# Sphere of arbitrarily polarized exceptional points with a single planar metasurface

Haoye Qin, Zijin Yang, Po-Sheng Huang, Xinyang Mu, Shih-Hsiu Huang, Yuzhi Shi, Wannian Zhao, Bo Li, Ji Zhou, Jesús Zúñiga-Pérez, et al.

► To cite this version:

Haoye Qin, Zijin Yang, Po-Sheng Huang, Xinyang Mu, Shih-Hsiu Huang, et al.. Sphere of arbitrarily polarized exceptional points with a single planar metasurface. Nature Communications, 2025, 16 (1), pp.2656. 10.1038/s41467-025-57737-2 . hal-05361876

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# Title: Sphere of arbitrarily polarized exceptional points with a single planar metasurface

**Authors:** Haoye Qin¹†, Zijin Yang¹†, Po-Sheng Huang²†, Shih-Hsiu Huang², Yuzhi Shi³*, Wannian Zhao¹, Bo Li¹,⁴, Ji Zhou⁵, Jesús Zúñiga-Pérez⁶,⁷,⁸, Patrice Genevet⁶,⁹*, Pin Chieh Wu²,¹⁰*, Qinghua Song¹*

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## Affiliations:

¹Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, 518055, China

²Department of Photonics, National Cheng Kung University, Tainan 70101, Taiwan

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³Institute of Precision Optical Engineering, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

⁴Suzhou Laboratory, Suzhou 215000, China

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⁵State Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China

⁶Université Côte d'Azur, CNRS, Centre de Recherche sur l'Hétéro-Epitaxie et ses Applications, Rue Bernard Gregory, Sophia Antipolis, 06560 Valbonne, France

⁷MajuLab, International Research Laboratory IRL 3654, CNRS, Université Côte d'Azur, Sorbonne Université, National University of Singapore, Nanyang Technological University, Singapore, Singapore

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⁸Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

⁹Physics department, Colorado School of Mines, 1523 Illinois Street, Golden, CO 80401, US

¹⁰Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan, 70101, Taiwan

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†These authors contributed equally to this work.

*Corresponding authors. Email: song.qinghua@sz.tsinghua.edu.cn; yzshi@tongji.edu.cn; Patrice.Genevet@mines.edu; pcwu@gs.ncku.edu.tw;

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**Abstract:** Exceptional points (EPs) are spectral singularities in non-Hermitian systems, where complex eigenvalues and eigenstates coalesce, reducing the Hilbert space dimension. Traditional photonic systems exhibit coalesced eigenstates corresponding to circular polarized light, restricting EPs to only two poles on the Poincaré sphere. Here, by combining optical anisotropy, chirality and non-Hermiticity in plasmonic metasurfaces we achieve a continuum of EPs, enabling coalesced eigenstates at any points on the Poincaré sphere. This breakthrough overcomes EPs' practical limitations, extending the associated topological phase to arbitrarily polarization. As a proof-of-concept, we construct topological metasurfaces tailored for linear, circular and elliptical polarized light. The emergence of these unconventional EPs not only promises disruptive

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applications in wavefront engineering and optical multiplexing, but also provide new fundamental insights into topological systems in general.

**One-Sentence Summary:** Exceptional Points (EPs) covering the entire Poincaré sphere are realized in non-Hermitian chiroptical metasurfaces.

**Main Text:** Light polarization underlies many aspects of our daily life, going from glare-reducing sunglasses to liquid crystal screens, and its exploitation has found numerous applications across scientific fields such as physics, biology, and health sciences, with polarized light microscopy being the most popular implementation. Its control and manipulation hold paramount significance. The vectorial nature of light polarization and its associated states find their most concise and geometrical intuitive representation in the Poincaré sphere, as every possible state of polarization corresponds to a point in the Poincaré sphere and vice versa (1, 2). Attaining complete control over light polarization thus necessitates manipulating light states across the entire Poincaré sphere.

