Complete asymmetric polarization conversion at zero-eigenvalue exceptional points of non-Hermitian metasurfaces

2.1 CAPC at chiral zero-eigenvalue EPs

First, we examine the Jordan form of a non-Hermitian Jones matrix J at the EP, which can be expressed using a coalesced polarization eigenstate σ and a Jordan vector μ . These vectors satisfy the conditions J − t e I μ = σ and μ σ = 0 , where I is the 2 × 2 identity matrix [32], [33]. The Jones matrix can be represented as,

where t _σμ and t _e denote the cross-polarized transmission from μ to σ and the degenerate eigenvalue (or eigentransmission), respectively. The Jordan form at the EP inherently signifies APC with conditions t _μσ = 0, t _σμ ≠ 0 and equal co-polarized transmissions (t _e = t _σσ = t _μμ ). Furthermore, this Jordan form indicates that at the EPs, the output polarization can be expressed as a weighted sum of the coalesced eigenstate and the associated Jordan vector, provided the eigentransmission is non-zero. However, when the eigentransmission becomes zero, only the polarization component parallel to the coalesced eigenstate appears at the output.

Figure 1 illustrates the dependence of polarization conversion (PC) on eigentransmission at chiral EPs. When the eigentransmission at the chiral EP is non-zero (t _e ≠ 0, see Figure 1(a)) in the circular polarization basis ( σ = L and μ = R ), APC is observed alongside non-zero co-polarized transmission (e.g., t _RL = 0, t _LR ≠ 0, and t _{RR =} t _LL ≠ 0₎. In contrast, when the eigentransmission at the chiral EP is zero (t _e = 0, see Figure 1(b)), CAPC occurs without co-polarized transmission (t _RL = 0, t _LR ≠ 0, and t _RR = t _LL = 0), as suggested by the simplified Jones matrix J = t L R R L . In the subsequent section, we propose a non-Hermitian metasurface situated on a gain substrate as a potential platform for realizing CAPC at the chiral zero-eigenvalue EP.

Figure 1:

Eigentransmission-dependent polarization conversion at chiral EPs. (a) Asymmetric polarization conversion (APC) at the chiral nonzero-eigenvalue EP, (b) complete asymmetric polarization conversion (CAPC) at the chiral zero-eigenvalue EP.

2.2 Conditions for achieving CAPC

To illustrate a viable approach for realizing CAPC, we examine a non-Hermitian metasurface with unit cells composed of two coupled resonators, situated on a gain substrate. The two constituent resonators are assumed to have the same resonance frequency of ω ₀/2π, with their major axes aligned along the x- and y-directions, respectively (see Figure 2(a)). Using temporal coupled-mode theory (TCMT) [22], [34], the non-Hermitian Jones matrix can be expressed in the linear polarization basis as follows:

Figure 2:

Analytic modeling of gain-incorporated non-Hermitian metasurfaces. (a) Schematic illustration of a gain-incorporated non-Hermitian metasurface consisting of two coupled resonators on a gain substrate. Two coupled resonators (drawn with cyan and magenta colors) are assumed to be connected to four ports. Two incoming wave amplitudes s ₁₊ (x-polarization) and s ₂₊ (y-polarization) and two outgoing wave amplitudes s ₃₋ (x-polarization) and s ₄₋ (y-polarization), which are relevant to the evaluation of the Jones matrix, are shown. (b) Eigentransmission amplitude plotted in a two-dimensional parameter space spanned by normalized incident frequency ( ω ̄ = ω / ω E P ) and normalized gain rate ( g ̄ = g / g E P ), where ω _EP and g _EP represent the incident angular frequency and gain rate at the chiral EP, respectively. (c) Polarization eigenstates are plotted on a Poincaré sphere as a function of ω ̄ and g ̄ , with color shading used to denote different gain rates. Frequency-dependent polarization eigenstates are represented by scatters of the same color. (d) Uniformly distributed input polarization states mapped on Poincaré sphere (left). All input polarization states are converted to the LCP state at the output.

