Abstract
NonHermitian degeneracies, also known as exceptional points (EPs), have been the focus of much attention due to their singular eigenvalue surface structure. Nevertheless, as pertaining to a nonHermitian metasurface platform, the reduction of an eigenspace dimensionality at the EP has been investigated mostly in a passive repetitive manner. Here, we propose an electrical and spectral way of resolving chiral EPs and clarifying the consequences of chiral mode collapsing of a nonHermitian gated graphene metasurface. More specifically, the measured nonHermitian Jones matrix in parameter space enables the quantification of nonorthogonality of polarisation eigenstates and halfinteger topological charges associated with a chiral EP. Interestingly, the output polarisation state can be made orthogonal to the coalesced polarisation eigenstate of the metasurface, revealing the missing dimension at the chiral EP. In addition, the maximal nonorthogonality at the chiral EP leads to a blocking of one of the crosspolarised transmission pathways and, consequently, the observation of enhanced asymmetric polarisation conversion. We anticipate that electrically controllable nonHermitian metasurface platforms can serve as an interesting framework for the investigation of rich nonHermitian polarisation dynamics around chiral EPs.
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A physical system describable with a nonHermitian Hamiltonian may host exceptional points^{1,2,3,4} (EPs), i.e., branching point singularities at which two or more eigenstates coalesce in parameter space. Unlike the degeneracies in Hermitian systems, for which an orthogonal set of eigenstates can be constructed, the eigenstates coalesce at the EP and become selforthogonal, leading to a defective eigenspace of reduced dimensionality. These singular features have been observed and utilised in various quantum and classic systems, including electronic spins^{5}, superconducting qubits^{6}, condensed excitonpolaritons^{7}, electronic circuits^{8}, thermotic systems^{9} and active matter^{10}. Particularly in photonic systems^{11,12,13,14}, the ease of precise loss and/or gain control has facilitated the discovery of a plethora of EPassociated exotic behaviours, with some representative examples including chiral mode transfer with or without encircling around EPs^{15,16}, controlled electromagnetically induced transparency^{17}, a ring (or a pair) of EPs in momentum space^{18,19}, and coupling to the missing dimension at an EP^{20}. In line with these advancements, we have also witnessed a series of promising EPenabled functionalities, such as paritytime (PT) symmetrybroken lasing^{21}, exceptional topological phase engineering^{22}, electrical winding number switching^{23}, exceptional sensing^{24,25} and coherent perfect absorption^{26}.
For the exploration of nonHermitian physics and the application of EPenabled functionalities, metasurfaces^{22,23,27,28,29} are now being considered one of the most versatile platforms because their constituent metaatoms are inherently constructed from lossy coupled subwavelengthscale resonators. Generally, any change in the polarisation state of light transmitted through the nonHermitian metasurface can be characterised by a nonHermitian Jones matrix that plays the role of an effective Hamiltonian^{28,29,30,31,32,33}. In contrast to the prevailing cases^{34}, the nonHermitian Jones matrix, of which the complexvalued elements can be engineered by geometrical and materials design of the metaatoms, enables the utilisation of polarisation eigenstates for the examination of EPrelated phenomena. Interestingly, at THz frequencies, the metasurface platform has been the only one that allows for the experimental observation of EPs, inheriting all the generic advantages of subwavelengthscale metaphotonics. However, until now, experimental probing of a branching point singularity in the parameter space has mostly been demonstrated in a passive way by repeatedly fabricating metasurfaces with varying metaatom designs^{28,29}. Furthermore, even with a series of repeated preparations, unavoidable errors from fabrications and/or measurements have made it difficult to observe relevant nonHermitian dynamics around/at EPs. It is thus highly desirable to have precise realtime control of the parameters for access to an EP in a single metasurface platform^{23,35}.