Among the various approaches employed so far to engineer the polarization of light, traditional ones using calcite crystals and waveplates suffer from bulky optical components and lack of multifunctionality. More recently, highly customizable metasurfaces have provided ability to achieve continuous and even non-uniform polarization distributions (3–9), in addition to controlling other optical degrees of freedom such as phase (10–12), amplitude (13, 14), momentum (5, 6, 15) or frequency (16–18). This versatility, achieved thanks to the precise manipulation of subwavelength elements, empowers metasurfaces to surpass traditional methods and achieve more complex and accurate polarization control. Exceptional points (EPs), which appear in non-Hermitian systems for certain combinations of parameters with spectral degeneracies, (19, 20), have shown exotic polarization properties. However, despite their potential, the EPs properties have not been yet harnessed for polarization control. In optical systems the non-Hermiticity can originate from finely engineered spatial distribution of material loss/gain, but it can also naturally arise from the intrinsic radiative properties of all photonic systems. In the context of photonics, EPs have been largely studied through coupled mode theory in various textbook systems, notably coupled waveguides and resonators (19–21). Lately, EPs have been implemented in more complex systems and treated using the more general scattering matrix formalism, providing new insights into Huygens resonators (22), reflectionless scattering modes as well as coherent perfect absorbers (23–27). In general, any complex-valued linear operator representing the coupling of input and output channels of a photonic system (i.e., the scattering matrix) might support EPs (28–30), as it
is the case too for the conventional Jones matrix describing output versus input polarization
changes of light. EPs manifest as a zero-intensity spectral point in one of the off-diagonal terms of
the matrix. Considering the specific case of Jones matrix, the EP governs the polarization
conversion between the two corresponding polarization eigenstates (10, 31–35). Building upon the
general theory of EPs, it was recently shown that by encircling an EP in the metasurface parameter
space, a novel exceptional topological (ET) phase ranging from 0 to 2π, and different from the
usual Pancharatnam-Berry phase, can be harvested for wavefront shaping (12). So far, the reported
coalesced states at EPs have always corresponded to chiral states (36–39), which correspond to
circular polarized degenerate eigenstates for the specific case of the Jones matrix studied in this
manuscript. This intriguing behavior had resulted in helicity dependent scattering dynamics, chiral
protected transport at chiral edge states, chiral phase accumulation, chiral state transfer and
electromagnetic induced transparency of chiral light, essentially restricting the observation of these
effects to only two antipodal points -the poles- on the Poincaré sphere (12, 28–30, 40). These
phenomena have never been observed for any other state than the two circular polarized states,
and no report mentions the existence of EPs lying at arbitrary positions on the Poincaré sphere. To
elucidate the physics underlying this limitation we first analyze a textbook case, the Jones matrix
describing a linear system, which can be easily translated to many other optical systems. Then,
and building upon the previous analysis, we break this fundamental and technological constraint
by engineering the reflection matrix of a non-Hermitian metasurface and realize thereby EPs with
arbitrary polarization.

Our approach is relatively simple. Indeed, when calculating the eigenvalues and eigenvectors of
the Jones matrix we noticed that any arbitrarily polarized degenerated eigenstate could be achieved
if all four elements of the Jones matrix, which are generally related to each other by symmetry
constraints, could be controlled independently. This is achievable thanks to chiroptical effects, that
is, by introducing an anisotropic response to decouple J<sub>xx</sub> from J<sub>yy</sub> and through optical chirality
to decouple J<sub>xy</sub> from J<sub>yx</sub> (41), where J<sub>mn</sub> represents the polarization conversion from an *n*-polarized input channel to an *m*-polarized output channel. In naturally-occurring materials
adjusting the contribution of these two properties is in general difficult, as they are fixed by the
atomic structure, but it can be accomplished by leveraging the optical response of nanoresonators.
We thus designed metasurfaces in which EPs can exist at linear, circular, and elliptical
polarization states. We also demonstrated a robust 2π phase winding around each of these
arbitrarily polarized singularities, which is subsequently exploited to modulate the wavefront of arbitrary polarized light beams. To illustrate the degree of control as well as the simplicity of the required metasurfaces we designed and fabricated three metasurfaces-based phase-holograms using the ET phase emerging around EPs, which project images of “T”, “H” and “U” for linear, circular, and elliptical polarizations, respectively, leaving the cross-polarization conversion channel unaffected.

**Design principle of arbitrary polarized EP with extrinsic chirality.** The conventional Jones matrix under linear x- and y-polarization basis for an achiral system can be generally written as (41):

$$J_{achiral} = \begin{pmatrix} J_{xx} & J_{xy} \\ J_{yx} & J_{yy} \end{pmatrix} = \begin{pmatrix} a & \mathbf{b} \\ \mathbf{b} & d \end{pmatrix} \qquad (1)$$

with identical off-diagonal components, and a, b, d being complex values. The lack of chirality in the system imposes the EP eigenstate to be circular polarized, i.e., $[1, \pm j]^T$, as shown in Fig. 1A. Under this achiral condition, and upon applying a basis change, the ET phase is found only in one polarization conversion channel, corresponding to the conversion efficiency for a circular polarization state of well-defined handedness. To control independently the values of the Jones matrix off diagonal terms we rely on optical chirality (41),

$$J_{chiral} = \begin{pmatrix} J_{ss} & J_{sp} \\ J_{ps} & J_{pp} \end{pmatrix} = \begin{pmatrix} a & \mathbf{b} \\ \mathbf{c} & d \end{pmatrix} \qquad (2)$$

where s and p denote the in-plane and out-of-plane polarization vectors, respectively, both being orthogonal to the propagation wave vector k. It should be noted that the introduced chirality can be either intrinsic chirality, under normal incidence (42, 43), or extrinsic chirality, due to oblique incidence (44, 45). As a proof of principle, and for sake of simplicity in the experimental realization, the latter case is adopted in this paper (Fig. 1B). The extrinsic chirality provides a new degree of freedom, as it enables to construct a more general degenerate eigenstate at the EP given by $[\sqrt{b}, \pm i\sqrt{c}]^T$. Thus, by controlling the complex values of the coefficients b and c of the Jones matrix we can engineer linear, circular, or even elliptical polarized degenerated eigenstates. Interestingly, these general considerations applying to a non-Hermitian photonic system can be extended to systems described by non-Hermitian Hamiltonians that, under certain circumstances (see Supplementary Note S6), can give rise also to arbitrarily polarized EPs.
**Structure design of non-Hermitian plasmonic metasurface.** The plasmonic metasurface for arbitrarily polarized diffraction EPs is composed of three layers, where a dielectric layer is sandwiched between a metallic ground plane and a metallic structured layer (see more details in Supplementary fig. S1). The metal ground blocks all the transmission, so that the metasurface works in the reflective regime only. For this reason, in the following sections, the Jones or scattering matrix refers specifically to its reflection matrix. However, we will still use “J” to represent the reflection matrix. The structured layer is made of repeated unit cells containing two aluminum nanorods, which can be seen as two dipole moments whose in-plane orientation is defined by angles $β_1$ and $β_2$ (46, 47). The two nanorods are shifted by a distance of $P_x/4$ and $5P_x/8$ from the right boundary. Considering a light beam impinging on the unit cell at normal incidence, the reflection matrix is given by (see more details in Supplementary Note S1),

