Non-Hermitian chiral coalescence associated with polarization in optical scattering systems has been routinely realized and applied. However, the scattering exceptional points (EPs) associated with polarization obtained by modes with broadband response and high dissipation constrain its further application in narrowband optics. Here, as a scheme, distinct from the implementation pathway of traditional EPs, we introduce a quasi-bound state in the continuum based on a quadrupole mode as the response environment for EP generation, achieving an EP with high-quality-factor characteristics on a metasurface. Furthermore, we demonstrate the robustness of EPs as intrinsic features of non-Hermitian systems, independent of specific parameter choices. Finally, we present an example of direct chirality detection and display, showcasing the maximum spin-selective property induced by EPs. Our approach unveils the potential of scattering chiral EPs for applications in narrowband optics.
【AIGC One Sentence Reading】:Non-Hermitian chiral coalescence in optical scattering is limited by broadband response and high dissipation. We introduce a quasi-bound state in the continuum via a quadrupole mode, achieving a high-quality-factor EP on a metasurface, demonstrating EP robustness and chirality detection.
【AIGC Short Abstract】:Non-Hermitian chiral coalescence linked to polarization is widely used in optical systems, but scattering exceptional points (EPs) with broadband response and high dissipation limit narrowband applications. We propose a new approach, using a quasi-bound state in the continuum based on a quadrupole mode to create an EP with high-quality-factor on a metasurface. This EP is robust, independent of specific parameters, showcasing non-Hermitian system features. We also demonstrate direct chirality detection and display, highlighting the EP-induced spin-selective property, opening potential applications in narrowband optics.
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1. INTRODUCTION
Tailorable spin-selective manipulation based on polarization conversion is essential for precisely controlling optical fields [1,2]. Metasurfaces, as integrated platforms, have demonstrated significant potential in regulating polarization conversion in various optical phenomena, including scattering and absorption [3–6], imaging and holography [7–11], sensing [12], quantum state modulation [13,14], and on-chip integration [15]. However, maximizing spin-dependent transmission asymmetry remains challenging, primarily due to the disorder introduced by the unguided modulation of numerous coupling components and degrees of freedom. A promising solution to this issue lies in the exploitation of non-Hermitian chiral coalescence within scattering systems [16–26]. In realistic open optical systems, non-Hermitian corrections become non-negligible. Exceptional points (EPs), unique to non-Hermitian systems, underlie optical phenomena like electromagnetically induced transparency [27], unidirectional reflectionlessness [28–32], extreme sensing [33–36], and coherent perfect absorption [37], distinguishing them from Hermitian systems.
Building on the pioneering work of Lawrence et al. [18], there has been growing interest in investigating scattering EPs based on metasurfaces associated with polarization. Recent advances, as demonstrated by Kang et al. [21], have simplified the realization of EPs by decoupling the design from the necessity of parity-time symmetry, showing that parity-time symmetry is not a prerequisite for EPs [38]. With the introduction of a new parameter space, supported by the Fermi energy of graphene [20,25], the mechanisms for implementing EPs are now well established. Hence, the focus has shifted towards exploiting EP features for new applications, such as sensing [39,40], vortex light generation [24], asymmetric holography [22,41–46], and full Stokes detection [23]. However, existing EPs typically exhibit a broad response lineshape, limiting their utility in applications requiring narrow bandwidth responses. Alternatively, bound states in the continuum (BICs)—dark modes in the scattering region of photonic platforms—can degenerate into quasi-BIC modes with both high quality () factors and narrowband radiative behavior when weakly perturbed [47]. These high- quasi-BIC modes are particularly advantageous for enhancing light-matter interactions [48–58], opening novel avenues for the design of scattering EPs.
In this work, we provide a scheme to derive polarization-associated scattering EPs from a quasi-BIC based on a quadrupole mode, thereby enabling the EP to inherit high- characteristics. We introduce a perturbed rectangular ring with a gap that supports a chiral quasi-BIC mode induced by extrinsic chirality. By meticulously adjusting the asymmetry within the structure plane, non-Hermitian chiral coalescence emerges from these quasi-BIC modes. Furthermore, by leveraging the structure design of bulk Dirac semimetal (BDS)—the three-dimensional (3D) counterparts of graphene with comparable tunability of Fermi energy [59], we demonstrate that EPs can be consistently located in the parameter space defined by the Fermi energy and wavelength, or the structural perturbations and wavelength. This finding suggests that EPs are intrinsic signatures of the non-Hermitian system rather than mere outcomes of parameterization. Finally, as an application of the maximum spin-selective asymmetry induced by EPs, we highlight the effectiveness of the designed metasurface in direct chiral detection and display. Our results open up new possibilities for implementing non-Hermitian chiral coalescence based on a narrower bandwidth response, representing a significant extension of the EP design.
