.\n46). We fully acknowledge that Ref. 42 represents a major milestone in demonstrating that BIC-induced\ntopological singularities can be harnessed for structured thermal emissions.\nWhile Fig. 1 in our original manuscript was specifically designed to illustrate the distinction\nbetween vectorial-polarized thermal emission from the conventional BIC system (shown in Fig.\n1a) and our proposed non-Hermitian approach (shown in Fig. 1b), we recognize that the original\nphrasing in the Introduction may have been ambiguous. Accordingly, we have refined the Introduction\nsection to mention Ref. 42 as a precedent for thermal vortex generation, ensuring its contributions are\nfairly acknowledged while clearly delineating the additional physical capabilities our platform\nintroduces.\nAs established by the comparison in Fig. 1, the fundamental physics of our study represent a significant\ndeparture from the traditional BIC regime utilized in Ref. 42:\n While Ref. 42 relies on the intrinsic scaling properties of traditional BIC modes, our work\nintroduces determistic non-Hermitian engineering—specifically the manipulation of\nexceptional points—to transform the radiative loss landscape.\n By tailoring these non-Hermitian dynamics, we can decouple the emission properties from the\nfixed dispersion of the BIC. This enables us to shift the critical coupling point significantly\ncloser to the point (as visualized in Fig. 1b), allowing for highly directional, rainbow-free\nemission—features that remain a substantial challenge for conventional systems like the one in\nRef. 42.\nWe emphasize that the central contribution of this work is not the demonstration of an isolated\npolarization vortex per se, but rather the integration of topological photonics with the perspective of\nnon-Hermitian physics to achieve controllable, application-oriented thermal emission\nfunctionalities. Through deliberate structural design, our approach enables high-purity, doughnut-\nshaped, and highly directional vectorial-polarized coherent thermal emission, addressing practical\nconstraints such as angular selectivity and emission purity. In this sense, our work advances beyond\nproof-of-concept demonstrations toward a platform that is directly relevant for realistic thermal\nphotonic applications. Regarding the Q factor, we appreciate the reviewer's perspective on the inherent\nchallenges of the >10 μm spectral window. We agree that direction comparison off Q factors across\ndifferent wavelengths can be influenced by frequency-dependent material losses and fabrication\ntolerances. Consequently, we have re-centered our discussion in the revised manuscript to focus on the\ntemporal coherence of the structured thermal emission. We demonstrate that our non-Hermitian\napproach achieves state-of-the-art performance for vectorial-polarized emission in the 3–5 μm\natmospheric window, emphasizing that our mechanism effectively controls emission characteristics\neven in the presence of substantial material absorption.\nWe hope this clarification helps to better distinguish our work from Ref. 42. In response to the reviewer's\ncomments, we have revised the manuscript to more clearly highlight the additional physics and\ncapabilities introduced in our study. We again thank the reviewer for the constructive feedback, which\nhas significantly helped improve the manuscript.\nUpdated main text:\n''…Although metal-insulator-metal absorbing metasurfaces can endow individual meta-atoms with\ntailored polarization states to yield vectorial-polarized ThE, the inherent plasmonic losses severely\ndegrade temporal coherence. To the best of our knowledge, vectorial-polarized or any other form of\nstructured ThE with high temporal coherence (Q > 50) has been scarcely demonstrated. While Zhang\net al. recently reported the realization of polarization vortices in ThE42, their system maintains near-\ncritical coupling over a broad range of wavevectors and wavelengths. The persistence of this critical\ncoupling condition along the dispersive BIC band (Fig. 1a) inevitably leads to broad emission angles\nand the characteristics rainbow effect. In contrast, our non-Hermitian approach enables us to confine\nthe critical coupling state to a specific, narrow angular and spectral window (Fig. 1b). This ensures\nhighly directional, rainbow-free, doughnut-shaped vectorial-polarized ThE, which is essential for\nmeeting the demands of practical applications.'' \n''…Consequently, although conventional architectures based on dispersive BIC modes can facilitate the\ngeneration of vectorial polarizations in ThE42, they typically exhibit broadband critical coupling\nbehavior. In such systems, high emissivity persists across a wide range of angles as the resonance\nwavelength shifts, necessitating the use of auxiliary ultra-narrowband filters with omnidirectional\nresponses to achieve precise spectral selection.'' \n2. Evidence for the \"mini-flat band\": From Fig. 2, the band in panel (j) does not appear significantly\nflatter than that in (g). I suggest providing quantitative evidence (e.g., group velocity) to substantiate\nthis claim. Additionally, a 2D band structure plot would help support the claim better.