$$ J_0 = \begin{pmatrix} J_{xx} & J_{xy} \\ J_{yx} & J_{yy} \end{pmatrix} = \begin{pmatrix} \cos^2 \beta_1 e^{i\delta_1} + \cos^2 \beta_2 e^{i\delta_2} & \sin \beta_1 \cos \beta_1 e^{i\delta_1} + \sin \beta_2 \cos \beta_2 e^{i\delta_2} \\ \sin \beta_1 \cos \beta_1 e^{i\delta_1} + \sin \beta_2 \cos \beta_2 e^{i\delta_2} & \sin^2 \beta_1 e^{i\delta_1} + \sin^2 \beta_2 e^{i\delta_2} \end{pmatrix} \quad (3) $$

where $δ_{1,2}$ are the detour phases induced by the displacement of the two nanorods. As expected, at normal incidence the off-diagonal terms of the reflection matrix are always equal to each other, resulting in circular polarized degenerate eigenstates at the EP. Impinging on the structure at oblique incidence, characterized by the azimuth angle ($α$) and the altitude angle ($θ$) with respect to the normal vector (**Fig. 1B**), introduces extrinsic chirality into the reflection matrix, which breaks the equality condition on the off-diagonal terms (42, 48, 49). In this work we focus only on the 1st order diffracted light. In general, with a nonzero $α$ and $θ$ excitation, the diffraction will be three-dimensional, i.e., the diffracted wave vector depends on the principal orientation axis of the grating and, generally, does not belong to the plane formed by the normal vector and the incident vector. The final reflection matrix under oblique incidence, $J_{oblique}$, written in the local ($s, p$) polarization basis can be obtained by using matrix transformation as (see more details in Supplementary Note S2),

$$ J_{oblique} = \begin{pmatrix} J_{ss} & J_{sp} \\ J_{ps} & J_{pp} \end{pmatrix} = [O_{out}^{-1}J_0O_{in}]_{2 \times 2} \quad (4) $$

where $O_{in,out}$ consists of unit vectors associated to the local orthogonal basis {$s, p, k$} for incident and output light. The combination of $O_{in,out}$ matrices dependent on the propagation direction with
an anisotropic reflection matrix, which describes the metasurface (Eq. 3), breaks all symmetry constraints between the four reflection matrix components, thus decoupling the off-diagonal terms. In other words, combining chiral and anisotropic responses leads to non-circular polarization eigenstates at the EPs. Moreover, the dependence of the reflection matrix $J_{oblique}$ on the incidence angle can be efficiently exploited to engineer the polarization state of the coalesced eigenstates at the EP, and this can be done just by changing $\alpha$ and $\theta$, which become additional external degrees of freedom to be combined with the geometric internal ones defining the metasurface unit cell.

**Fig. 1C** demonstrates the eigenstate evolution of the reflection matrix versus wavelength with $(\beta_1, \beta_2, \alpha, \theta) = (40.3^\circ, 43^\circ, 0^\circ, 78^\circ)$. The two eigenstates coalesce at an EP whose position on the Poincare sphere attests of an arbitrary elliptical polarization. To show the controllability of the EP position on the Poincaré sphere, we illustrate its evolution as function of the incidence angles ($\alpha$ and $\theta$). At fixed altitude angle of $\theta = 78^\circ$, varying $\alpha$ enables to move the polarization of the coalesced eigenstates along the latitude of the Poincaré sphere, as shown in **Fig. 1D**. Similarly, the elliptical angle of the coalesced polarization eigenstates at the EP can be finely tuned by changing $\theta$, as shown in **Fig. 1E**. Therefore, by tuning the incidence angles ($\alpha$ and $\theta$) of the incident light we modify the extrinsic chirality and, thus, the polarization conversion reflection matrix, which in turn dictates the polarization of the coalesced eigenstates. Eventually, by varying $\alpha$ and $\theta$ in a large enough range one can cover the entire Poincaré sphere, as illustrated in **Fig. 1F** by the homogeneous cloud of orange points enveloping the sphere.