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2. CHIRAL QUASI-BICS AS THE ENVIRONMENT FOR ASYMMETRIC CIRCULAR POLARIZATION RESPONSE
A. Geometry Structure and Theoretical Model
To realize the EP originating from the quasi-BIC mode, we illustrate a tunable metasurface based on BDS in Fig. 1, which consists of periodically arranged split-ring resonators (SRRs) with broken symmetry. Although graphene is widely used for its Fermi energy tunability, its two-dimensional (2D) material nature makes it unsuitable for constructing the three-dimensional (3D) structure of a metasurface. Instead, BDS, which also offers tunable Fermi energy, serves as an excellent 3D alternative. The BDS can be modeled by complex conductivity using the Kubo formalism under the random phase approximation (RPA) in the long-wavelength limit [59], where the real and imaginary parts of the conductivity, , are given by with the complex frequency related to the Fermi energy of BDS, and the Fermi momentum . denotes the Heaviside step function with a frequency threshold for the conductivity contribution. The BDS selected here is AlCuFe quasi-crystal [60], with the degeneracy factor , the reduced cutoff energy , and the relaxation time . Considering the interband electronic transitions, the equivalent permittivity of the BDS can be expressed as [59] where and is the vacuum permittivity. The near-terahertz band, which exceeds the plasma resonance frequency of the BDS, is selected as the frequency band of interest. The subwavelength unit cell is designed with specific dimensions for the length and for the width of the outer () and inner () rectangles, respectively. The scattering properties of this structure are simulated using the 3D finite-difference time-domain (FDTD) method. Periodic boundary conditions are utilized in the direction of periodicity, while perfectly matched layers are applied in the -direction to absorb outgoing waves. It is important to note that the schemes and outcomes discussed here are not material-specific, and alternative materials are also generally applicable. As a theoretical model, the entire scattering response of the non-Hermitian system is modeled as two contributions, including direct scattering and resonant scattering. Utilizing temporal coupled-mode theory (TCMT) [21,61,62], we theoretically derive the physical processes associated with EPs and BICs, as well as the phenomena occurring in their vicinity (see Appendix A).
Figure 1.(a) Schematic representation of the unit cells of the metasurface. The unit cell period is defined as . The rectangular rings are characterized by their outer and inner dimensions: the outer rectangle has a length of and a width of , while the inner rectangle has a length of and a width of . (b) Eigenmode analysis of the metasurface with a rectangular ring in the absence of perturbation (), illustrating the transition of a symmetry-protected BIC (SP-BIC at ) to a quasi-BIC. Error bars indicate the linewidth of the eigenmodes. (c) Schematic diagram of spawning an EP from a quasi-BIC. Based on the quasi-BIC of the quadrupole mode, high- characteristics are imparted to the EP. The white arrow denotes the mode schematic. The inset illustrates the LCP and RCP responses at the EP, where the metasurface completely suppresses the scattering channel from RCP to LCP.
Starting with the rectangular ring structure, we calculate its eigenmode, as illustrated in Fig. 1(b). Along the direction of the first Brillouin zone, a transverse electric (TE)-like band structure is obtained. Away from the point, the radiative rate of the eigenmode of interest can be evidently identified, characterized by the full width at half maximum (FWHM) and marked as error bars. The symmetry-protected BIC is captured at the point, where the FWHM vanishes, manifesting an infinite factor in the absence of intrinsic losses. This reveals that quasi-BIC modes with both a high factor and radiative properties can be achieved by using oblique incidence as an extrinsic perturbation. Conversely, as an intrinsic way, the in-plane symmetry breaking can also perturb the BIC mode. As shown schematically in Fig. 1(c), this perturbation is realized by introducing a defect into the rectangular ring, transforming it into a split ring with a gap denoted by . Subsequent discussions will be based on the quasi-BIC mode derived from this approach.
In prior studies, quasi-BIC modes have been endowed with chiral features via in-plane symmetry breaking. We note that the modes with distinct chirality excitations exhibit a slight spectral separation. Furthermore, the EP of the non-Hermitian systems can be localized at a perturbation level, displaying a maximum asymmetry, i.e., , between polarization conversion processes for circularly polarized (CP) incidences with opposite chirality.
For a clear depiction of the evolution of the BIC mode to the quasi-BIC mode due to geometric symmetry breaking, Fig. 2 shows the response of a rectangular ring with an incremental gap under opposite chiral incidences. To provide a generalized description of the BIC family, a dimensionless parameter is introduced to characterize the degree of geometric asymmetry in the structure, which in turn defines the extent of symmetry breaking. Figure 2(a) illustrates the transmission spectra observed under left-hand circular polarization (LCP) and right-hand circular polarization (RCP) excitations. As increases, the resonance linewidth increases, accompanied by a blueshift in the mode frequency.