\nOur response: We thank the reviewer for the insightful comment and for suggesting the inclusion of\nquantitative metrics. In response, we have performed additional calculations to substantiate the\nexistence and properties of the “mini-flatband”. Fig. S4 in the revised Supplementary Information now\npresents the 2D band structure and the calculated group velocity for different δ values, providing a direct\ncomparison between the modes shown in Figs. 2(g) and (j).\nThe quantitative results reveal a fundamental difference in the emission mechanisms between the two\nregimes:\n1) Case 1: δ = 20 nm. As observed, QGM exhibits significant dispersion. While QGM shows lower\nA B\ndispersion in both the k and k directions, the band folding induced by this small perturbation\nx y\nresults in extremely low radiative loss (high Q factor) across the entire band, as shown in Fig.\nrad\n2(g). Consequently, the critical coupling condition is met at larger emission angles, extending\nbeyond the low-dispersion range. This inevitably results in the typical rainbow thermal emission\nbehavior.\n2) Case 2: δ = 173 nm. In this optimized mini-flatband regime, the strong coupling between multiple\nmodes leads to the formation of two mini-flatbands near the  point, characterized by significantly\nsuppressed group velocity. Benefiting from the EP effect, the critical coupling occurs at smaller\nwavevectors, as indicated by the black circles in Fig. S4. Consequently, on one hand, the maximum\nemissivity is strictly confined to these circles within the flatband region, ensuring that the near-\ncritical coupling occurs at a single frequency. On the other hand, thermal emission at larger\nwavevectors is significantly suppressed due to the rapid decaying of Q . This mechanism allows\nrad\nuse to achieve doughnut-shaped, coherent, vectorial-polarized thermal emission with high-purity\nand extreme directionality, avoiding the spectral-angular dispersion seen in the =20 nm case.\nWe have updated the Supplementary Information as follows:\n''…Fig. S4 presents the numerical results of the 2D band structure and group velocity calculated for\ndifferent perturbation parameter of δ. When δ = 20 nm, as shown in Figs. S4a-c, the QGM exhibits\nA\nsignificant dispersion. While the QGM shows lower dispersion in both the k and k directions, the\nB x y\nband folding induced by this small perturbation leads to an exceptionally high radiative Q-factor across\nthe entire band2,3. As a result, critical coupling occurs only at larger angles, extending beyond the low-\ndispersion range. This inevitably leads to the typical rainbow thermal emission behavior.\nFor the optimized case (δ = 173 nm), the strong coupling between multiple modes leads to the\nformation of two distinct mini-flatbands. Benefiting from the EP effect, critical coupling occurs at\nsmaller wavevectors, as indicated by the black circles in Fig. S4. On one hand, the maximum emissivity\nis strictly confined to these circles within the flatband region, ensuring that the near-critical coupling\noccurs at a single frequency. On the other hand, thermal emission at larger wavevectors is significantly\nsuppressed due to the rapid decay of Q . This mechanism allows us to achieve doughnut-shaped,\nrad\ncoherent, vectorial-polarized thermal emission with high spectral purity and extreme directionality.\nFig. S4 | Numerical Characterization of 2D band structure and group velocity. a-c, Calculated 2D band\nstructures and group velocities for different modes at δ = 20 nm. d- f, Corresponding results for the optimized\ndesign with δ = 173 nm. The black circles indicate the distribution of the critical coupling angles.'' \nUpdated references in the Supplementary Information:\n2. Sun, K., Wang, W. & Han, Z. High- Q resonances in periodic photonic structures. Phys. Rev. B\n109, 085426 (2024).\n3. Sun, K., Cai, Y., Kivshar, Y. & Han, Z. Flatband high-Q metasurfaces inspired by coupled-resonator\noptical waveguides. Adv. Photonics 7, 056008 (2025).\n3. Critical coupling and spectral signatures: The numerical results show critical coupling occurring\nwithin a narrow bandwidth, but this feature is not clearly visible in the emission spectrum. As perfect\nabsorption is typically used to identify critical coupling, can the authors measure transmission and\nreflection to confirm this? Also, the dispersion shown in Fig. 2 (including real and imaginary parts)\ncould likely be extracted using the same experimental setup, which would strengthen the manuscript.\nOur response:\nTo address the identification of critical coupling by perfect absorption, we first refer to the transmission\nspectrum provided in Fig. S14 of the Supplementary Information. Thee measurement results confirm\nthat the bottom gold film is optically thick enough to suppress light transmission (T≈0). Consequently,\nin accordance with Kirchhoff’s law (E=A) and energy conservation for an opaque system (A=1−R),\nthe measured thermal emission is directly complementary to the reflection (E=1−R). Thus, a peak in thermal\nemission corresponds directly to a dip in reflection (critical coupling).