**Confirming the topological nature of the coalescence points.** Assuming the degenerate eigenstates and eigenvalues of $J_{oblique}$ at the EP are $|e_1\rangle$ and $\lambda_1$, i.e., $J_{oblique}|e_1\rangle = \lambda_1|e_1\rangle$, implies that for an input polarization state $|e_1\rangle$ the output polarization is necessarily proportional to $|e_1\rangle$. In other words, the EP creates a reflection zero for the polarization conversion channel from $|e_1\rangle$ to its orthogonal polarization state $|e_2\rangle$, i.e., $J_{e2,e1} = 0$. We previously showed that an EP-induced reflection zero, $J_{e2,e1} = 0$, enables accumulating a topological phase covering the full $0 - 2\pi$ range by encircling the EP within a rationally-chosen two-dimensional parameter space. In the following we demonstrate that relying on the topological phase accumulated while encircling an arbitrarily polarized EP (linear, circular and elliptical polarized cases correspond to the first, second, and third column in **Fig. 2**, respectively), we can manipulate the phase of light on a single polarization channel for any arbitrary polarization. **Fig. 2A** displays the four components of the reflection matrix $J^{\text{TP}}$ as a function of wavelength. A reflection singularity of $J_{e2,e1}$ is observed, as
expected by design. To prove that this reflection zero is associated to an EP, we compute the eigenvalues and eigenstates of the reflection matrix. The corresponding eigenvalues, obtained from the reflection matrix $J^{\text{TP}}$, are shown in **Fig. 2D**, with coinciding amplitudes and phases at the spectral location of the reflection zero. **Fig. 2G** shows the spectral evolution of the two eigenstates of $J^{\text{TP}}$ on the Poincaré sphere in the surroundings of the reflection zero (red star). Both trajectories converge to the red star located along the equator of the Poincaré sphere, i.e., linear polarization. These concomitant eigenvalues and eigenstates coalescences demonstrate that, indeed, the spectral reflection zero is associated to an EP of the reflection matrix. To demonstrate the control over the position of the EPs on the Poincaré sphere we modify the in-plane rotation angles of the metallic nanorods with respect to the previous example and we sweep again the wavelength. Circular polarized EPs, i.e., located on the pole of the Poincaré sphere, are also realized, as shown in **Figs. 2B, 2E and 2H**. **Figs. 2C, 2F and 2I** show an example of elliptical polarized EP, characterized by $\chi = 24^\circ, \phi = -76^\circ$. Finally, these three examples highlight the complete, and previously unattainable, control over the polarization state of EPs.

**Topological properties of EP in a two-dimensional parameter space.** To observe the EP complex-square-root dependency of the eigenvalue surfaces we plot the real and imaginary parts of the eigenvalues in the two-dimensional parameter space of $L = (\beta_1, \beta_2)$, corresponding to the rotation angles of the two metallic nanorods. These two parameters are expected to control the co- and cross-polarization ratio (46, 47). By sweeping them in the vicinity of the parameters corresponding to the linear polarized EP position, the real (Fig. 3A) and imaginary (Fig. 3D) parts of the eigenvalues of the reflection matrix present two intersecting double Riemann surfaces, which coalesce at the EP (black dots). Furthermore, for these two rotation-angles the reflection $J_{e2,e1}$ coefficient is zero (singularity), as shown in Fig. 4A, consistent with the EP nature of this coalescence point. As we previously pointed out, we obtain a winding of the phase around the EP singularity that realizes a full $0 - 2\pi$ phase coverage, of interest for the design of polarization converting metasurfaces. It should be noted that due to its topological nature, the accumulated phase always covers $2\pi$ regardless of the trajectory followed in parameter space, as long as the loop encircles the reflection zero. EP induced topological phase can thus be employed for hologram construction by choosing four unit-cell designs, corresponding to four different rotation angles around the singularity center (indicated by the black triangles in Fig. 4A). EPs with circular polarized coalesced eigenstates are also realized, as presented in the second column of Fig. 3,
where we obtain similar degeneracy of eigenvalues (Figs. 3B and 3E) and an equivalent topological phase pattern (Fig. 4B). For completeness, the last column of Fig. 3 (Figs. 3C and 3F) shows an example of EP with elliptical polarized coalesced eigenstates, characterized by $\chi = 24^\circ$, $\phi = -76^\circ$. The topological phase still shows the $2\pi$ winding in Fig. 4C. It is worth mentioning that, for a fixed dipole geometry, arbitrarily polarized topological phase can also be realized under different incident conditions (see more details in Supplementary Note S5). In contrast, the other cross-polarization channel of reflection matrix reveals no topological phase, as illustrated in Figs. 4D-4F, corresponding to the linear, circular, and elliptical polarized EPs, respectively. Overall, this demonstrates that the topological phase accompanying an EP in a two-dimensional parameter space can be designed to act on any arbitrary polarization state and, thus, allows to modulate the wavefront of any polarized beam. With the detailed four structures around each EP summarized in Fig. 4G (black triangles in Figs. 4A-4F), a $2\pi$ phase variation can be covered and used for wavefront manipulation. Building on this versatility and capitalizing on the $2\pi$ ET phase associated to each EP, one can produce now holograms based on arbitrarily polarized EPs (Fig. 4H).