Figure 2.Quasi-BIC supported by the metasurface with broken in-plane symmetry. (a) Transmission characteristics for different chiral incidences as the asymmetry parameter increases. The response modes indicated by black arrows within the dashed box correspond to the structure’s behavior under LCP and RCP incidence, at an asymmetry value of . (b) -factors extracted from the spectrum in (a). An inverse quadratic fit, represented by the solid gray line, is characteristic of quasi-BICs with broken symmetry. (c) Distribution of surface currents and the -component of the electric field for the modes corresponding to black arrows in (a). Both configurations exhibit behavior indicative of a magnetic quadrupole mode.
While the two spectral responses excited by different polarizations exhibit a similar trend, they do not overlap due to the presence of chirality. The discrepancy in the lineshapes of the two spectra can be explained by the Fano resonance, which attributes the differences to the distinct interference effects experienced by the narrowly separated resonances associated with opposite chiralities. These interference effects result in non-overlapping Fano profiles.
Furthermore, as a significant characteristic of BICs, the factors of the response modes related to LCP and RCP incidences are presented in Fig. 2(b). In systems with Fano resonance, the factor can be reasonably assessed through [49] where represents the resonant frequency of the mode; and denote peaks and dips of transmission, respectively, with the resonance width . At an asymmetry parameter or , the factor attains 312.40 under RCP, while it reaches 541.23 for LCP incidence. As the structure becomes increasingly symmetric, the factors tend toward infinity, causing the data points for various values to converge to a single transmission spectrum. At the same time, the factor exhibits an inverse quadratic relationship with the asymmetry parameter, as illustrated by the fitting curves represented by the gray solid lines in Fig. 2(b). This relationship is a characteristic feature of symmetry-protected BICs [63].
To gain deeper insight into the physics of the chiral quasi-BIC, Fig. 2(c) concentrates on the -component of the electric field distribution and the surface current distribution for opposite chiral incidences at . Given that our structures are composed of orthogonal antenna combinations, symmetry incompatibility allows for the direct extraction of electromagnetic multilevel modes from the surface current distributions [64], thus bypassing radiative coupling [18]. At the characteristic positions represented by the black arrows in Fig. 2(a), the electric field results indicate that a quadrupole mode is manifested under both LCP and RCP, which is a typical subradiant mode indicative of high factors. From the perspective of surface currents, perturbed quasi-magnetic quadrupole modes can be identified near the black arrows in Fig. 2(a). The currents converging towards the diagonals are akin to the combination of two electric quadrupole modes, effectively equivalent to two reversed magnetic dipoles. The Fano lineshapes are the consequence of the interaction of these modes with the background dipole mode. In symmetry-protected structures, perfect quadrupole modes without a net dipole moment cannot be excited, and therefore only background dipole radiation leakage is observed, which manifests as a BIC.
3. EXCEPTIONAL POINT SPAWNING FROM QUASI-BIC
A. Maximal Asymmetric Polarization Conversion
We proceed to engineer the chiral coalescence at an EP in non-Hermitian systems with tailored symmetry breaking, leveraging the quasi-BIC mode as a foundation. This design enables the achievement of the maximum spin-selective asymmetric transmission via polarization conversion. By traversing the parameter space , which comprises the variables of wavelength () and the gap parameter (), we identify the maximal asymmetric polarization conversion occurring at the EP, as illustrated in Fig. 3. The disappearance of the polarization conversion from RCP to LCP at the EP suggests that, for the RCP incidence, the polarization state is maintained in the emitting light. Conversely, no such preservation is observed for incident light with the opposite chirality. Moreover, as indicated by the color map in Fig. 3, the asymmetry of the phase variation in the parameter space implies a difference of the mode response for distinct chiralities, which underpins the asymmetry of the polarization conversion. Meanwhile, the phase distribution in the parameter space is characterized by a geometric phase [65], meaning that encircling the EP will always be accompanied by a -phase accumulation. This phase behavior is a distinctive feature of systems with non-Hermitian dynamics and symmetry breaking, carrying significant implications for the control and manipulation of light at the EP.
Figure 3.Transmission coefficients and for circular polarization conversion in the parameter space. The maximum asymmetry of the polarization conversion is detected at . The colormap of indicates a vortex phase profile adjacent to the vanishing polarization conversion.
To concretize the simplicial behavior of the non-Hermitian system around EP, we demonstrate the distribution and evolution of the eigenvalues and eigenstates of this scattering system. As a consequence of Fig. 3 for the off-diagonal elements of the Jones matrix, the polarization transitions from RCP to LCP will disappear at . Similarly, at , () from Eq. (A16). A redshift is observed in comparison to the simulation result . With the introduction of the coefficients and to as Eq. (A4), the conservation of energy is more precisely described, leading to .