To circumvent the aforementioned problems, we hybridise gated graphene microribbons with nonHermitian metasurfaces and demonstrate the electrically controlled probing of polarisation eigentransmission surfaces along with the corresponding eigenstates. Notably, this probing methodology utilises time domain spectroscopy that makes use of a broadband pulse, which in combination with a continuous gate tuning capability enables highresolution access to chiral EPs in twoparameter space. Here, chiral EPs refer specifically to the nonHermitian degeneracy at which a circularly polarised state becomes the only eigenstate as a result of coalescence. The measured nonHermitian Jones matrix in the parameter space enables a systematic investigation of nonorthogonality between polarisation eigenstates and atypical linkage between input and output polarisation states at the chiral EP. Specifically, we show that, for a specific incident polarisation, augmenting dimensionality at the chiral EP can be solely revealed at the output. It is also found that the maximal nonorthogonality assured by the defective Jones matrix at the chiral EP leads to the observation of enhanced asymmetric polarisation conversion. Last but not least, the examination of polarisation eigenstates in parameter space reveals a vortex structure, from which half integer topological charges at the chiral EP are clarified.
Results
Design of nonHermitian gated graphene metasurfaces
To map eigentransmission surfaces and investigate their structure near a chiral EP, we designed a nonHermitian gated graphene metasurface consisting of an array of pairs of coupled split ring resonators (SRRs) with a graphene microribbon bridging the SRRs (Fig. 1a). The paired SRRs have their splits opened in orthogonal directions and are characterised by distinct external loss rates (Fig. 1b). Then, by employing temporal coupledmode theory (TCMT)^{31,33,36}, a parameterdependent nonHermitian Jones matrix of the designed metasurface can be derived (see Methods). The two coupled SRRs are modelled as two orthogonally oriented resonators with a resonance (angular) frequency of ω_{0}, a coupling rate of κ, and intrinsic and external loss rates of γ_{iμ} and γ_{eμ} (μ = x, y), all of which can be adjusted to a certain degree by the geometry and materials constituting the unit cell (Fig. 1a, c). Under steadystate conditions, a 2×2 nonHermitian Jones matrix T_{l} can be written in a linear polarisation basis. The matrix can be expressed as a sum of uncoupled and coupled parts (\({{{\mathbf{T}}}}_l = {{{\mathbf{T}}}}_{lu} + {{{\mathbf{T}}}}_{lc}\)), only the latter of which is relevant for investigating the coalescing behaviour near/at the chiral EP. Specifically, the coupled part is found to be proportional to the following matrix:
where the dimensionless parameters are introduced for the simplicity of expression (see Methods for details): \(\Omega _y = (\omega _0  \omega )/\gamma _{ey}\), \(\Omega _x = (\omega _0  \omega )/\gamma _{ex}\), \(\Gamma = \left( {\gamma _{iy}/\gamma _{ey}  \gamma _{ix}/\gamma _{ex}} \right)/2\), and \({{{\mathrm{K = }}}}  \!\kappa /\sqrt {\gamma _{ex}\gamma _{ey}}\). An inspection of the eigenvalues and eigenvectors of the above matrix shows the presence of a pair of chiral EPs when the following conditions are satisfied:
The first equality specifies a onedimensional subspace satisfying PT symmetry, while the second equalities further identify the two chiral EPs as singularities distinguishing a PT exact phase from a broken one on the subspace. The chirality of a coalesced eigenstate at each EP is determined by the sign in the second equality (i.e., + for RCP and − for LCP). In this work, the system is parameterised by two variables: the frequency ω of an input wave and the gate voltage V_{g} (or the Fermi level E_{F} in simulations) that control the optical conductivity of graphene microribbons (Fig. 1c). The gating of graphene disproportionately adjusts the intrinsic loss rate of each SRR, resulting in a change in the Γ value and correspondingly the potential fulfilment of the second equality (here, in this experimental work, Γ = −Κ). From the simulations, it is found that the external loss rates are almost invariant within the gating range of interest. It is worthwhile to note that the pair of chiral EPs connected by a PT broken phase in the parameter space of our metasurface platform is analogous to the pair of EPs linked by a bulk Fermi arc in the momentum space of a twodimensional nonHermitian photonic crystal^{19,37,38}.