**Experiments on asymmetric meta-holograms based on arbitrarily polarized topological phase.** These four topological phase-delaying elements are spatially disposed according to the spatial phase distribution of meta-holograms calculated using the traditional Gerchberg-Saxton algorithm (31, 50). The three letters “T”, “H”, and “U” are encoded into three reflective metasurfaces with differently polarized EPs. The SEM images of the fabricated plasmonic metasurfaces are shown in Figs. 5A-5C. Fig. 5D and 5G show the reflected images when linear polarized light is shined on a metasurface designed to operate around a linear polarized EP. A letter “T” can be observed when the polarization corresponds to that of the EP (Fig. 5D), while only the specularly-reflected beam is detected when impinging the device with cross-polarization (Fig. 5G). This confirms that the topological phase at the EP operates only on one of the off-diagonal reflection matrix terms. We repeated this experiment with a second and a third metasurface encoding the letters “H” and “U” for circular and elliptical polarizations, respectively, which can be observed in Figs. 5E and 5F, respectively. Similarly, shining these devices with cross-polarization (i.e., addressing the complementary off-diagonal term), does not project any holographic images, and only specular reflection occurs, as shown in Figs. 5H and 5I respectively.
**Concluding remarks.** In conclusion, we have proposed and demonstrated the existence of arbitrarily polarized EPs on the reflection matrix of a non-Hermitian metasurface, and we have successfully leveraged on the associated topological phase surrounding them in a two-dimensional parameter space to modulate arbitrarily polarized light beams. To this end, a non-Hermitian plasmonic metasurface that incorporates optical anisotropy and extrinsic chirality under oblique incidence is employed to decouple the four elements of the Jones matrix, lifting the circular polarized constraint of the EP. This arbitrarily polarized diffraction EPs in plasmonic metasurfaces cover thus the whole Poincaré sphere, and provide new design degrees of freedom and topologically-protected robustness, bringing topological concepts closer to the realm of technological applications. Beyond technology, the controllability and the design versatility of our non-Hermitian plasmonic metasurface will certainly ease the way for further investigations combining topology, nonlinearities, and higher-dimensional spaces and its working-principle can be translated to any system described by a non-Hermitian Hamiltonian.

**References and Notes**

1. H. Poincaré, *Théorie mathématique de la lumière* (Georges Carré).

2. M. Born, E. Wolf, "ELECTROMAGNETIC POTENTIALS AND POLARIZATION" in *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light* (Cambridge University Press, Cambridge, ed. 7, 1999).

3. X. Gao, W. L. Yang, H. F. Ma, Q. Cheng, X. H. Yu, T. J. Cui, A Reconfigurable Broadband Polarization Converter Based on an Active Metasurface. *IEEE Trans. Antennas Propag.* **66**, 6086–6095 (2018).

4. M. Burresi, R. J. P. Engelen, A. Opheij, D. Van Oosten, D. Mori, T. Baba, L. Kuipers, Observation of Polarization Singularities at the Nanoscale. *Phys. Rev. Lett.* **102**, 033902 (2009).

5. H. Qin, Z. Su, M. Liu, Y. Zeng, M.-C. Tang, M. Li, Y. Shi, W. Huang, C.-W. Qiu, Q. Song, Arbitrarily polarized bound states in the continuum with twisted photonic crystal slabs. *Light Sci. Appl.* **12**, 66 (2023).

6. H. Qin, Y. Shi, Z. Su, G. Wei, Z. Wang, X. Cheng, A. Q. Liu, P. Genevet, Q. Song, Exploiting extraordinary topological optical forces at bound states in the continuum. *Sci. Adv.* **8**, eade7556 (2022).

7. X. Yin, M. Schäferling, A.-K. U. Michel, A. Tittl, M. Wuttig, T. Taubner, H. Giessen, Active Chiral Plasmonics. *Nano Lett.* **15**, 4255–4260 (2015).
5

8. Y. Shi, Q. Song, I. Toftul, T. Zhu, Y. Yu, W. Zhu, D. P. Tsai, Y. Kivshar, A. Q. Liu, Optical manipulation with metamaterial structures. *Appl. Phys. Rev.* **9**, 031303 (2022).

9. Q. Song, S. Khadir, S. Vézian, B. Damilano, P. de Mierry, S. Chenot, V. Brandli, R. Laberdesque, B. Wattellier, P. Genevet, Printing polarization and phase at the optical diffraction limit: near- and far-field optical encryption. *Nanophotonics*. **10**, 697–704 (2021).

10. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, Z. Gaburro, Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction. *Science*. **334**, 333–337 (2011).

10

11. D. Lin, P. Fan, E. Hasman, M. L. Brongersma, Dielectric gradient metasurface optical elements. *Science*. **345**, 298–302 (2014).

12. Q. Song, M. Odeh, J. Zúñiga-Pérez, B. Kanté, P. Genevet, Plasmonic topological metasurface by encircling an exceptional point. *Science*. **373**, 1133–1137 (2021).

13. N. Liu, M. Mesch, T. Weiss, M. Hentschel, H. Giessen, Infrared Perfect Absorber and Its Application As Plasmonic Sensor. *Nano Lett.* **10**, 2342–2348 (2010).

15

14. Q. Song, W. Zhang, P. C. Wu, W. Zhu, Z. X. Shen, P. H. J. Chong, Q. X. Liang, Z. C. Yang, Y. L. Hao, H. Cai, H. F. Zhou, Y. Gu, G.-Q. Lo, D. P. Tsai, T. Bourouina, Y. Leprince-Wang, A.-Q. Liu, Water-Resonator-Based Metasurface: An Ultrabroadband and Near-Unity Absorption. *Adv. Opt. Mater.* **5**, 1601103 (2017).

20

15. Q. Ma, C. B. Shi, G. D. Bai, T. Y. Chen, A. Noor, T. J. Cui, Beam-Editing Coding Metasurfaces Based on Polarization Bit and Orbital-Angular-Momentum-Mode Bit. *Adv. Opt. Mater.* **5**, 1700548 (2017).

16. S. S. Kruk, L. Wang, B. Sain, Z. Dong, J. Yang, T. Zentgraf, Y. Kivshar, Asymmetric parametric generation of images with nonlinear dielectric metasurfaces. *Nat. Photonics*. **16**, 561–565 (2022).