\nWhile the reflection can be commonly measured in the near-infrared or visible regimes, directly\nmeasuring the reflection spectrum in the Mid-infrared with collimated incidence, however, presents a\nfundamental challenge when characterizing high-Q, narrow-angle resonances using standard infrared\nmicroscopy. As illustrated in Fig. R1, our FTIR microscope employs a high-numerical-aperture (NA)\nobjective to ensure a sufficient signal-to-noise ratio. This setup inherently utilizes an excitation light\ncone spanning an angular range exceeding 10° and is restricted to capturing spectra centered at oblique\nangles (around ∼20°).\nBecause the critical coupling phenomenon in our non-Hermitian system is confined to an extremely\nnarrow angular and spectral window, the use of such a broad excitation cone results in an angularly\naveraged response. This averaging effect \"washes out\" the sharp spectral dip associated with critical\ncoupling by simultaneously exciting multiple resonances within the collection cone, making it difficult\nto resolve the intrinsic critical coupling feature. Given these instrumental constraints—and supported\nby the fact that transmission is negligible—we maintain that our direct angle-resolved thermal emission\nmeasurement represents the most rigorous and reliable method for characterizing the system's\nperformance at highly collimated angles.\nFurthermore, regarding the extraction of the real and imaginary parts of the dispersion parameters,\nwe agree that this would be a valuable addition. In practice, however, experimentally quantifying the\nimaginary part (total loss rate) is complicated by fabrication-induced scattering losses. The sidewall\nroughness introduces an extrinsic scattering loss channel that contributes to the non-radiative decay,\nmaking it extremely challenging to distinguish intrinsic material absorption from scattering losses under\nthe current experimental conditions. Consequently, attempting to extract the precise complex dispersion\nparameters from the experimentally broadened spectra would likely yield inaccurate results.\nWe hope this explanation clarifies the physical and technical rationales behind our measurement\napproach and highlights the reliability of the thermal emission data presented in the manuscript.\nFig.R1| Schematic of the infrared microscope reflection measurement setup(a), along with the reflection\nspectrum(b) of the sample measured by this setup.\nUpdated main text:\n''…Due to the optically thick bottom layer of gold, which prevents light transmission (T(λ) = 0;\nexperimental transmission spectra are shown in Fig. S14), this expression simplifies to E(λ) = A(λ) = 1\n− R(λ)…''\nUpdated Supplementary Information:\n''…We measured the transmission spectrum of the four-hole metasurface device, as shown in Fig. S14.\nThe overall transmission is near zero across the spectral range, confirming that the gold film is optically\nthick enough. This configuration ensures the structure emits only upward, where high emissivity is\nachieved precisely at the critical coupling conditions, as predicted by the temporal coupled mode theory4.\nFig. S14 | Transmission spectra results. a, Measured transmission spectrum of the sample. b, Longitudinally\nmagnified view of the transmission spectrum shown in a…''\nReference in the Supplementary Information:\n4. Sun, K., Cai, Y., Huang, L. & Han, Z. Ultra-narrowband and rainbow-free mid-infrared thermal\nemitters enabled by a flat band design in distorted photonic lattices. Nat. Commun. 15, 4019 (2024).\n4. Quality of experimental data (Fig. 5): The measured emission appears asymmetric, especially in Fig.\n5(b). The authors should explain why this occurs—is it due to fabrication deviations or measurement\nmisalignment? Furthermore, the polarization plots in (d) and (e) seem to be numerical rather than\nexperimental. Since the polarization vortex is a key result of this work, experimental validation is\nneeded.\nOur response: We sincerely appreciate the reviewer's keen observation regarding the experimental data\nquality.\nRegarding the asymmetry in Fig. 5b, we acknowledge the presence of this feature in the measured\nemission. We attribute this primarily to minor optical misalignment (slight sample tilt relative to the\noptical axis) during the angle-resolved measurement, which is challenging to completely eliminate in\nour setup. Secondarily, fabrication imperfections such as sidewall roughness and minor structural non-\nuniformity across the fabricated sample area, can lead to uneven scattering losses, contributing to the\nobserved intensity variations between positive and negative angles. We have added a brief discussion\non these experimental factors in the revised manuscript to ensure transparency.