Manifestation of super chiral exceptional points in a plasmonic metasurface

Haojie Li; Guoxia Yang; Anwen Jiang; Min Ni; Qianwen Jia; Fengzhao Cao; Jiayi Zhang; Bokun Lyu; Dahe Liu; Jinwei Shi

doi:10.1364/PRJ.537395

1. INTRODUCTION

Non-Hermitian systems satisfying parity-time (PT) symmetry exhibiting real eigenvalues were first proposed by Bender and Boettcher and have attracted widespread attention in various fields [1,2]. Unlike diabolic points (DPs) in Hermitian systems where only eigenvalues degenerate, at PT-symmetry phase transition points, known as exceptional points (EPs), both eigenvalues and eigenvectors degenerate [3]. In 2007, El-Ganainy et al. extended the PT symmetry to optical systems exploiting the mathematical equivalence between the single-particle Schrödinger equation and the paraxial approximate electromagnetic wave equation [4]. Therefore, photonic systems, with their complex refractive indices, provide a fertile ground to investigate PT symmetry and EP, leading to many intriguing phenomena, such as single-mode lasing [5,6], non-reciprocal light transmission [7 –9], asymmetric optical response [10 –13], coherent perfect absorber [14 –17], PT-symmetry-induced bound state in the continuum [18], EP-enhanced sensing [19 –22], unidirectional non-reflection [23 –25], and chiral response [26 –36]. Chirality plays a key role in biology, chemistry, medicine, and communication. However, chiral analysis through optical techniques is often challenging due to the weak chiral signals in natural materials [37,38]. Metasurfaces are artificial two-dimensional electromagnetic materials consisting of periodic or non-periodic arrangements of subwavelength units. By finely tailoring the geometric size of the unit cells, the polarization, amplitude, phase, and propagation mode of light can be flexibly controlled [39,40]. The development of chiral nanostructured metasurfaces has become a practical necessity, and various methods have been explored to break metasurface mirror symmetry, achieving high circular dichroism (CD), such as monolayer asymmetric patterns [41], three-dimensional chiral metasurfaces [42,43], and moiré metasurfaces [44]. Non-Hermitian metasurfaces, characterized by gain and loss difference, offer a promising platform for realizing chiral response at EP [26 –36].

In 2014, Lawrence et al. first designed a terahertz non-Hermitian metasurface composed of orthogonally oriented split-ring resonators (SRRs) of different metal materials with the zero detuning but different losses [26]. In this system, EP with a single circularly polarized state is observed by varying separation of the SRRs, where a specific circularly conversion component disappears. Following a similar principle, Park et al. proposed three ways to observe EPs in 2020 [27]. In addition, a range of tunable materials have been applied to the design of non-Hermitian metasurfaces, including type II superconductor niobium nitride [28], chalcogenide glass GeTe [29], graphene [30], and vanadium dioxide [31]. These materials facilitate active regulation of the PT-symmetric phase transition by parameters such as temperature and voltage. In 2021, Song et al. demonstrated an intriguing non-Hermitian bilayer metasurface, utilizing a single material to realize holography by exceptional phase and geometric phase [32]. Recently, we designed a PT-symmetric nonlocal metasurface featuring orthogonal gold nanorods and achieved chiral response of EP in the telecom band [34]. Despite these advancements, achieving ideal CD remains challenging due to the coexistence of the conversion and parallel components.

In this work, we propose a PT-symmetric active metasurface composed of orthogonal orientation nano resonators with local gain. Circular transmission response of the metasurface is analyzed analytically by the coupled-mode theory and transmission matrix. By varying the coupling strength and gain–loss ratio between the resonators, we achieve PT-symmetric phase transition and a super chiral EP (SCEP). At SCEP, only one circular transmission component exists, resulting in not only near-ideal circular conversion dichroism (CCD) but also near-ideal CD, accompanied by an abrupt phase flip of the transmitted wave. Additionally, we realize unidirectional invisibility, with the light exhibiting opposite chirality in forward and backward incident directions. For structures satisfying $c_{2 y}$ symmetry, we observe a topological phase transition point (TP) featuring a linear polarization state, enabling a near-ideal circular polarization converter. Finally, leveraging SCEP and TP, we attain polarization states across the entire Poincaré surface. Our findings open new avenues for applications in optics and photonics, including polarizing optical elements, holography, logic gates, chiral molecular detection, ultrasensitive sensing, and polarization-sensitive imaging.