25

17. D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D.-Y. Choi, Y. Kivshar, Third-Harmonic Generation in Photonic Topological Metasurfaces. *Phys. Rev. Lett.* **123**, 103901 (2019).

18. T. Santiago-Cruz, S. D. Gennaro, O. Mitrofanov, S. Addamane, J. Reno, I. Brener, M. V. Chekhova, Resonant metasurfaces for generating complex quantum states. *Science*. **377**, 991–995 (2022).

30

19. Ş. K. Özdemir, S. Rotter, F. Nori, L. Yang, Parity–time symmetry and exceptional points in photonics. *Nat. Mater.* **18**, 783–798 (2019).

20. M.-A. Miri, A. Alù, Exceptional points in optics and photonics. *Science*. **363**, eaar7709 (2019).

35

21. J. Wiersig, Review of exceptional point-based sensors. *Photon. Res.*, PRJ **8**, 1457–1467 (2020).
5

22. R. Colom, E. Mikheeva, K. Achouri, J. Zuniga-Perez, N. Bonod, O. J. F. Martin, S. Burger, P. Genevet, Crossing of the Branch Cut: The Topological Origin of a Universal $2\pi$-Phase Retardation in Non-Hermitian Metasurfaces. *Laser Photonics Rev.* **17**, 2200976 (2023).

23. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, X. Zhang, Demonstration of a large-scale optical exceptional point structure. *Opt. Express.* **22**, 1760–1767 (2014).

24. Y. Huang, Y. Shen, C. Min, S. Fan, G. Veronis, Unidirectional reflectionless light propagation at exceptional points. *Nanophotonics.* **6**, 977–996 (2017).

25. Q. Liu, B. Wang, S. Ke, H. Long, K. Wang, P. Lu, Exceptional points in Fano-resonant graphene metamaterials. *Opt. Express.* **25**, 7203–7212 (2017).

10

26. C. Wang, W. R. Sweeney, A. D. Stone, L. Yang, Coherent perfect absorption at an exceptional point. *Science.* **373**, 1261–1265 (2021).

27. P. Genevet, E. Mikheeva, R. Colom, K. Achouri, A. Overvig, F. Binkowski, J.-Y. Duboz, S. Cueff, S. Fan, S. Burger, A. Alu, Asymmetric phase modulation of light with parity-symmetry broken metasurfaces. *Optica Open* (2023), doi:10.1364/opticaopen.22828976.v1.

15

28. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, S. Zhang, Manifestation of PT Symmetry Breaking in Polarization Space with Terahertz Metasurfaces. *Phys. Rev. Lett.* **113**, 093901 (2014).

29. M. Kang, J. Chen, Y. D. Chong, Chiral exceptional points in metasurfaces. *Phys. Rev. A.* **94**, 033834 (2016).

20

30. S. H. Park, S.-G. Lee, S. Baek, T. Ha, S. Lee, B. Min, S. Zhang, M. Lawrence, T.-T. Kim, Observation of an exceptional point in a non-Hermitian metasurface. *Nanophotonics.* **9**, 1031–1039 (2020).

31. Q. Song, X. Liu, C.-W. Qiu, P. Genevet, Vectorial metasurface holography. *Appl. Phys. Rev.* **9**, 011311 (2022).

25

32. C.-W. Qiu, T. Zhang, G. Hu, Y. Kivshar, Quo Vadis, Metasurfaces? *Nano Lett.* **21**, 5461–5474 (2021).

33. S. Wang, Z.-L. Deng, Y. Wang, Q. Zhou, X. Wang, Y. Cao, B.-O. Guan, S. Xiao, X. Li, Arbitrary polarization conversion dichroism metasurfaces for all-in-one full Poincaré sphere polarizers. *Light Sci. Appl.* **10**, 24 (2021).

30

34. N. Meinzer, W. L. Barnes, I. R. Hooper, Plasmonic meta-atoms and metasurfaces. *Nat. Photonics.* **8**, 889–898 (2014).

35. M. W. Knight, N. S. King, L. Liu, H. O. Everitt, P. Nordlander, N. J. Halas, Aluminum for Plasmonics. *ACS Nano.* **8**, 834–840 (2014).

35

36. K. L. Zhang, L. Jin, Z. Song, Helical resonant transport and purified amplification at an exceptional point. *Phys. Rev. B.* **100**, 144301 (2019).
37. H. Nasari, G. Lopez-Galmiche, H. E. Lopez-Aviles, A. Schumer, A. U. Hassan, Q. Zhong, S. Rotter, P. LiKamWa, D. N. Christodoulides, M. Khajavikhan, Observation of chiral state transfer without encircling an exceptional point. *Nature*. **605**, 256–261 (2022).

5

38. C. Wang, X. Jiang, G. Zhao, M. Zhang, C. W. Hsu, B. Peng, A. D. Stone, L. Jiang, L. Yang, Electromagnetically induced transparency at a chiral exceptional point. *Nat. Phys.* **16**, 334–340 (2020).

39. A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, M. C. Rechtsman, Experimental realization of a Weyl exceptional ring. *Nat. Photonics*. **13**, 623–628 (2019).

10

40. Z. Li, G. Cao, C. Li, S. Dong, Y. Deng, X. Liu, J. S. Ho, C.-W. Qiu, Non-Hermitian Electromagnetic Metasurfaces at Exceptional Points. *Prog. Electromagn. Res.* **171** (2021).