B. Evolution of Eigenvalues and Eigenstates of Non-Hermitian Systems in Parameter Space
The eigenvalues, i.e., the eigen-transmission coefficients, are solved by the Jones matrix. A more explicit picture of these features is presented in Figs. 4(a) and 4(b), which are calculated using TCMT based on the eigen-parameters obtained by fitting the numerical data. Our focus is on the region surrounding the EP. The blue and red sheets of the Riemann surface denote the two eigenvalues of the matrix. We can intuitively recognize the anti-crossing and crossing properties of the amplitudes and phases of the eigenvalues. The white point is the projection of the EP onto this parameter space. The eigenvalue amplitude evolves from a merged state to a split state as decreases, while the eigenvalue phase exhibits an opposite trend. The coupling strength , extracted from the fitting is found to be inversely proportional to . This suggests that the observed evolution of the eigenvalues with decreasing is consistent with the system transitioning from a weak to a strong coupling regime [66]. Indeed, it is natural for the coupling strength to decrease as the gap widens, a consequence of structure disentanglement. The structure can be visualized as a direct coupling between a C-ring with a horizontal opening and a vertical bar with variable length, which governs the responses of the - and -directions, respectively. As increases, the vertical response weakens, leading to a reduction in the coupling between the two elements.
Figure 4.Theoretical analysis of eigenvalue and eigenstate dynamics in the parameter space of non-Hermitian systems. Amplitude (a) and phase (b) distribution of the eigenvalues in the parameter space, where the blue and red denote the two distinct eigenvalues of the non-Hermitian system. The yellow pentagram indicates the locations of the EPs. The representative evolution of eigenstate in the parameter space: encircling EP (c), evolution with for fixed (d), and evolution with for fixed (e). (c) Extracted eigenvalue trajectories encircling the EP on Riemann surfaces with corresponding eigenstate tracking. “A”–“G”: selected characteristic states along the trajectory for illustrative demonstration. The black scatter in the Poincaré sphere is RCP. (d), (e) Eigenstate responses under parameter variations with fixed EP-corresponding wavelength (d) or geometry (e). Extended parameter ranges from (a) and (b) used for comprehensive understanding.
The eigenstates associated with the eigenvalues are characterized as the polarization eigenstates. In the scattering system described by the Jones matrix, the non-orthogonal eigenstates correspond to two polarization bases that have orthogonal long axes but share the same direction of rotation. For the sake of clarity and without loss of generality, we have identified three paths in the parameter space related to the EP, as depicted in Figs. 4(c)–4(e), to scrutinize the evolution of eigenstates. We begin by selecting an elliptical path with a major axis and a minor axis , encircling the EP. Starting from the initial point on this path, after a half-period, the trajectory crosses the branch cut of the Riemann surface, transitioning from one sheet to another, as illustrated by the shift from red to blue in Fig. 4(c). Intriguingly, in contrast to the Hermitian system, the eigenstate does not return to its original state after a single loop around the EP. The half-integer topological charge of the EP, defined by the loop integral of the ellipse angle , i.e., [67], indicates that the eigenstate only reverts to its initial state after two complete loops. The sign of the topological charge is linked to the chirality of the quasi-BIC modes. Furthermore, in the lower panel of Fig. 4(c), we probe the mapping of eigenstate evolution along the trajectory onto the Poincaré sphere. The detected closed path, comprising two encirclements on the Riemann surface, provides direct evidence for the topological feature of the EP. By analyzing the perturbation induced by the bottom-up slit with a definition of , we can identify an EP with opposite chirality in the parameter space, which will be explored further in Section 4. Additionally, the evolutions of traversing EP are illustrated in Figs. 4(d) and 4(e). These figures demonstrate a 45° rotation of the eigenstates before and after passing through the EP. At the EP itself, the eigenstates coalesce to RCP, signifying the collapse of the two-state space dimensionality. Further, to provide a comprehensive view of the eigenstate evolution along different paths in the parameter space, we present a comparative visualization of eigenstate behaviors of the cross sections of Riemann surfaces near the EP, as presented in Section 5.B.