Mapping of the eigentransmission surface and identification of chiral EP
NonHermitian graphene metasurfaces were prepared by standard microfabrication techniques and a CVDgrown graphene transfer method and characterised by terahertz time domain spectroscopy (THzTDS, see Methods). Here, a chargeneutral point of the graphene is located at a gate voltage of ~ −1.1 V. From the measured co and crosspolarised complex amplitude transmission through the fabricated metasurface, a set of nonHermitian Jones matrices T_{l}, each of which is specified on a rectangular grid in the twoparameter space, can be obtained. The eigentransmission of T_{l} clearly reveals selfintersecting Riemann surface structures (Fig. 2a, b); the cusp at the end of the line of intersection is identified as the chiral EP and found to be located at (f_{EP},V_{g},_{EP}) = (0.55 THz, −0.2 V). To support the experimentally measured topological structure, we performed numerical simulations using a finite element method and extracted the eigentransmission (see Methods). As shown in Fig. 2c, d, the numerical simulations capture the key features of coupling and phase transitions across the chiral EP, which is in topological agreement with experimental results. First, a transition from weak (Γ > −Κ, \(V_g \,>\,V_{g,{{{\mathrm{EP}}}}}\)) to strong (Γ < −Κ, \(V_g \,<\, V_{g,{{{\mathrm{EP}}}}}\)) coupling between polarisation eigenstates can be seen by sampling eigentransmission surfaces at consecutively decreasing values of V_{g} across the chiral EP (corresponding to three cut lines on the Riemann surfaces shown in Fig. 2a). As more clearly seen in Fig. 2e, f, a crossing (anticrossing) to anticrossing (crossing) transition is clearly observable in the spectrally resolved eigentransmission magnitudes (phases). Second, an exceptional phase transition is observed in the gatevoltagedependent eigentransmission magnitudes (phases) sampled along a onedimensional subspace satisfying PT symmetry (\({{\Omega }}_x = {{\Omega }}_y\)); in these plots, exact and broken PT phases appear on either side of the chiral EP (Γ =−Κ, Fig. 2a–d). Here, it is also worth noting that the coupling crossover is observed to be concomitant with the exceptional phase transition across the chiral EP, as in other nonHermitian systems^{23,39}.
Nonorthogonality of eigenstates and halfinteger topological charge of chiral EP
To visualise the eigenstate coalescing behaviour, numericallycalculated and experimentallyextracted polarisation eigenstates are mapped on a Poincaré sphere (see Fig. 3a and Methods). For clarity, the polarisation eigenstates corresponding to different values of gate voltage (or Fermi levels) are colourcoded. As seen in Fig. 3a, except at the chiral EP, the polarisation eigenstates exist in pairs and appear symmetrically with respect to the south pole. It is noteworthy that the paired polarisation eigenstates are not represented on the Poincaré sphere by antipodal points, which is indicative of the characteristic nonorthogonality of general nonHermitian systems. As the chiral EP is approached in parameter space, the paired polarisation eigenstates move towards the south pole and eventually coalesce into the left circularly polarised state. To quantify the degree of nonorthogonality and coalescence, a Petermann factor (F_{p}) is calculated based on the left and right polarisation eigenstates extracted from the measurement (see Methods)^{40,41}. Ideally, selforthogonality and maximal nonorthogonality at the EP lead to the divergence of F_{p}, of which the experimental quantification can be done by plotting an inverse of F_{p} in parameter space (Fig. 3b). In the plot, smaller values of F_{p}^{1} are seen along a onedimensional subspace satisfying PT symmetry, and the value sharply drops down to ~3×10^{−4} near the chiral EP. This sharp decrease illustrates a singular sensitivity near the chiral EP to a variation in parameters. It is interesting to note that, in addition to the investigation of nonorthogonality, the polarisation eigenstate mapping in parameter space enables the characterisation of topological charges associated with the chiral EPs. For this purpose, we monitored the cyclic variation of the ellipse orientation of polarisation eigenstates along an encircling path around the chiral EP on the Riemann surface and quantified the topological charge defined by \(q = \frac{1}{{2\pi }}\mathop {\oint }\nolimits d\chi\), where χ is the ellipse angle (Fig. 3c, d)^{42,43,44}. The cyclic variation of the ellipse orientation reveals a polarisation vortex centre at the chiral EP along with a half integer topological charge (q = +1/2). While not observable in this work due to the maximum gate voltage limit, the existence of the other chiral EP in parameter space with a half integer topological charge of q = −1/2 can be confirmed in the analytic calculation^{37,38,45}.