2. RESULTS

A. Theoretical Model

The coupled-mode theory and transmission matrix are employed to study the transmission response of a non-Hermitian chiral metasurface. The resonant modes and transmitted fields in the two orthogonal directions ( ${\tilde{a}}_{x} = a_{x} e^{i ω t}$ and ${\tilde{a}}_{y} = a_{y} e^{i ω t}$ ) of a chiral meta-atom can be expressed as [45,46] $\frac{d {\tilde{a}}_{x}}{d t} = (i ω_{x} - γ_{x} - Γ_{x}) {\tilde{a}}_{x} + i g {\tilde{a}}_{y} + i \sqrt{γ_{x}} {\tilde{E}}_{x}^{in},$ (1) $\frac{d {\tilde{a}}_{y}}{d t} = (i ω_{y} - γ_{y} - Γ_{y}) {\tilde{a}}_{y} + i g {\tilde{a}}_{x} + i \sqrt{γ_{y}} {\tilde{E}}_{y}^{in},$ (2) $(\begin{matrix} {\tilde{E}}_{x}^{out} \\ {\tilde{E}}_{y}^{out} \end{matrix}) = (\begin{matrix} {\tilde{E}}_{x}^{in} \\ {\tilde{E}}_{y}^{in} \end{matrix}) + (\begin{matrix} i \sqrt{γ_{x}} & 0 \\ 0 & i \sqrt{γ_{y}} \end{matrix}) (\begin{matrix} {\tilde{a}}_{x} \\ {\tilde{a}}_{y} \end{matrix}) = t (\begin{matrix} {\tilde{E}}_{x}^{in} \\ {\tilde{E}}_{y}^{in} \end{matrix}),$ (3)where $ω_{x} (ω_{y})$ , $γ_{x} (γ_{y})$ , and $Γ_{x} (Γ_{y})$ denote the resonance frequency, radiation loss, and dissipative loss of the resonance ${\tilde{a}}_{x} ({\tilde{a}}_{y})$ , respectively; $g$ represents the coupling strength between two sets of resonators; ${\tilde{E}}_{x}^{out} ({\tilde{E}}_{y}^{out})$ and ${\tilde{E}}_{x}^{in} ({\tilde{E}}_{y}^{in})$ are the transmitted and incident $x$ ( $y$ )-polarized electric fields, respectively; and $t$ is the transmission matrix under linear polarization basis. The scattering field $S ψ = λ ψ$ can be written in a form analogous to the Schrödinger equation $H ψ = E ψ$ [30]. Equations (1) and (2) can be rewritten as $(\begin{matrix} (ω_{x} - ω) γ_{y} + i γ_{y} (γ_{x} + Γ_{x}) & g \sqrt{γ_{x}} \sqrt{γ_{y}} \\ g \sqrt{γ_{x}} \sqrt{γ_{y}} & (ω_{y} - ω) γ_{x} + i γ_{x} (γ_{y} + Γ_{y}) \end{matrix}) (\begin{matrix} i \sqrt{γ_{x}} a_{x} \\ i \sqrt{γ_{y}} a_{y} \end{matrix}) = - i γ_{x} γ_{y} (\begin{matrix} E_{x}^{in} \\ E_{y}^{in} \end{matrix}) .$ (4)In this work, $δ_{x} = ω_{x} - ω = 0$ and $δ_{y} = ω_{y} - ω = 0$ . We can perform a gauge transformation on the left part of Eq. (4) as follows [47]: $S = S_{0} + S_{1} = (\begin{matrix} E_{0} & 0 \\ 0 & E_{0} \end{matrix}) + (\begin{matrix} \frac{1}{2} i (γ_{y} Γ_{x} - γ_{x} Γ_{y}) & g \sqrt{γ_{x}} \sqrt{γ_{y}} \\ g \sqrt{γ_{x}} \sqrt{γ_{y}} & - \frac{1}{2} i (γ_{y} Γ_{x} - γ_{x} Γ_{y}) \end{matrix}),$ (5)where $E_{0} = i γ_{x} γ_{y} + i (γ_{y} Γ_{x} + γ_{x} Γ_{y}) / 2$ . It is evident that $S_{1}$ satisfies PT symmetry. The eigenvalues can be obtained straightforwardly: $λ_{\pm} = E_{0} \pm \frac{1}{2} \sqrt{4 γ_{x} γ_{y} g^{2} - {(γ_{y} Γ_{x} - γ_{x} Γ_{y})}^{2}} .$ (6)