C. Causal Relationship between Chiral Quasi-BIC Modes and EPs
Figure 5 elucidates the physical mechanism linking chiral BICs to EPs through Fano resonance and energy conservation principles. As illustrated in Fig. 5(a), circularly polarized illumination excites a quadrupolar quasi-BIC mode that interacts with a dipolar broadband mode, generating a chiral Fano interference pattern. The red and blue curves respectively represent resonance lineshapes under RCP/LCP excitation, with gray dashed lines marking the quasi-BIC spectral positions. Structural symmetry breaking induces controlled spectral shifts along the direction indicated by the yellow arrow. Detailed surface current analysis reveals two critical phenomena: (1) the quadrupole-dipole interaction establishes the fundamental Fano resonance framework, and (2) RCP excitation generates stronger mode coupling than LCP, creating intrinsic chiral asymmetry in the Fano lineshape. Neither mode exhibits pure circular polarization radiation, instead emitting hybrid states composed of linear combinations of and basis states—a prerequisite for extrinsic chirality emergence under symmetry perturbation. Figure 5(b) quantitatively verifies these mechanisms at the EP condition. The total transmittance spectrum confirms chiral excitation asymmetry, while the co-polarized contribution reveals the change in the dominant input state after chiral incidence. Taking RCP incidence as an example, at the peak on the right side of EP, the proportion of the state increases, while at the dip on the left side, its proportion decreases. Energy conservation mandates complementary behavior in cross-polarized conversion, indicating that the state decreases on the right side of EP and increases on the left side, corresponding to the red line in the rightmost panel of Fig. 5(b), and the total energy of polarization conversion is relatively low. When the state decreases to zero, the output state is completely . Maximum polarization conversion asymmetry () coincides with EP formation, with LCP’s weaker mode coupling amplifying this dichotomy. Mirroring this analysis under LCP illumination reveals reversed chiral behavior, completing the demonstration of parity-controlled EP characteristics.
Figure 5.(a) Framework for EP emergence from quasi-BIC modes via Fano resonance. The incident CP light excites a quadrupole quasi-BIC mode () coupled with a dipole broad mode (), forming a Fano lineshape. Red/blue represents RCP/LCP-induced responses, with the gray dashed line marking quasi-BIC spectral positions. The yellow arrows indicate symmetry-breaking-induced spectral shifts. (b) EP-specific transmission spectra analysis: (left) total transmittance under chiral illumination; (middle) co-polarized component conversion efficiency; (right) cross-polarized conversion efficiency. Gray dashed lines denote EP resonance positions.
To gain a more comprehensive insight into the non-Hermitian systems supporting EP, we maintain the geometry of the structure while varying the Fermi energy. This enables us to monitor the evolution of nondiagonal elements of the Jones matrix and the corresponding eigenvalues. This approach leverages the tunable Fermi energy of BDS, which can be achieved by tuning the gate voltage [68] or performing surface doping with alkali metals [60]. Figures 6(a) and 6(b) present the real and imaginary parts of the permittivity of BDS as the Fermi energy varies from 0.12 eV to 0.16 eV, respectively. With increasing Fermi energy, the transmission coefficients of polarization conversion from RCP to LCP are shown in Fig. 6(c). It is observed that the suppression of polarization conversion is generally incomplete except at the Fermi energy of 0.15 eV (indicated by the blue line), where vanishes apparently. Recalling the response near the EP by adjusting the geometrical parameter , we further investigate the eigenvalues for different Fermi energies, illustrating their amplitudes in Fig. 6(d). Notably, the two Riemann surfaces begin to split once the Fermi energy exceeds 0.15 eV, a hallmark of an EP. As a consequence, by analyzing the evolution of the eigenvalues in the parameter space upon variation of the Fermi energy, we find that a stable EP persists despite the transformation of the parameter space. This observation reveals that the EP is an intrinsic property of the system, signifying the dimensional collapse of the non-Hermitian system rather than being dependent on a particular choice of parameters. Therefore, for a given set of system parameters at an EP, a fixed EP can always be identified by varying any two parameters to construct a 2D parameter space while keeping the others constant.
Figure 6.Construction of an alternative parameter space with tunable Fermi energy of BDS. (a) As Fermi energy increases, the real part of the permittivity of BDS increases. (b) Concurrently, the imaginary part of the permittivity decreases. (c) The CP conversion coefficients vary with distinct Fermi energies. The blue solid line, indicating complete suppression of the polarization conversion, corresponds to a Fermi energy , which is the EP discussed previously. (d) The eigenvalues of the system evolve with growing Fermi energy, with the gray dashed line illustrating the resonance positions on the upper and lower Riemann surfaces.
4. MAXIMAL ASYMMETRIC SPIN-SELECTIVITY FOR CHIRAL DETECTION AND DISPLAY
Last but not least, as an application of maximum asymmetry in spin-selective transmission supported by EPs spawning from quasi-BIC, we demonstrate chiral detection and display. The structure was optimized with implementation feasibility in mind, as discussed in Appendix B. For near-field detection, the key parameter is mode resonance. At the EP, based on maximum spin selection, only RCP activates the quasi-quadrupole mode, while LCP does not. This implies in a disparity in the strength of the resonant response for different chiralities, as evidenced in Figs. 12(b) and 12(c). Figures 12(d) and 12(e) suggest that mirror symmetry induces a chiral transition in the response, driven by another EP of opposite chirality in the parameter space. The bright state of the strong resonance is detected in the near field and labeled as “1”, while the response of the other chirality is the dark state, tagged as “0”. Thus, the chirality of the incident field can be directly detected based on the structure corresponding to the bright “1” state.