Revealing the missing dimension in a reduced polarisation eigenspace at chiral EP
Polarisation eigenstate coalescence and the corresponding reduction of an eigenspace dimensionality at the chiral EP also lead to singular behaviours in the transmission of waves through the nonHermitian metasurface. More specifically, the left and right polarisation eigenstates become selforthogonal at the chiral EP so that the output polarisation state \(\left {\psi _o} \right \rangle\) is described with the single eigenpolarisation state \(\left L \right \rangle\) and the associated Jordan vector, i.e., in our case, \(\left J \right. \rangle= \left R \right.\rangle\),
where t’s are elements of the 2×2 nonHermitian Jones matrix at the chiral EP written in a circular polarisation basis and \(\left {\psi _i} \right.\rangle\) is the input polarisation state. Note that the matrix is in Jordan form at the chiral EP with its elements indicating co and crosspolarised transmission (\(t_{RR,{{{\mathrm{EP}}}}} = t_{LL,{{{\mathrm{EP}}}}}\) and \(t_{RL,{{{\mathrm{EP}}}}} = 0\)). Three representative cases are schematically shown (left panels in Fig. 4) along with their corresponding Poincaré sphere representations for input (middle panels) and output polarisation states (right panels) extracted from the measured Jones matrix at the chiral EP: (i) For RCP incidence (\(\left {\psi _i} \right.\rangle = \left J \right.\rangle\), orthogonal to the polarisation eigenstate at the chiral EP, see Fig. 4a), the output polarisation state becomes a superposition of the polarisation eigenstate and the Jordan vector (\(\left {\psi _o} \right. \rangle= t_{LL,{{{\mathrm{EP}}}}}\left J \right. \rangle+ t_{LR,{{{\mathrm{EP}}}}}\left L \right.\rangle\)). (ii) For LCP incidence (\(\left {\psi _i} \right.\rangle = \left L \right.\rangle \), the polarisation eigenstate at the chiral EP, see Fig. 4b), the output polarisation state contains only the LCP component (\(\left {\psi _o} \right.\rangle = t_{LL,{{{\mathrm{EP}}}}}\left L \right.\rangle\)). Note that the coalescence of polarisation eigenstates prohibits simultaneous nulling of both crosspolarised transmissions, which eventually leads to asymmetric polarisation conversion, as will be discussed below. (iii) Of particular interest is the case where the output polarisation state is completely devoid of the component parallel to the coalesced polarisation eigenstate (Fig. 4c); more specifically, preferential conversion to the Jordan vector (\(\left {\psi _o} \right. \rangle=  t_{LL,{{{\mathrm{EP}}}}}^2/t_{LR,{{{\mathrm{EP}}}}}\left J \right.\rangle\)) can be achieved by setting the input polarisation states to \(\left {\psi _i} \right.\rangle =  t_{LL,{{{\mathrm{EP}}}}}/t_{LR,{{{\mathrm{EP}}}}}\left J \right.\rangle + \left L \right.\rangle\). This counterintuitive outcome is the accidental revelation of the missing dimension through the destructive interference of two LCP components: one from copolarised transmission and the other from crosspolarised transmission of the prescribed input state. It is also worth mentioning that the solid angles subtended by output polarisation states are slightly smaller than those of input states (Fig. 4a–c) due to the nonunitary transformation performed by the metasurface^{46}. More interestingly, the perfect nulling of crosspolarised transmission t_{RL} at the chiral EP leads us to observe the signature of a Pancharatnam–Berry phase during gatecontrolled coupling crossover^{47} (Fig. 5a, b). The phase of t_{RL} at high frequencies sharply changes by 2π (see Fig. 5b), implying zerotoone topological winding number switching^{23} by gating around the chiral EP (Fig. 5c). This winding number switching and the associated phase jump across the chiral EP can also be employed for enhanced sensing and monitoring of chemical and biological events^{48}.