41. C. Menzel, C. Rockstuhl, F. Lederer, Advanced Jones calculus for the classification of periodic metamaterials. *Phys. Rev. A*. **82**, 053811 (2010).

15

42. A. Y. Zhu, W. T. Chen, A. Zaidi, Y.-W. Huang, M. Khorasaninejad, V. Sanjeev, C.-W. Qiu, F. Capasso, Giant intrinsic chiro-optical activity in planar dielectric nanostructures. *Light Sci. Appl.* **7**, 17158–17158 (2017).

43. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, M. Wegener, Gold Helix Photonic Metamaterial as Broadband Circular Polarizer. *Science*. **325**, 1513–1515 (2009).

20

44. E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, N. I. Zheludev, Metamaterials: Optical Activity without Chirality. *Phys. Rev. Lett.* **102**, 113902 (2009).

45. E. Plum, V. A. Fedotov, N. I. Zheludev, Extrinsic electromagnetic chirality in metamaterials. *J. Opt. A: Pure Appl. Opt.* **11**, 074009 (2009).

25

46. Z. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. Guan, A. Alù, G. Li, X. Li, Full-Color Complex-Amplitude Vectorial Holograms Based on Multi-Freedom Metasurfaces. *Adv. Funct. Mater.* **30**, 1910610 (2020).

47. Z. Deng, Q. Tu, Y. Wang, Z. Wang, T. Shi, Z. Feng, X. Qiao, G. P. Wang, S. Xiao, X. Li, Vectorial Compound Metapixels for Arbitrary Nonorthogonal Polarization Steganography. *Adv. Mater.* **33**, 2103472 (2021).

30

48. Y. Chen, H. Deng, X. Sha, W. Chen, R. Wang, Y.-H. Chen, D. Wu, J. Chu, Y. S. Kivshar, S. Xiao, C.-W. Qiu, Observation of intrinsic chiral bound states in the continuum. *Nature*. **613**, 474–478 (2023).

49. G. Yun, K. Crabtree, R. A. Chipman, Three-dimensional polarization ray-tracing calculus I: definition and diattenuation. *Appl. Opt.* **50**, 2855 (2011).

35

50. Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. de Mierry, S. Khadir, P. Ferrand, P. Genevet, Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces. *Nat. Commun.* **11**, 2651 (2020).
5

51. C. W. Peterson, S. Kim, J. T. Bernhard, G. Bahl, Synthetic phonons enable nonreciprocal coupling to arbitrary resonator networks. *Sci. Adv.* **4**, eaat0232 (2018).

52. L. Zhang, Y. Yang, Y. Ge, Y.-J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S.-Q. Yuan, H.-X. Sun, H. Chen, B. Zhang, Acoustic non-Hermitian skin effect from twisted winding topology. *Nat. Commun.* **12**, 6297 (2021).

53. D. L. Sounas, A. Alù, Non-reciprocal photonics based on time modulation. *Nat. Photonics.* **11**, 774-783 (2017).

10

54. X.-L. Zhang, T. Jiang, C. T. Chan, Dynamically encircling an exceptional point in anti-parity-time symmetric systems: asymmetric mode switching for symmetry-broken modes. *Light Sci. Appl.* **8**, 88 (2019).

55. E. Brinkmeyer, Forward-backward transmission in birefringent single-mode fibers: interpretation of polarization-sensitive measurements. *Opt. Lett.* **6**, 575-577 (1981).

Acknowledgments:

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**Funding:** Q.S. acknowledges the funding support from the National Natural Science Foundation of China (No. 12204264), the Shenzhen Stability Support Program (No. WDZC20220810152404001), and the startup funding in Tsinghua Shenzhen International Graduate School (SIGS), Tsinghua University (No. 01030100006). Y.S. acknowledges the funding support from the National Natural Science Foundation of China (No. 62205246), the Fundamental Research Funds for the Central Universities, and Shanghai Pilot Program for Basic Research, Science and Technology Commission of Shanghai Municipality (22ZR1432400). P.C.W. gratefully acknowledges the use of advanced focused ion beam system (EM025200) of NSTC 112-2731-M-006-001 and electron beam lithography system belonging to the Core Facility Center of National Cheng Kung University (NCKU). P.C.W. acknowledges the support from the National Science and Technology Council (NSTC), Taiwan (Grant number: 111-2112-M-006-022-MY3; 111-2124-M-006-003), and in part from the Higher Education Sprout Project of the Ministry of Education (MOE) to the Headquarters of University Advancement at NCKU. P.C.W. also acknowledges the support from the Ministry of Education (Yushan Young Scholar Program), Taiwan. The research is also supported in part by Higher Education Sprout Project, Center for Quantum Frontiers of Research & Technology (QFort) at NCKU. P.G. acknowledges financial support by the French National Research Agency ANR Project DILEMMA (ANR-20-CE09-0027) and support from the European Innovation Council (EIC) project TwistedNano (under the grant agreement number Pathfinder Open 2021-101046424).

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**Author contributions:** H.Q. and Q.S. conceived the ideas and designed the research; H.Q. performed the numerical calculations; P.-S.H., S.-H.H. and P.C.W. fabricated the devices; Z.Y. and W.Z. performed the measurement; H.Q., J.Z.P., P.G., P.C.W. and Q.S. wrote the manuscript; Q.S., Y.S., P.G., and P.C.W. supervised and coordinated the project; all authors discussed the results and approved the paper.