\nAs for the polarization plots in Figs. 5(d) and (e), we wish to explicitly clarify that these plots are derived\nentirely from experimental measurements, not numerical simulations. To generate these figures, we\nexperimentally measured the thermal emission intensity as a function of the polarizer angle α. The data\npoints represent the extracted peak intensities from the measured spectra at the target wavevectors. The\nclose agreement with the theoretical prediction serves as robust validation of the polarization vortex\nnature of our device.\nWe thank the reviewer again for this opportunity to clarify our experimental procedures and validate\nthe authenticity of our results.\nUpdated main text:\n''…In contrast, the right panel of Fig. 5c shows that the QGM mode achieves peak emission at α = 45°,\nA\nindicating that its emission alternates between azimuthal and radial polarization at 45° intervals in\nradiation space. These experimental results are in excellent agreement with the simulations in Fig. 3\nand Fig. S5. It should be noted that the slight asymmetry observed in the angle-resolved ThE\nmeasurements in Fig. 5 can be attributed to a combination of experimental factors. Primarily, minor\noptical misalignment—specifically, a slight tilt of the sample relative to the optical axis during the\nangle-resolved capture—induces the observed intensity deviations between symmetric wavevectors.\nAdditionally, fabrication imperfections, such as sidewall roughness and minor defects in the\nmetasurface structure, may introduce non-uniform scattering losses…''\nMinor Issues：\n1.The phrase “convert incoherent thermal fluctuations into coherent vectorial-polarized beams” is\ninaccurate. Vectorial-polarized beams generally imply stable phase relationships in different directions,\nbut thermal emission is inherently incoherent with random phase. Unless the authors provide\nexperimental evidence demonstrating directional phase coherence, this statement should be revised.\nOur response: We sincerely appreciate the reviewer's rigorous assessment of the terminology. We fully\nagree that unlike coherent laser sources, thermal emission originates from spontaneous fluctuations with\nrandom phases. However, the high-Q resonance in our system acts as a spectral and spatial filter,\neffectively imparting temporal coherence and defined polarization structures to the radiated field at\nthe resonant wavelength.\nImportantly, while individual thermal emission events carry random phases, the polarization state at a\ngiven frequency and emission angle is fixed by the non-degenerate resonant mode profile. As a result,\nensemble-averaged measurements reveal stable polarization vortices and well-defined vectorial\npolarization patterns.\nTo avoid any ambiguity regarding the nature of the source, we have revised the phrase to: \"sculpt\nincoherent thermal fluctuations into coherent vectorial-polarized emission at a specific frequency.\"\nThis revision clarifies that the observed coherence—manifested as stable polarization vortices—is\nestablished in the spectral domain via the high-Q resonance, distinguishing our structured emission\nfrom broadband, incoherent thermal background.\n2. I think it is better to show the numerically calculated dispersion in Fig. 5(b) and (c) for direct\ncomparison.\nOur response: We sincerely appreciate the reviewer's thoughtful suggestion to include the numerical\ncomparisons directly in Fig. 5. We carefully considered this layout; however, since Fig. 5 is already\ndensely populated with multiple panels of experimental data, we have opted to place the corresponding\nnumerical calculations in the Supplementary Information (Fig. S5) to preserve the visual clarity of the\nmain text. This approach avoids overloading Fig. 5 while still providing the detailed numerical data\nnecessary for a direct side-by-side comparison with the experimental results. We have explicitly\nreferenced Fig. S5 in the main text to guide the reader to this comparison.\nUpdated main text:\n''…Due to the structure's high C point group symmetry, we measured the emission spectra at different\n4V\nazimuthal angles (β = 0, 22.5, 45, 180, 202.5, and 225°) and the results are presented in Figs. 5a-c (the\ncorresponding numerical results are presented in Fig. S5). The color amplitude represents the intensity\nof ThE…''\n''…In contrast, the right panel of Fig. 5c shows that the QGM mode achieves peak emission at α = 45°,\nA\nindicating that its emission alternates between azimuthal and radial polarization at 45° intervals in\nradiation space. These experimental results are in excellent agreement with the simulations in Fig. 3\nand Fig. S5…''\nUpdated Supplementary Information:\n''…For δ= 173 nm, we calculated the angle-resolved emission spectra of the four-hole metasurface under\ndifferent polarizations and azimuthal angles, as shown in Fig. S5a, b, and c. The emissivity of the QGM\nA\nmode alternates between azimuthal and radial polarization in radiation space at 45° intervals. In contrast,\nthe QGM mode maintains azimuthal polarization throughout the radiative space. These results are\nB\nconsistent with the experimental findings shown in Fig. 5a, b, and c.