Figure 1(a) depicts the schematic of a typical PT-symmetric metasurface employed in this study, composed of orthogonal silver nanorods with local gain on the $x$ -oriented nanorods, embedded in a 500 nm thick silica layer. The pitch of the metasurface is $P_{x} = P_{y} = 800 nm$ . The geometric dimensions of $x$ - and $y$ -oriented nanorods differ ( $l_{1} = 260 nm$ , $s_{1} = 50 nm$ , $l_{2} = 281 nm$ , and $s_{2} = 90 nm$ ), with the same height $h_{1} = h_{2} = 50 nm$ . The gap widths $w_{1}$ and $w_{2}$ determine the coupling strength between the nanorods. The resonance mode of the metasurface can be tuned in a wide range by modifying the length of the nanorod. A gain material is introduced formally by a Lorentz function $[ε (λ) = 1 + ε_{Lorentz} ω_{0}^{2} / (ω_{0}^{2} - 2 i δ_{0} 2 π c / λ - {(2 π c / λ)}^{2}]$ with a thickness of 10 nm, where the center frequency $ω_{0}$ is set to $1.358 \times 10^{15} rad / s$ , matching the resonant frequency of the nanorods. The Lorentz permittivity $ε_{Lorentz}$ , which determines the gain amplitude, is set to $- 0.2$ , and the Lorentz linewidth $δ_{0}$ is set to $1 \times 10^{14} rad / s$ . The corresponding dielectric function graph can be found in Appendix A. The gain material’s complex refractive index is denoted as $n = n^{'} - i n^{''}$ , and the amplification coefficient of light intensity is given by $k = 4 π n^{''} / λ$ . At resonance ( $λ = 1398 nm$ ), $k$ is $5.66 \times 10^{4} {cm}^{- 1}$ , where $n^{''}$ represents the gain coefficient [48]. This value of $k$ falls within the range reported in the literature [12,48 –63]. Several materials are potential candidates for achieving this moderate gain level, including dye molecules such as rhodamine-6G [49], T5oCx [50], semiconductor colloidal quantum wells CdSe@CdS core@shell [51], quantum dots such as InAs/GaAs [52], rare earth ions such as ${Er}^{3 +}$ [53 –56], and InAs quantum dots [57,58]. Figure 1(b) illustrates the corresponding schematic coupling principle of this system. The evolution of corresponding eigenvalues is depicted by two intersecting Riemannian sheets centered around chiral EP formed in non-Hermitian system, as illustrated in Figs. 1(c) and 1(d), where $δ = ω_{x} - ω_{y}$ . At the EP, both the eigenvalues $(λ_{\pm} = E_{0})$ and the chiral eigenstates $(Ψ_{\pm} = \hat{x} - i \hat{y})$ of the system degenerate, and only one chiral state can be found in the converted component of transmission. The EP can be found under the condition of $2 \sqrt{γ_{x} γ_{y}} | g | = | γ_{y} Γ_{x} - γ_{x} Γ_{y} | .$ (7)The typical EP curve (i.e., PT-symmetric phase transition curve) that satisfies Eq. (7) is shown by the blue curve in Fig. 1(e), where $Δ Γ = Γ_{x} - Γ_{y}$ and $γ_{x} = γ_{y} = 0.5$ . The white and green areas represent the PT-symmetric phase and the PT-symmetry broken phase, respectively.