Maximal asymmetric polarisation conversion
The asymmetric nonHermitian Jones matrix of the fabricated graphene metasurface also leads to gatecontrolled asymmetric polarisation conversion (Fig. 5d). The proposed nonHermitian metasurface can be classified as a planar chiral structure with broken inplane mirror symmetry that induces intrinsic chirality at normal incidence^{49,50}. When compared with a forward propagation case, a backward propagating wave is incident on the metasurface consisting of unit cells that are mirrorreflected with a line of symmetry connecting the centres of two constituting SRRs. This guarantees that the offdiagonal elements of the Jones matrix are exchanged, i.e., \(t_{LR}^b = t_{RL}^f\) and \(t_{RL}^b = t_{LR}^f\), where superscripts specify the direction of propagation. Specifically, at the chiral EP of the fabricated metasurface, \(t_{LR}^b\) becomes zero, while \(t_{RL}^b\) remains nonzero, which suggests that a large difference in offdiagonal elements can be observed. To quantitatively characterise the effect, a normalised difference in asymmetric polarisation conversion is defined here as \(\delta _t \equiv ( { {t_{RL}^b} ^2   {t_{RL}^f} ^2} )/( { {t_{RL}^b} ^2 +  {t_{RL}^f} ^2})\), the value of which generally ranges from −1 to 1. Figure 5e shows the values of parameterdependent δ_{t} extracted from the transmission measurement performed on the fabricated metasurface. It is clearly seen that the values are nonzero near the chiral EP and approach the maximum value of 1 at the chiral EP. While the asymmetric polarisation conversion has been intensively investigated and experimentally demonstrated in the context of planar chirality, it is worth to note that a chiral EP should first be accessed to guarantee maximal asymmetric polarisation conversion. Therefore, the gatecontrolled access to the chiral degeneracy enables us to easily satisfy this necessary condition for the maximal asymmetric polarisation conversion that is hard to be achieved in passive nonHermitian metasurfaces due to the sensitivity of EPs to fabrication errors.
Conclusions
In this work, we experimentally demonstrated the potential of a nonHermitian gated graphene metasurface platform for the clarification and characterisation of chiral EPs in parameter space. The proposed platform stands among other recently implemented tunable nonHermitian photonic systems^{23} while distinguishing itself from others by utilising a nonHermitian Jones matrix for the manipulation of polarisation states. Specifically, in addition to the wellknown general features such as nonorthogonality and mode coalescence, the nonHermitian Jones matrix, especially written in Jordan form at the chiral EP, leads to an unusual nonunitary relation between input and output polarisation states. One such manifestation is the preferential polarisation conversion into the state represented by a Jordan vector of the nonHermitian Jones matrix. This implies that the output polarisation state can be made independent of the coalesced eigenstate of the metasurface being transmitted, which is contrary to our usual conception. We further experimentally clarified halfinteger topological charges of a nonHermitian chiral degeneracy and topological winding number switching by gating. We believe that the proposed tunable metasurface platform may become an essential tool in the investigation of dynamic phenomena related to nonHermitian chiral degeneracies and serve as a testbed for realising artificial nonHermitian effective matter.