\nFig. S5 | Numerical angle-resolved thermal emission spectra. a, b, and c, Calculated emission spectra\ncorresponding directly to the experimental measurements shown in Fig. 5a, b, and c of the main text.'' \n3.In Fig. 2(b), how is the angle-resolved emission spectrum computed? Is it actually calculated from\nthe absorption spectrum? Please clarify.\nOur response: We thank the reviewer for the insightful question. In our work, the angle-resolved\nemission spectrum shown in Fig. 2(b) is numerically derived from the absorption spectrum. Specifically,\nthe emissivity is calculated using Kirchhoff's law of thermal radiation, which states that the emissivity\nand absorptivity of a thermally balanced object are identical. For our structure, the emissivity is\nexpressed as E(λ) = A(λ) = 1-R(λ)-T(λ), where R(λ) and T(λ) represent reflectivity and transmissivity,\nrespectively. Given the optically thick gold layer in our design, which prevents light transmission (i.e.,\nT(λ) = 0), this simplifies to E(λ) = A(λ) = 1-R(λ). Therefore, the emission spectrum is directly related\nto the absorption spectrum, or the reflectivity R(λ) calculated for each configuration, with the emissivity\nsubsequently determined by subtracting the reflectivity from 1.\nWe have added the following explanation in the revised manuscript:\n''…We conducted numerical simulations of the structure based on the final geometric design,\nincorporating realistic absorption losses to validate its ThE characteristics. According to Kirchhoff's law\nof thermal radiation, the emissivity and absorptivity of an object in thermal equilibrium are equivalent.\nConsequently, the emissivity can be characterized by the calculated absorptivity of the structure, i.e.,\nE(λ) = A(λ) = 1 − R(λ) − T(λ). Due to the optically thick bottom layer of gold, which prevents light\ntransmission (T(λ) = 0; experimental transmission spectra are shown in Fig. S14), this expression\nsimplifies to E(λ) = A(λ) = 1 − R(λ). For comparison, we first designed a conventional metasurface with\nC point group symmetry that supports BIC system with a dispersive band, as illustrated in the right\n4V\npanel of Fig. 3a…''\n4.Colorbars in Fig. S4 of the Supplementary Information are missing.\nOur response: We thank the reviewer for pointing out the missing colorbars in Fig. S4. We have now\nadded the colorbars in the revised figure, which is now Fig. S5.\nOnce again, we sincerely appreciate the reviewer's constructive feedback, which has significantly\nimproved the quality of the manuscript.\nResponse to Reviewers’ Comments\nIn this document the reviewers' comments are pasted in blue, our response in black and the parts newly\nadded to the manuscript are in red.\nReviewer #1:\nThe authors have addressed my concerns. I am happy to recommend it for publication in Nature\nCommunications.\nOur response: We sincerely thank the reviewer for his/her positive evaluation and for recommending\nour manuscript for publication in Nature Communications. We are glad that our revisions have\naddressed the reviewer’s concerns.\nReviewer #2:\nIn the revised manuscript, the problems I concern and my suggestions have been addressed\naccordingly. I will recommend a publication in current version in Nature Communications.\nOur response: We sincerely thank the reviewer for his/her positive evaluation. We are pleased that our\nrevisions addressed the reviewer’s concerns and suggestions, and we appreciate the reviewer’s\nrecommendation to publish in Nature Communications.\nReviewer #3:\nI would like to thank the authors for their detailed responses to the referees’ comments. My main\nconcerns have now been well addressed, and the revised manuscript has been significantly improved. I\ntherefore recommend this paper for publication. One suggestion is that, since the critical coupling plays\na central role in this work, it would be beneficial to include the total quality factor Q_total, as well as\nits radiative and absorptive components (Q_rad and Q_abs), for different values of the structural\nparameter \\delta shown in Fig. 2(g–h). These data could be provided in the Supplementary Information\nto further clarify the role of non-Hermitian engineering.\nOur response: We sincerely thank the reviewer for the positive evaluation of our revised manuscript\nand for recommending it for publication. We are pleased that the main concerns have been satisfactorily\naddressed.\nFollowing the reviewer’s helpful suggestion, we have now included a systematic analysis of the total\nquality factor Q , together with its radiative and absorptive components Q and Q , for different\ntotal rad abs\nvalues of the structural parameter delta corresponding to Fig. 2(g–h). These results are provided in the\nSupplementary Information (see Fig. S15), which further elucidate the role of non-Hermitian\nengineering and critical coupling in our system.