Figure 1.Design and characteristics of PT-symmetric metasurface. (a) Schematic of PT-symmetric metasurface composed of orthogonal oriented sliver nanorods with local gain. Typical parameters: $P_{x} = P_{y} = 800 nm$ , $l_{1} = 260 nm$ , $s_{1} = 50 nm$ , $l_{2} = 281 nm$ , $s_{2} = 90 nm$ , and $h_{1} = h_{2} = 50 nm$ . (b) Illustration of a system with two coupled resonators where radiation loss and dissipative loss are distinguished. (c),(d) Riemann surfaces of the complex eigenvalues of the matrix $\hat{S}$ versus the $(g, δ)$ parameters, showing (c) real and (d) imaginary parts of the eigenvalues. (e) Evolution of EP (blue curve) and ZP (red curve), with the super chiral EP (SCEP) denoted by a red solid star and the topological phase transition point (TP) by a hollow star. The white and green areas represent the PT-symmetric phase and the PT-symmetry broken phase, respectively.

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Circular dichroism is defined as $CD = \frac{T_{lcp} - T_{rcp}}{T_{lcp} + T_{rcp}} = \frac{T_{r l} - T_{l r} + T_{l l} - T_{r r}}{(T_{l l} + T_{r l}) + (T_{r r} + T_{l r})} \overset{2 D}{\Rightarrow} \frac{T_{r l} - T_{l r}}{(T_{r l} + T_{l r}) + (T_{l l} + T_{r r})},$ (8)where the subscripts $r$ and $l$ represent RCP and LCP components, respectively. $T_{i j} (i, j = r, l)$ is the transmission ratio of the $i$ outgoing component at the $j$ input. To achieve ideal CD, both of the following equations should be satisfied: $T_{r l} = 0 or T_{l r} = 0 but T_{r l} \neq T_{l r},$ (9) $T_{r r} = T_{l l} = 0 .$ (10)

Obviously, the EP condition in Eq. (7) can only guarantee Eq. (9) [ideal CCD $(T_{r l} - T_{l r}) / (T_{r l} + T_{l r})$ ], but not Eq. (10); hence Eq. (7) by itself is not sufficient to result in ideal CD.

The chiral EP determined by Eq. (7) is a well-studied property of the PT-symmetry non-Hermitian system. However, another property of the non-Hermitian system, often overlooked, is the existence of a zero-transmission point of parallel component (ZP), i.e., $T_{l l} = T_{r r} = 0$ . From Eqs. (3) and (4), the transmission matrix $t$ can be calculated as $t = I - i γ_{x} γ_{y} S^{- 1} = (\begin{matrix} 1 - \frac{i γ_{x} γ_{y}}{\det (S)} [i γ_{x} (γ_{y} + Γ_{y})] & \frac{i g γ_{x} γ_{y} \sqrt{γ_{x}} \sqrt{γ_{y}}}{\det (S)} \\ \frac{i g γ_{x} γ_{y} \sqrt{γ_{x}} \sqrt{γ_{y}}}{\det (S)} & 1 - \frac{i γ_{x} γ_{y}}{\det (S)} [i γ_{y} (γ_{x} + Γ_{x})] \end{matrix}),$ (11)and the Jones matrix under circular polarization basis is outlined below: $t_{circ} = (\begin{matrix} t_{l l} & t_{l r} \\ t_{r l} & t_{r r} \end{matrix}) = \frac{1}{2} (\begin{matrix} (t_{x x} + t_{y y}) + i (t_{x y} - t_{y x}) & (t_{x x} - t_{y y}) - i (t_{x y} + t_{y x}) \\ (t_{x x} - t_{y y}) + i (t_{x y} + t_{y x}) & (t_{x x} + t_{y y}) - i (t_{x y} - t_{y x}) \end{matrix}) .$ (12)