Methods
Jones matrix of nonHermitian metasurfaces
A parameterdependent nonHermitian Jones matrix can be obtained by modelling gated graphene metasurfaces with temporal coupledmode theory (TCMT). The analysis results in the following 2×2 nonHermitian Jones matrix in a linear polarisation basis.
where I is the identity matrix, \({{{\mathbf{D}}}} = \left[ {\begin{array}{*{20}{l}} {\root {j} \of {{\gamma _{ex}}}} \hfill & 0 \hfill \\ 0 \hfill & {\root {j} \of {{\gamma _{ey}}}} \hfill \end{array}} \right]\), \({{{\mathbf{H}}}} = j\left( {{{{\mathbf{\Omega }}}}  \omega {{{\mathbf{I}}}}} \right)  {{{\mathbf{\Gamma }}}}_{{{\boldsymbol{e}}}}  {{{\mathbf{\Gamma }}}}_{{{\boldsymbol{i}}}}\), \({{{\mathbf{\Gamma }}}}_{{{\boldsymbol{e}}}} = \left[ {\begin{array}{*{20}{c}} {\gamma _{ex}} & 0 \\ 0 & {\gamma _{ey}} \end{array}} \right]\), \({{{\mathbf{\Gamma }}}}_{{{\boldsymbol{i}}}} = \left[ {\begin{array}{*{20}{c}} {\gamma _{ix}} & 0 \\ 0 & {\gamma _{iy}} \end{array}} \right]\), and \({{{\mathbf{\Omega }}}} = \left[ {\begin{array}{*{20}{c}} {\omega _0} & \kappa \\ \kappa & {\omega _0} \end{array}} \right]\). Then, the nonHermitian Jones matrix can be expressed as
Here, for simplicity, we introduce two dimensionless parameters \(\xi = 1  \chi /\det \left( {{{\mathbf{H}}}} \right)\) and \(\eta = j\gamma _{ex}\gamma _{ey}/\det \left( {{{\mathbf{H}}}} \right)\), where \(\chi = \gamma _{ex}\gamma _{ey} + \left( {\gamma _{ex}\gamma _{iy} + \gamma _{ey}\gamma _{ix}} \right)/2\). In a circular polarisation basis, the matrix can be written as
Numerical simulations
To numerically calculate eigentransmission surfaces of the gated graphene metasurfaces, we use a commercial simulation tool that employs the commercial finiteelement method solver CST Microwave Studio. In the frequency range of interest, the dielectric constant for gold is tabulated in ref. ^{51} and can be fitted by using the Drude model with a plasma frequency ω_{P} = 1.37 × 10^{13} rad s^{1} and a collision frequency γ = 4.07 × 10^{13} rad s^{1}. The complex index of the silicon substrate is extracted experimentally by measuring the transmission of the THz wave through the substrate. The optical conductivity of graphene is modelled by Kubo’s formula^{52} with an experimentally fitted scattering time of 20 fs.
Device fabrication
The gated graphene nonHermitian metasurfaces are prepared by employing standard microelectromechanical fabrication techniques. All the metallic structures are made of 200nmthick gold and attached to the substrate with a 20nmthick chrome adhesion layer. To bridge the gap between SRRs with a graphene microribbon, CVDgrown graphene is first transferred to the substrate with previously patterned SRRs. The transfer of graphene is accomplished by using PMMA (C2, Microchem) as a supporting layer. The transferred largearea graphene is then patterned by UV lithography with bilayered photoresists (PMGI and HKT 501). After UV exposure and development, the part of graphene uncovered by photoresists is etched by a plasma asher. As shown in Fig. 1c, the graphene microribbon can be electrically doped by utilising an iongel gate dielectric with inplane gate and ground electrodes patterned on an undoped silicon substrate.
THzTDS measurement
To retrieve nonHermitian Jones matrices, a conventional THz time domain spectroscopy (THzTDS) system is employed. The main part of the system consists of a Ti:sapphire femtosecond laser (MaiTai, Spectraphysics) operating at a repetition rate of 80 MHz with a centre wavelength of 800 nm, a GaAs photoconductive antenna (iPCA, BATOP) for the generation of a THz signal and a 1mmthick ZnTe crystal for detection. The generated THz signal covers a spectral range from ~0.1 to ~2.0 THz. The co and crosspolarised complex amplitude transmissions are measured by employing two wiregrid terahertz polarisers and used to retrieve the nonHermitian Jones matrix.
References
Berry, M. V. Physics of nonhermitian degeneracies. Czechoslovak J. Phys. 54, 1039–1047 (2004).