To construct this integrated metasurface platform, as shown in Fig. 7(a), we first map the 2-bit grayscale image to a numerical matrix, where bright and dark pixels are assigned values of one and zero, respectively. Then, by matching the bright and dark pixels to unit structures with strong and weak resonance, respectively, we obtain the designed metasurface. The outer frame in the schematic on the right side of Fig. 7(a) is intended for clamping or support, with dimensions adjusted according to the stability exhibited by the metasurface structure. The EP considered in this work corresponds to RCP, and as an example, the unit cells for configurations unmanipulated and rotated 180° around the -axis are defined as “1” and “0”, respectively. When detecting the total field directly with unknown chirality of the incidence, the illumination of the alphabets consisting of the structures for the “1” state enables identification of RCP, while the illumination of the corresponding structures of the “0” state confirms LCP, as shown in Figs. 7(b) and 7(c). When the incident chirality is known, polarization conversion filtering can provide higher contrast, a result of the complete suppression of one-sided polarization conversion by the EP. This offers greater potential in display scenarios requiring more detail, as demonstrated in Figs. 7(c) and 7(d).
Figure 7.Integrated chiral detection-display platform on a metasurface utilizing maximum spin transmission asymmetry induced by EP. (a) Schematic of the metasurface design process. The designed 2-bit grayscale image is mapped to a numerical matrix, and by matching unit structures, a metasurface achieving near-field display functionality is obtained. The symbol “1” represents the unit cell that exhibits strong resonance response in conversion, while “0” corresponds to the unit cell that shows strong resonance in conversion. The platform’s dual functionality is demonstrated in (b)–(e): (b), (c) chiral detection via total field analysis and (d), (e) high-contrast chiral display through polarization-filtered field display.
In this section, we further discuss three aspects: the identification of EP, eigenvalue evolution on the Poincaré sphere, and potential advantages of weak chirality.
A. Evolution on the Complex Plane and Parameter Addressing
The evolution paths of the conversion coefficient in the 2D complex plane under various structural perturbations are shown in Fig. 8. Scattering EPs manifest as purely zero-valued points in space. As the parameter increases, the vanishing cross-polarized transmission coefficients, denoted as in Fig. 8(a), encircle the origin in the parameter space. This shift corresponds to the topological winding number transition from zero to one. In comparison with the evolution of depicted in Fig. 8(b), only one of the two polarization conversions undergoes an abrupt change of the winding number, while the other does not. This serves as the mechanism behind the maximum asymmetric spin-selective transmission, ensuring the robustness of the asymmetric transmission against structural perturbations. Such distinct behavior is a direct consequence of the topological nature of the system. Consequently, we point out that, based on the trajectory asymmetry, the approach towards the origin can aid in addressing the system parameters corresponding to the EPs.
Figure 8.Evolution of the CP conversion coefficient with wavelength for fixed geometry in the complex plane of (a) and (b) . The black spot in the center of the plane is EP, and the yellow line is the case .
The eigenstates mapped onto the Poincaré sphere are not always distributed at the ends of the diameter of the sphere, which provides compelling evidence that non-Hermitian systems have non-orthogonal eigenstates. Figures 9(a) and 9(b) illustrate the evolution of the eigenstates with (fixed wavelength) and wavelength (fixed ), respectively, where color gradients (from light to dark) indicate evolution direction. Prior to reaching the EP, the two eigenstates tend to converge toward the north pole (i.e., RCP) before diverging, until EP almost reaches the pole. After the EP, however, the continuous evolution abruptly transitions into a different state, a behavior attributed to the branch cut of the Riemann surface.
Figure 9.Polarization eigenstate evolution of distinct paths modulated. (a) Evolution of the eigenstates with increasing geometrical defect for a fixed wavelength , where corresponds to the process in Fig. 4(d). (b) Evolution of the eigenstates with increasing for a fixed , where corresponds to the process in Fig. 4(e). Blue and red denote the two eigenstates. The color gradient indicates the running direction of the parameter, as shown by the direction of the colored arrows. The along the black dashed arrows in (a) represent 1 μm, 2 μm, 2.10 μm (), 2.25 μm, and 3.10 μm, while the black dashed arrows in (b) relate to of 29 μm, 29.18 μm, 29.25 μm (), 29.31 μm, and 29.50 μm.