The circular transmission component can be obtained straightforwardly: $t_{l l} = t_{r r} = 1 - \frac{2 γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x}}{2 (γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x} + Γ_{x} Γ_{y} + g^{2})},$ (13) $t_{l r} = \frac{γ_{y} Γ_{x} - γ_{x} Γ_{y} - 2 g \sqrt{γ_{x}} \sqrt{γ_{y}}}{2 (γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x} + Γ_{x} Γ_{y} + g^{2})},$ (14) $t_{r l} = \frac{γ_{y} Γ_{x} - γ_{x} Γ_{y} + 2 g \sqrt{γ_{x}} \sqrt{γ_{y}}}{2 (γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x} + Γ_{x} Γ_{y} + g^{2})} .$ (15)

Therefore, the condition for $T_{l l} = T_{r r} = 0$ (ZP) is $γ_{x} Γ_{y} + γ_{y} Γ_{x} + 2 Γ_{x} Γ_{y} + 2 g^{2} = 0 .$ (16)

It is worth noting that ZP is a universal property of the non-Hermitian system, and this condition works not only for a PT-symmetry-type system but also for other types of non-Hermitian systems. For example, if $γ_{x} = γ_{y} > 0$ , $Γ_{x} = Γ_{y} < 0$ , the system is a non-PT-symmetry-type non-Hermitian system, and Eq. (16) still has a solution. However, this condition cannot be satisfied in a Hermitian system that has no gain or loss. In Fig. 1(e), the red curve denotes the ZP solution of Eq. (16) with $γ_{x} = γ_{y} = 0.5$ .

According to Eqs. (9) and (10), by combining Eqs. (7) and (16), one can achieve ideal CD at a point where only one circular component can survive in the transmission. This point is defined as super chiral EP (SCEP), as indicated by the solid red star in Fig. 1(e). This provides a new degree of freedom for achieving ideal CD.

B. PT-Symmetric Phase Transition and Realization of SCEP

Taking the PT-symmetric metasurface shown in Fig. 1 as an example, we demonstrate the existence and property of SCEP by full-wave simulation. By tailoring the coupling strength and gain–loss ratio, PT-symmetric phase transition and SCEP of a single circularly polarized state are observed. The transmission spectra of the metasurfaces, obtained from full-wave simulations (the simulation settings can be found in Appendix A), are presented in Figs. 2(a)–2(f).

Figure 2.PT-symmetric phase transition and SCEP. (a)–(f) Chiral-component transmission spectra and total chiral transmission spectra of the system (a), (d) in PT-symmetric phase, (b), (e) at SCEP, and (c), (f) in PT-symmetry broken phase. The Rabi splitting, resulting from strong coupling of orthogonally oriented nanorods, is clearly observed in PT-symmetric phase. At SCEP, the circular components $T_{r r}$ , $T_{l l}$ , and $T_{l r}$ almost decrease to zero at 1398 nm in (e). (g) Results of electric field amplitude and phase difference for $x$ -polarization incident light. (h) Near-ideal CCD [ $(T_{r l} - T_{l r}) / (T_{r l} + T_{l r})$ ] and CD with values of 0.995 and 0.98 are realized at SCEP, respectively. (i) The phase flip of $T_{l r}$ is observed near SCEP with $w_{1} = 186 nm$ .