\nWe believe that this additional information strengthens the physical interpretation of the critical\ncoupling mechanism and improves the completeness of the manuscript.\nUpdated Supplementary Information:\n“We further calculated the variations of Q , Q , and Q as a function of wavevector for the two\ntotal rad abs\nmodes (QGM and QGM ) at δ = 20 nm and 150 nm. As shown in Fig. S15, the results demonstrate\nB A\nthat, for a smaller δ, Q remains high over a broader wavevector range, indicating that near-critical\nrad\ncoupling occurs at larger output angles.\nFig. S15 | Calculated Q factors for different δ values. a and b show the variations of Q , Q , and Q as\ntotal rad abs\na function of wavevector for the two modes at δ = 20 nm. c and d show the corresponding results for the two\nmodes at δ = 150 nm.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed my concerns. I am happy to recommend it for publication in Nature Communications.", "reviewer_2": "In the revised manuscript, the problems I concern and my suggestions have been addressed accordingly. I will recommend a publication in current version in Nature Communications.", "reviewer_3": "I would like to thank the authors for their detailed responses to the referees’ comments. My main concerns have now been well addressed, and the revised manuscript has been significantly improved. I therefore recommend this paper for publication.\nOne suggestion is that, since the critical coupling plays a central role in this work, it would be beneficial to include the total quality factor Q_total, as well as its radiative and absorptive components (Q_rad and Q_abs), for different values of the structural parameter \\delta shown in Fig. 2(g–h). These data could be provided in the Supplementary Information to further clarify the role of non-Hermitian engineering."}, "author_response": "Response to Reviewers’ Comments\nIn this document the reviewers' comments are pasted in blue, our response in black and the parts newly\nadded to the manuscript are in red.\nReviewer #1:\nThe authors have addressed my concerns. I am happy to recommend it for publication in Nature\nCommunications.\nOur response: We sincerely thank the reviewer for his/her positive evaluation and for recommending\nour manuscript for publication in Nature Communications. We are glad that our revisions have\naddressed the reviewer’s concerns.\nReviewer #2:\nIn the revised manuscript, the problems I concern and my suggestions have been addressed\naccordingly. I will recommend a publication in current version in Nature Communications.\nOur response: We sincerely thank the reviewer for his/her positive evaluation. We are pleased that our\nrevisions addressed the reviewer’s concerns and suggestions, and we appreciate the reviewer’s\nrecommendation to publish in Nature Communications.\nReviewer #3:\nI would like to thank the authors for their detailed responses to the referees’ comments. My main\nconcerns have now been well addressed, and the revised manuscript has been significantly improved. I\ntherefore recommend this paper for publication. One suggestion is that, since the critical coupling plays\na central role in this work, it would be beneficial to include the total quality factor Q_total, as well as\nits radiative and absorptive components (Q_rad and Q_abs), for different values of the structural\nparameter \\delta shown in Fig. 2(g–h). These data could be provided in the Supplementary Information\nto further clarify the role of non-Hermitian engineering.\nOur response: We sincerely thank the reviewer for the positive evaluation of our revised manuscript\nand for recommending it for publication. We are pleased that the main concerns have been satisfactorily\naddressed.\nFollowing the reviewer’s helpful suggestion, we have now included a systematic analysis of the total\nquality factor Q , together with its radiative and absorptive components Q and Q , for different\ntotal rad abs\nvalues of the structural parameter delta corresponding to Fig. 2(g–h). These results are provided in the\nSupplementary Information (see Fig. S15), which further elucidate the role of non-Hermitian\nengineering and critical coupling in our system.\nWe believe that this additional information strengthens the physical interpretation of the critical\ncoupling mechanism and improves the completeness of the manuscript.\nUpdated Supplementary Information:\n“We further calculated the variations of Q , Q , and Q as a function of wavevector for the two\ntotal rad abs\nmodes (QGM and QGM ) at δ = 20 nm and 150 nm. As shown in Fig. S15, the results demonstrate\nB A\nthat, for a smaller δ, Q remains high over a broader wavevector range, indicating that near-critical\nrad\ncoupling occurs at larger output angles.\nFig. S15 | Calculated Q factors for different δ values. a and b show the variations of Q , Q , and Q as\ntotal rad abs\na function of wavevector for the two modes at δ = 20 nm. c and d show the corresponding results for the two\nmodes at δ = 150 nm.”"}]

Decision letter