C. Behavior of Non-reciprocal Forbidden Polarization Conversion in Parameter Space under Oblique Incidence
To better understand the origin and evolution of the EP, we expand our parameter space to three dimensions by introducing the incident angle as an additional degree of freedom. We define non-reciprocal forbidden polarization conversion (NR-FPC) as the complete suppression of polarization conversion in one direction while maintaining it in the other. Extracting NR-FPC characteristics from this 3D parameter space, Fig. 10(a) reveals their evolution behavior in the 2D subspace angle of incidence versus structural perturbation. The analysis identifies four evolution trajectories, with Figs. 10(b1)–10(b8) showing corresponding Jones matrix elements at trajectory inflection points. Notably, oblique incidence breaks reciprocity (), limiting clear EP identification to zero-angle incidence where three EPs emerge in the upper half-space—with Fig. 10(b5) serving as the basis for our previous discussion. The observed resonance frequency shifts and geometric parameter deviations reflect boundary condition modifications and meshing adjustments under oblique incidence.
Figure 10.(a) Evolution trajectory of NR-FPC in the parameter space when both incident angle and geometric perturbation are varied simultaneously. The subscripts L/R indicate the corresponding chiral incidence that yields NR-FPC. Inset: configuration of oblique incidence in the plane, with the incident angle defined as positive on the positive -direction side. (b1)–(b8) Jones matrix transmission coefficients at the inflection points of the trajectory.
Intriguingly, the NR-FPC trajectory under RCP illumination forms a closed loop with minimal side-dependent angular effects, suggesting structural perturbation dominates over angular variations. LCP illumination reveals more complex behavior, divisible into two phases: pre-Fig. 10(b2) evolution (trajectories A and B) showing angular-dominated scattering; post-Fig. 10(b2) evolution where increasing geometric perturbation induces side-dependent divergence. Trajectory B maintains its original evolution (structure-independent), while trajectory A twists toward plane, culminating in a unique [Fig. 10(b3)]. To elucidate the underlying physics, Fig. 13(d) in Appendix C tracks transmission changes along the four trajectories. Trajectories A/C/D exhibit similar mode interaction dominated by quadrupole mode 1 versus dipole mode 2 competition, primarily influenced by incident angle-geometric perturbation balance. Trajectory B presents distinct behavior where unwanted mode 3 redshifts continuously, compressing mode 1’s initial blueshift to eventual redshift, explaining trajectory A’s directional turning in Fig. 10(a).
D. Unlocking the Application Potential of Weakly Chiral Systems
In Section 4, we have already mentioned some existing approaches to circular polarization recognition and subsequent applications, all driven by the pursuit of enhanced chiral chemical components or optical systems. Recently, optical methods based on mode resonance to amplify light-matter interactions have emerged as highly effective means for achieving stronger chirality [5,52,53,57,69–72]. The non-Hermitian metasurface designed in our framework utilizes the contrast of the polarization conversion process to effectively detect and further leverage chirality, even though the overall system is only weakly chiral. As shown in Fig. 11, we analyzed the chirality in the parameter space around the EP, characterized by circular dichroism (CD), defined as ). Despite the weak chirality, the system demonstrates strong performance in the application examples presented in Section 4. Furthermore, the quasi-BIC mode supports significant near-field modal intensity. This design approach highlights the potential of weakly chiral systems for advanced chiral applications.
6. CONCLUSIONS
In conclusion, the EP associated with polarization in the scattering system is effectively realized within the quasi-BIC mode. We have demonstrated that the chiral coalescence in non-Hermitian systems originates from the quasi-BIC mode. These two distinctive phenomena are coupled through the designed metasurface, which provides a viable route for the engineering of EPs. Furthermore, by examining the distinct parameter spaces defined by Fermi energy and geometrical parameters, we have uncovered that an EP can stably reside at the same position within the non-Hermitian system, even when different parameters are utilized to construct the parameter space. Finally, as a direct application of the maximum spin-selective asymmetric transmission induced by chiral EP, we have proposed an integrated platform for direct chiral detection and display. The non-Hermitian chiral coalescence spawned by quasi-BIC mode breaks the conventional constraint of realizing chiral EPs in broadband systems. This development opens up a wide range of applications in chiral sensing, lasers, and optical forces, expanding the horizons for innovation in these fields.
APPENDIX A: TEMPORAL COUPLED-MODE THEORY
The mode resonances are inscribed by the Hamiltonian of the two-port system, and the total scattering matrix connects the scattering amplitudes and incident fields . Within the weak coupling approximation, the mode response and scattering in the vicinity of the resonance can be formulated by where the Hamiltonian of a system consisting of two resonators with coupling strength is a complex matrix while the total loss () is composed of radiation loss and intrinsic loss of material. depicts the direct transmission and absorption loss of the incidence in absence of resonances, as
The coupling between resonances and the outgoing planewaves is respectively determined by which is constrained by mirror symmetry, energy conservation, and time-reversal symmetry of the metasurface (i.e., , , ).