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At $w_{1} = 100 nm$ , the system is in PT-symmetric phase with $2 \sqrt{γ_{x} γ_{y}} | g | > | γ_{y} Γ_{x} - γ_{x} Γ_{y} |$ . The Rabi splitting, formed by strong coupling of orthogonally oriented meta-atoms, can be clearly observed [Figs. 2(a) and 2(d)], where the lower frequency branch corresponds to a bonding state and the upper frequency branch represents an anti-bonding state. Upon reaching $w_{1} = 186 nm$ , the system enters SCEP, where the bonding and anti-bonding states undergo degeneracy at 1398 nm, and the circular transmission components $T_{r r}, T_{l l}$ and $T_{l r}$ almost decrease to zero at 1398 nm as shown in Fig. 2(e). This implies that for an RCP beam, there is negligible transmission through the metasurface, while an LCP beam transmits through the metasurface, converting to RCP. Consequently, a linearly polarized (LP) beam also transmits through the metasurface, only converting to RCP [Fig. 2(g)], where $Δ φ = φ (E_{x}) - φ (E_{y})$ . Near-ideal CCD and CD with values of 0.995 and 0.98 are simultaneously realized at 1398 nm, as shown in Fig. 2(h). The system also exhibits another SCEP with opposite chirality when $w_{2} = 186 nm$ ( $w_{1} = 264 nm$ ); details can be found in Appendix B. Interestingly, an abrupt phase flip of $T_{l r}$ is observed near SCEP as shown in Fig. 2(i). The phase dispersion sensitivity of $T_{l r}$ arises from the sensitivity of eigenvalues and eigenstates in close proximity to EP [22,27], which originates from the topological singularity nature of EP [32]. It should be noted that near-ideal CD in 2D structures without non-Hermitian EP has also been reported, but there is no topological phase transition [64,65]. Notably, the gain coefficient required to achieve SCEP can be reduced through the following three methods: (1) increasing the thickness of the gain material, (2) tuning the radiation loss and coupling strength of nanorods, and (3) utilizing the nonlocal characteristics of the nonlocal metasurface [34,66]. In addition, the resonant wavelength of the metasurface can be tuned across a wide range by modifying the nanorod length, offering greater flexibility in selecting gain materials. For practical applications, materials such as ${Er}^{3 +}$ and InAs/GaAs quantum dots, may be effective gain materials for achieving SCEP in our system. When $w_{1} = 210 nm$ , the system enters PT-symmetry broken phase (satisfying the condition $2 \sqrt{γ_{x} γ_{y}} | g | < | γ_{y} Γ_{x} - γ_{x} Γ_{y} |$ ). In this regime, $T_{l r}$ is no longer equal to zero and gradually approaches $T_{r l}$ , as illustrated in Figs. 2(c) and 2(f). In addition, compared to systems with the same geometry but no gain–loss difference, non-Hermitian systems exhibit significantly enhanced chiral responses (see Appendix C).

C. Unidirectional Invisibility of Circularly Polarized Light and Polarization Converter

Two- and three-dimensional chiral structures as well as EP of non-Hermitian systems can exhibit asymmetric transmission response of circularly polarized light [34,44]. Here, because only one circular component exists in transmission, SCEP shows that RCP and LCP components can exhibit unidirectional invisibility in opposite incident directions, as illustrated in Fig. 3(a). The transmission spectra of backward incidence ( $- z$ direction) are shown in Figs. 3(b)–3(d). Due to the reciprocity of the system, at SCEP, $T_{b - r r} = T_{r r} = 0; T_{b - l l} = T_{l l} = 0; T_{b - r l} = T_{l r} = 0; T_{b - l r} = T_{r l} \neq 0$ , where $T_{b - i j} (i, j = r, l)$ is the transmission ratio of the $i$ outgoing component at the backward $j$ input. In other words, if an LCP (RCP) light beam is incident on the metasurface in the backward (forward) direction, there is no transmission. However, when the LCP (RCP) light beam is incident on the metasurface in the forward (backward) direction, there is significant transmission. This phenomenon illustrates a typical unidirectional invisibility of circularly polarized light, which holds promising applications in fields such as information encryption and holographic imaging.

Figure 3.Unidirectional invisibility and polarization converter. (a) Schematic of unidirectional invisibility at SCEP, where only one chiral component can transmit through the metasurface. (b) Total transmission spectra for circularly polarized incident light, (c) transmission spectra for RCP and LCP components, and (d) amplitude and phase difference of transmitted electric fields for backward incidence ( $- z$ direction) at SCEP. (e) Schematic of a polarization converter at the topological phase transition point (TP). (f) Transmission spectra for RCP and LCP components for forward incidence at TP. Amplitude and phase difference of transmitted electric fields for (g) RCP incidence and (h) LCP incidence. $Δ φ = φ (E_{x}) - φ (E_{y})$ .