Heiss, W. D. The physics of exceptional points. J. Phys. A: Math. Theor. 45, 444016 (2012).
Bender, C. M. Making sense of nonhermitian hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007).
Ashida, Y., Gong, Z. P. & Ueda, M. NonHermitian physics. Adv. Phys. 69, 249–435 (2020).
Ren, Z. J. et al. Chiral control of quantum states in nonHermitian spin–orbitcoupled fermions. Nat. Phys. 18, 385–389 (2022).
Abbasi, M. et al. Topological quantum state control through exceptionalpoint proximity. Phys. Rev. Lett. 128, 160401 (2022).
Gao, T. et al. Observation of nonHermitian degeneracies in a chaotic excitonpolariton billiard. Nature 526, 554–558 (2015).
Choi, Y. et al. Observation of an antiPTsymmetric exceptional point and energydifference conserving dynamics in electrical circuit resonators. Nat. Commun. 9, 2182 (2018).
Xu, G. Q. et al. Observation of Weyl exceptional rings in thermal diffusion. Proc. Natl Acad. Sci. USA 119, e2110018119 (2022).
Fruchart, M. et al. Nonreciprocal phase transitions. Nature 592, 363–369 (2021).
Feng, L., ElGanainy, R. & Ge, L. NonHermitian photonics based on paritytime symmetry. Nat. Photonics 11, 752–762 (2017).
Özdemir, Ş. K. et al. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).
Miri, M. A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).
Parto, M. et al. NonHermitian and topological photonics: Optics at an exceptional point. Nanophotonics 10, 403–423 (2020).
Yoon, J. W. et al. Timeasymmetric loop around an exceptional point over the full optical communications band. Nature 562, 86–90 (2018).
Nasari, H. et al. Observation of chiral state transfer without encircling an exceptional point. Nature 605, 256–261 (2022).
Wang, C. Q. et al. Electromagnetically induced transparency at a chiral exceptional point. Nat. Phys. 16, 334–340 (2020).
Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).
Cerjan, A. et al. Experimental realization of a Weyl exceptional ring. Nat. Photonics 13, 623–628 (2019).
Chen, H. Z. et al. Revealing the missing dimension at an exceptional point. Nat. Phys. 16, 571–578 (2020).
Feng, L. et al. Singlemode laser by paritytime symmetry breaking. Science 346, 972–975 (2014).
Song, Q. H. et al. Plasmonic topological metasurface by encircling an exceptional point. Science 373, 1133–1137 (2021).
Ergoktas, M. S. et al. Topological engineering of terahertz light using electrically tunable exceptional point singularities. Science 376, 184–188 (2022).
Chen, W. J. et al. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017).
Park, J. H. et al. Symmetrybreakinginduced plasmonic exceptional points and nanoscale sensing. Nat. Phys. 16, 462–468 (2020).
Wang, C. Q. et al. Coherent perfect absorption at an exceptional point. Science 373, 1261–1265 (2021).
Colom, R. et al. Crossing of the branch cut: the topological origin of a universal 2πphase retardation in nonHermitian metasurfaces. Print at https://doi.org/10.48550/arXiv.2202.05632 (2022).
Lawrence, M. et al. Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces. Phys. Rev. Lett. 113, 093901 (2014).
Park, S. H. et al. Observation of an exceptional point in a nonHermitian metasurface. Nanophotonics 9, 1031–1039 (2020).
Cerjan, A. & Fan, S. H. Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials. Phys. Rev. Lett. 118, 253902 (2017).
Kang, M., Chen, J. & Chong, Y. D. Chiral exceptional points in metasurfaces. Phys. Rev. A 94, 1–5 (2016).
Yu, S. et al. Lowdimensional optical chirality in complex potentials. Optica 3, 1025–1032 (2016).
Li, S. X. et al. Exceptional point in a metalgraphene hybrid metasurface with tunable asymmetric loss. Opt. Express 28, 20083–20094 (2020).
Makris, K. G. et al. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008).