Hence, based on Eqs. (A1) and (A2) and coefficient matrices Eqs. (A4) and (A5), the scattering matrix is derived as where the diagonal elements describe the scattering process for polarization maintaining, while the off-diagonal elements define the scattering with polarization conversion. Due to reciprocity, . Then, with the definition of (), the transmission matrix in the linearly polarized basis is extracted as where because of anisotropy, and on account of reciprocity.
Actually, this transmission matrix is equivalent to the Jones matrix in the language of polarization, which is a more appropriate description for chiral coalescence. For a better match with the spin states, the Jones matrix of the system can be re-expressed via the circular polarization basis with the basis vectors for LCP and for RCP.
Then, considering the reciprocity, the diagonal elements of Eq. (A8) are equal, from which we obtain its eigenvalues as . It is not hard to notice that the two eigenvalues degenerate when the root formula is zero, i.e., , which represents the generation of the EP. From Eqs. (A7) and (A8), the complex eigenvalues can be derived as where
Figure 11.The chirality of the system in the parameter space in the vicinity of the EP.
Figure 12.(a) Evolution of transmission coefficient versus unit cell connection width . (b)–(e) Surface current distributions for (b), (d) RCP incidence and for (c), (e) LCP illumination in (b), (c) original and (d), (e) -axis mirror-symmetric configurations.
Ignoring material loss in non-resonant direct scattering provides an effective approximation to simplify the discussion, corresponding to . We then obtain a pair of well-formed eigenvalues and corresponding eigenstates as where
This reveals the conditions for EP emergence: with the degeneracy of eigenvalues and the coalescence of eigenstates.
First focusing on the eigenvalue evolution, from Eq. (A15), an EP occurs at with from Eq. (A14).
Next, we analyze the behavior of the eigenstates. For simplicity, we rewrite Eq. (A12) as , where . Therefore, the eigenstates of the system at EP are with the condition from Eqs. (A14) and (A15). When (i.e., ) but | (i.e., ), substituting into Eq. (A12) gives where . Equation (A17) implies that the two eigen-polarization states have major axes at , respectively. Similarly, we can derive the set of eigenstates along 0° and 90° as with when . From these, an abrupt 45° rotation of the major axes before and after the EP can be observed when moderating the coupled system at a featured frequency.
APPENDIX B: FORMULATING AN EXTENSIBLE INTEGRATED METASURFACE THROUGH UNIT CELL CONNECTIONS
To enable experimental implementation, we engineered an extensible metasurface by interconnecting unit cells via BDS strips. This architecture supports both boundary clamping and perforated substrate mounting while preserving functionality. This design facilitates subsequent fixation and application through boundary clamping or mounting on a perforated substrate. As demonstrated in Fig. 12(a), systematic optimization of the connection width reveals a critical trade-off: excessive widths () perturb EP conditions, while insufficient widths () compromise structural stability. The optimal width maintains EP characteristics with minimal spectral shift to 29.02 μm. Figures. 12(b) and 12(c) demonstrate the optimized structure’s chiral response, exhibiting strong RCP-selective current excitation versus weak LCP coupling—a hallmark of EP-induced spin selectivity. Mirror symmetry inversion [Figs. 12(d) and 12(e)] systematically reverses this chirality, producing LCP-dominated resonance with a suppressed RCP response, thereby confirming the parity-controlled EP behavior.
APPENDIX C: INFLUENCE OF ANGLE OF INCIDENCE AND CHIRALITY ON MODE INTERACTIONS
Initial analysis of the unperturbed system () under incident angle variation reveals identical LCP/RCP responses in Fig. 13(a), confirming mirror symmetry preservation and chirality absence. Structural perturbation under normal incidence [Fig. 13(b)] demonstrates limited chirality generation, aligning with Fig. 10 conclusions. Gap enlargement induces characteristic mode behavior: mode 2 undergoes a rapid blueshift while experiencing compression from longer-wavelength modes, whereas mode 1 exhibits a gradual blueshift. This differential shift drives a coupling transition from Fano-like to EIT-like lineshapes. Phase analysis of in Fig. 13(c) corroborates this mechanism with the black dashed box highlighting the phase jump progression during perturbation increase. Introducing oblique incidence [Fig. 13(d)] reproduces these geometry-dominated trends while unveiling an angularity-induced third mode. This emergent mode critically governs the trajectory bifurcation observed in Fig. 10(a), particularly evident when geometric effects compete with angular perturbations. The additional mode’s spectral compression behavior explains trajectory A’s directional turning through modified mode interaction dynamics.
Figure 13.(a) Transmission under varying incident angles for CP incidences in the symmetric structure (). (b) Structural perturbation-dependent transmission under normal incidence. (c) Phase of the corresponding to (b). (d) Transmission evolution along the four trajectories in Fig. 10(a). Yellow markers denote the quadrupole mode of interest (mode 1), with blue/purple representing two other modes. Arrows indicate parametric mode migration.