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If the gap $w_{1}$ is further increased while keeping $w_{1} + w_{2}$ fixed, the system transitions into the PT-symmetry broken phase and can be depicted by a typical topological Su–Schrieffer–Heeger (SSH) model [18,67,68]. When the structure satisfies $c_{2 y}$ symmetry ( $w_{1} = w_{2} = 225 nm$ , i.e., $g = 0$ ), a topological phase transition point (TP) emerges. A near-ideal circular polarization converter is obtained in Fig. 3(e). The transmission spectra are presented in Fig. 3(f), where $T_{l r} = T_{r l} \neq 0$ , $T_{l l} = T_{r r} = 0$ , corresponding to the hollow star in Fig. 1(e). Obviously, the conversion efficiency is the same for different circular polarizations.

Furthermore, we extract the amplitude and phase differences of different components of the transmitted electric field to validate the circular polarization converter, and the results are shown in Figs. 3(g) and 3(h). For an RCP beam being incident forward on the metasurface, we find that $E_{x} = E_{y}$ , $Δ φ = φ (E_{x}) - φ (E_{y}) = 90 °$ . The same behavior is observed for LCP incident light.

D. Polarization Control

Leveraging two SCEPs with opposite chirality and TP, polarization modulation across the entire Poincaré surface can be achieved by tailoring the coupling strength and rotation angle of the metasurface unit cell.

First, for the $x$ -polarization incident light, the transmission amplitude under circular polarization representation is as follows: $(\begin{matrix} t_{l} \\ t_{r} \end{matrix}) = t_{cir} \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} t_{x x} - i t_{y x} \\ t_{x x} + i t_{y x} \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} 1 - \frac{γ_{x} γ_{y} + γ_{x} Γ_{y} + g \sqrt{γ_{x}} \sqrt{γ_{y}}}{γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x} + Γ_{x} Γ_{y} + g^{2})} \\ 1 - \frac{γ_{x} γ_{y} + γ_{x} Γ_{y} - g \sqrt{γ_{x}} \sqrt{γ_{y}}}{γ_{x} γ_{y} + γ_{x} Γ_{y} + γ_{y} Γ_{x} + Γ_{x} Γ_{y} + g^{2})} \end{matrix}),$ (17)where $t_{l}$ and $t_{r}$ represent the transmission of LCP and RCP components with an LP beam through the metasurface, respectively. The numerical calculation results of Eq. (17) are included in Appendix D. The results show that the transmission polarization states of the system are continuously modulated across the entire Poincaré meridian as the coupling strength varies. Figure 4 depicts transmission spectra through metasurfaces with tailored gap (rod-to-rod distance). At TP, the transmission is $x$ -polarized, indicated by the blue star in Fig. 4(g). In the region where $w_{1} < 225 nm$ , as $w_{1}$ decreases, at the wavelength of 1398 nm, the electric field component $E_{y}$ of the transmitted light gradually increases from zero, while the component $E_{x}$ decreases. The phase difference $Δ φ = φ (E_{x}) - φ (E_{y}) = 90 °$ . At this point, the transmitted light’s polarization becomes right-handed elliptically polarized, with the long axis along the $x$ direction, as shown in Fig. 4(f) (taking $w_{1} = 210 nm$ as an example). When $w_{1} = 186 nm$ , the system reaches right-handed SCEP, and the polarization of transmitted light becomes right-handed circularly polarized, as depicted by the green star in Fig. 4(e). As $w_{1}$ decreases further, the electric field components $E_{y}$ and $E_{x}$ of the transmitted light continue to increase and decrease, respectively. The phase difference $Δ φ$ remains at 90°, and the polarization of the transmitted light becomes right-handed elliptically polarized with the long axis along the $y$ direction, as illustrated in Fig. 4(d) (taking $w_{1} = 180 nm$ as an example). When $w_{1} = 164 nm$ , the component $E_{x}$ decreases to zero, resulting in $y$ -direction linear polarization, as depicted by the red star in Fig. 4(c). As the coupling strength between nanorods continues to increase, the system will further evolve around the Poincaré sphere meridian, as shown in Fig. 4(b).