Wang, D. Y. et al. Superconductive PTsymmetry phase transition in metasurfaces. Phys. Rev. Lett. 110, 021104 (2017).
Suh, W., Wang, Z. & Fan, S. H. Temporal coupledmode theory and the presence of nonorthogonal modes in lossless multimode cavities. IEEE J. Quantum Electron. 40, 1511–1518 (2004).
Król, M. et al. Annihilation of exceptional points from different Dirac valleys in a 2D photonic system. Nat. Commun. 13, 5340 (2022).
Su, R. et al. Direct measurement of a nonHermitian topological invariant in a hybrid lightmatter system. Sci. Adv. 7, eabj8905 (2021).
Seyranian, A. P., Kirillov, O. N. & Mailybaev, A. A. Coupling of eigenvalues of complex matrices at diabolic and exceptional points. J. Phys. A: Math. Gen. 38, 1723–1740 (2005).
Berry, M. V. Mode degeneracies and the petermann excessnoise factor for unstable lasers. J. Mod. Opt. 50, 63–81 (2003).
Lee, S. Y. et al. Divergent Petermann factor of interacting resonances in a stadiumshaped microcavity. Phys. Rev. A 78, 015805 (2008).
Zhang, Y. W. et al. Observation of polarization vortices in momentum space. Phys. Rev. Lett. 120, 186103 (2018).
Tang, W. Y. et al. Exceptional nexus with a hybrid topological invariant. Science 370, 1077–1080 (2020).
Zhou, H. Y. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).
Yang, Z. S. et al. Fermion doubling theorems in twodimensional nonhermitian systems for fermi points and exceptional points. Phys. Rev. Lett. 126, 086401 (2021).
BenAryeh, Y. Nonunitary squeezing and biorthogonal scalar products in polarization optics. J. Opt. B: Quantum Semiclassical Opt. 7, S452 (2005).
Mailybaev, A. A., Kirillov, O. N. & Seyranian, A. P. Geometric phase around exceptional points. Phys. Rev. A 72, 014104 (2005).
Kravets, V. G. et al. Singular phase nanooptics in plasmonic metamaterials for labelfree singlemolecule detection. Nat. Mater. 12, 304–309 (2013).
Fedotov, V. A. et al. Asymmetric propagation of electromagnetic waves through a planar chiral structure. Phys. Rev. Lett. 97, 16 (2006).
Singh, R. et al. Terahertz metamaterial with asymmetric transmission. Phys. Rev. B. 80, 15 (2009).
Ordal, M. A., Bell, R. J., Alexander, R. W., Long, L. L. & Querry, M. R. Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W. Appl. Opt. 24, 4493 (1985).
Ziegler, K. Minimal conductivity of graphene: Nonuniversal values from the Kubo formula. Phys. Rev. B. 75, 233407 (2007).
Acknowledgements
This work was supported by National Research Foundation of Korea (NRF) through the government of Korea (NRF2021R1C1C100631612 and 2017R1A2B3012364) and Institute of Information & communications Technology Planning and Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2022000624). The work was also supported by the center for Advanced MetaMaterials (CAMM) funded by Korea Government (MSIP) as Global Frontier Project (NRF2014M3A6B3063709). The work is partially funded by NRF (NRF2021R1A6A3A14044805) and the Institute for basic science (IBSR011D1).
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S.B., S.H.P., S.Z., B.M., and T.T. K conceived the original idea. S.B. developed the gated graphene metasurfaces and performed the THzTDS measurement. S.B., S.H.P., and D.O. developed the analytical model. All authors analysed the data and discussed the results. S.B., S.H.P., D.O., S.Z., B.M., and T.T.K. wrote the paper, and all authors provided feedback. B.M. and T.T.K. supervised the project.
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Baek, S., Park, S.H., Oh, D. et al. NonHermitian chiral degeneracy of gated graphene metasurfaces. Light Sci Appl 12, 87 (2023). https://doi.org/10.1038/s41377023011216
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DOI: https://doi.org/10.1038/s41377023011216
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