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**Competing interests:** The authors declare that they have no competing interests.
**Data and materials availability:** All data are available in the main text or the supplementary materials.

**Supplementary Materials**

Materials and Methods

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Supplementary Text

Figs. S1 to S8

References
<figure><img src="image_2.png" /><figcaption>A</figcaption></figure>

Jones Matrix:

$$ \begin{bmatrix} a & b \\ b & d \end{bmatrix} $$

Eigenstates at EP:
(circular polarization)

<figure><img src="image_3.png" /><figcaption>B</figcaption></figure>

<figure><img src="image_4.png" /></figure>

Extrinsic chirality

-1st order
diffraction

θ

α

Jones Matrix:

$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

Eigenstates at EP:
(arbitrary polarization)

<figure><img src="image_5.png" /><figcaption>C</figcaption></figure>

<figure><img src="image_6.png" /></figure>

EP on Poincare sphere

<figure><img src="image_7.png" /><figcaption>D</figcaption></figure>

$α$-dependent $φ$

<figure><img src="image_8.png" /><figcaption>E</figcaption></figure>

$θ$-dependent $χ$

<figure><img src="image_9.png" /><figcaption>F</figcaption></figure>

Varying $α$ and $θ$

<caption>**Fig. 1.** Plasmonic metasurface with controllable degenerated polarization eigenstate at EP.</caption>

(A) Perspective view of one unit-cell of the non-Hermitian plasmonic metasurface consisting of two aluminum nanorods sitting on a dielectric spacer and an aluminum ground, serving as two decoupled nano-scatterers with individual controllable rotation angles ($β_1, β_2$). For light impinging at normal incidence, the off-diagonal terms of the reflection matrix are equal to each other, resulting in circular polarized EPs. (B) The off-diagonal terms of the reflection matrix can be decoupled by introducing extrinsic chirality under oblique incidence on the metasurface, described by an azimuth angle of $α$ and an altitude angle of $θ$. The -1st order reflected signal is only considered herein. (C) Spectral dependence of the eigenstates of the reflection matrix on the Poincaré sphere. The two eigenstates coalesced at an EP represented by the red star. The polarization of the degenerate eigenstate can be varied by changing (D) $α$ and (E) $θ$. (F) For varying incident conditions, the full Poincaré sphere can be covered by EPs.

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<figure><img src="image_10.png" /><figcaption>Fig. 2. Optical response of the degenerate eigenstate with linear (first column), circular (second column), and elliptical (third column) polarization. (A–C) Four components of reflection matrix $J^{TP} = \begin{pmatrix} J_{e1,e1} & J_{e1,e2} \\ J_{e2,e1} & J_{e2,e2} \end{pmatrix}$ as a function of wavelength. The minimal dip happens for one of the off-diagonal terms $J_{e2,e1}$ of the reflection matrix, attesting the occurrence of an EP. (D–F) Calculated amplitude and phase of the eigenvalues (represented as Ev1 and Ev2) confirming the existence of EP at the coalescence point, corresponding to the zero-reflection dip in the spectrum. (G–I) Evolution of the two eigenstates (represented as Es1 and Es2) on the Poincaré sphere with coalesced eigenstates $|e_1\rangle$ (red stars) at each EP. The aluminum nanorods rotation angles $(\beta_1, \beta_2)$ for the three columns are (48.6°, 47°), (111°, 58°) and (41.3°, 48.7°), respectively.</figcaption></figure>

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<figure><img src="image_11.png" /><figcaption>Fig. 3. Exceptional points in the two-dimensional parameter space $(\beta_1, \beta_2)$, for degenerate eigenstates displaying linear (first column), circular (second column), and elliptical polarization (third column). Real (A–C) and imaginary (D–F) parts of the eigenvalues of the reflection matrix $J^{\mathrm{TP}}$ around EP in the parameter space of the nanorod rotation angles $(\beta_1, \beta_2)$, for three eigen-polarizations. At the three EPs (black dots), both real and imaginary parts of the eigenvalues coalesce.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 4.** Arbitrarily polarized EP-induced topological phase. Amplitude (of the reflection coefficient) and topological phase around the singularity (EP) as a function of the two rotation angles ($β_1, β_2$) for (A) linear, (B) circular, and (C) elliptical polarized eigenstates. (D-F) Cross-polarization channel corresponding to the first-row eigenstates showing the absence of topological phase. The triangular dots indicate four structures providing $2π$ phase coverage for hologram construction. (G) Topological phase of the selected four structures and their trivial counterparts. (H) The retrieved topological phase can be used for asymmetric holographic display working on a designated polarization conversion channel with no effects on its orthogonal counterpart due to the degeneracy of the eigenstates at EPs.</figcaption></figure>

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<figure><img src="image_13.png" /><figcaption>Fig. 5. Arbitrarily polarized EP topological phase for meta-hologram applications. (A-C) SEM images of the designed plasmonic metasurfaces (indicated in Fig. 4 by black triangles) for encoding holographic images of “T”, “H”, and “U”, employing (D) linear polarized, (E) circular polarized and (F) elliptical polarized EPs induced topological phase, respectively. (G-I) Signal at the cross-polarization channel. The scale bars in A-C are 1 µm.</figcaption></figure>