Figure 4.Generation of arbitrary polarization states on the Poincaré surface under an LP beam incident on metasurfaces. (a) Polarization states of the transmitted light for different values of $g$ are plotted on the Poincaré sphere at 1398 nm. $S_{i}$ ( $i = 1$ , 2, 3) denote the three Stokes parameters. (b)–(k) Electric field amplitude and phase difference of the transmitted light with varying separation between the nanorods. Polarization states across the entire Poincaré surface can be obtained by rotating the unit cell of metasurfaces.

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In the region where $w_{1} > 225 nm$ (i.e., $w_{2} < 225 nm$ ), when $w_{1}$ is increased, the polarization of the transmitted light is symmetrically distributed with respect to the previous case, but with opposite chirality. This is determined by the geometric symmetry of the structure [35]. Starting from TP (blue star), as $w_{2}$ decreases, the polarization of the transmitted light goes through left-handed elliptically polarized with the long axis along the $x$ direction [Fig. 4(h)], left-handed SCEP [Fig. 4(i)], left-handed elliptically polarized with the long axis along the $y$ direction [Fig. 4(j)], and, finally, $y$ polarization [Fig. 4(k)]. Therefore, we achieve polarization modulation across the entire Poincaré meridian by tuning the coupling strength between the two rods. Finally, the polarization can be further extended to the entire Poincaré sphere surface by rotating the metasurface unit cell (see Appendix E). Therefore, by leveraging two SCEPs with opposite chirality and TP, we achieved arbitrary polarization states by tailoring the gap and rotation angle of the unit cell. Compared with recently published work using two normal chiral EPs [35], our approach based on SCEP eliminates the need to separate the converted component from the parallel component by precise design of Pancharatnam–Berry (PB) phase. As a result, we can directly obtain the desired polarization state in the forward coaxial direction. Considering practical situations, the two-step electron-beam lithography (EBL) technique with a precise alignment step is a viable fabrication scheme.

3. CONCLUSION

In conclusion, our study explores the fascinating realm of chiral exceptional points and, particularly, the emergence of the super chiral EP in non-Hermitian parametric space. Through a comprehensive theoretical analysis, we established the conditions required for achieving ideal CD and unveiled the unique properties of SCEP. Leveraging a PT-symmetric metasurface with gain and loss, we successfully demonstrated the SCEP by tuning the coupling strength and gain–loss ratio, observing both near-ideal CCD (0.995) and CD (0.98) and a phase flip of the conversion component. The presented PT-symmetric metasurface exhibits distinctive behaviors, such as unidirectional invisibility and efficient polarization conversion, emphasizing its potential applications in various fields. Additionally, we demonstrated the metasurface’s capability to generate arbitrary polarization states across the entire Poincaré surface without utilizing PB phase to separate the parallel component. Our findings not only contribute to the understanding of non-Hermitian physics and chiral responses but also provide practical insights for designing advanced photonic devices with unprecedented functionalities. This work opens new avenues for innovative applications in diverse areas of optics and photonics, such as polarizing optical elements, holography, logic gates, chiral molecular detection, ultrasensitive sensing, and polarization-sensitive imaging.

Category: Surface Optics and Plasmonics

Received: Jul. 26, 2024

Accepted: Oct. 3, 2024

Published Online: Nov. 27, 2024

The Author Email: Jinwei Shi (shijinwei@bnu.edu.cn)

DOI:10.1364/PRJ.537395

CSTR:32188.14.PRJ.537395

Lorentz Function and Parameters		Lorentz Function Graph
Function	$ε (λ) = 1 + \frac{ε_{Lorentz} ω_{0}^{2}}{ω_{0}^{2} - 2 i \partial_{0} 2 π c / λ - {(2 π c / λ)}^{2}}$
$ε_{Lorentz}$	$- 0.2$
$ω_{0}$	$1.35 \times 10^{15} rad / s$
$\partial_{0}$	$1 \times 10^{15} rad / s$
$c$	$3 \times 10^{8} m / s$