To simplify the expression, we introduce the following dimensionless parameters: Ω x = ω x − ω / γ e x , Ω y = ω y − ω / γ e y , Γ = γ i y − g / 2 γ e y − γ i x − g / 2 γ e x , and K = − κ / γ e x γ e y . Here, ω represents the angular frequency of the incident light, while κ, g, γ _eμ and γ _iμ denote the coupling, gain, radiative loss, and intrinsic loss rates, respectively μ = x , y . In the analysis, two coupled resonators are connected to four ports: two ports describe x- and y-polarized waves on the input side of the metasurface, while the remaining two ports describe the same waves on the output side (Figure 2(a)). The incoming and outgoing waves are represented by s + = s 1 + , s 2 + , s 3 + , s 4 + T and s − = s 1 − , s 2 − , s 3 − , s 4 − T , respectively. To derive the Jones matrix, we analytically establish the relationship between the incoming wave amplitudes s ₁₊ (x-polarization) and s ₂₊ (y-polarization), and the outgoing wave amplitudes s ₃₋ (x-polarization) and s ₄₋ (y-polarization) (see Supplementary Information for details). The two constituent resonators are assumed to have different radiative loss rates (γ _ex ≠ γ _ey ), while experiencing equal influence from the gain substrate. Accounting for the substrate gain, the net intrinsic loss rates of the two resonators are modified to γ _ix − g and γ _iy − g, respectively.

By solving the matrix eigenvalue equation, T l ψ e = t e ψ e , we can obtain the eigentransmissions and the corresponding polarization eigenstates. Examining the eigentransmissions reveals a pair of chiral EPs. One EP has a coalesced LCP eigenstate, characterized by the condition Ω _x = Ω _y = Ω and Γ = −K. The other EP has a coalesced RCP eigenstate, characterized by Ω _x = Ω _y = Ω and Γ = K. Without loss of generality, in this work, we primarily focus on the chiral EP characterized by a coalesced LCP eigenstate. At this specific chiral EP, the eigentransmission is given by t _e = ξ + ηΩ, and the coalesced eigenstate is represented by ψ e = 1 2 1 j T (see Supplementary Information for details).

From the eigentransmission expression, it can be shown that the following relationships between a set of parameters are sufficient to ensure zero eigentransmission at the chiral EP (see Supplementary Information):ω = ω ₀ = ω _EP and γ e x / γ e y = − γ i x − g / γ i y − g = δ > 0 . The latter requirement indicates that the gain rate should be adjusted such that the net intrinsic loss rates, γ _ix − g and γ _iy − g, have opposite signs. This can be achieved if the two resonators are composed of materials with different intrinsic loss rates (γ _ix ≠ γ _iy ), and the gain rate is set to a value between γ _ix and γ _iy . To illustrate this, we plot the eigentransmission amplitude in a two-dimensional parameter space (Figure 2(b)), defined by the normalized incident frequency ( ω ̄ = ω / ω E P ) and the normalized gain rate ( g ̄ = g / g E P ). Here, ω _EP and g _EP represent the incident angular frequency and gain rate at the chiral EP, respectively. In this example, the following parameters are used to model the non-Hermitian metasurface: ω _EP /2π = 193 THz, γ _ex /ω _EP = 0.06, γ _ey /ω _EP = 0.1, γ _ix /ω _EP = 0.02, γ _iy /ω _EP = 0.04, κ/ω _EP = 0.00968, and g _EP /ω _EP = 0.0275. As shown in Figure 2(b), the eigentransmission approaches zero at the chiral EP, specifically at ω ̄ , g ̄ = 1,1 . The polarization eigenstates are plotted on a Poincaré sphere as a function of ω ̄ and g ̄ , with color shading indicating different gain rates to enhance visualization (Figure 2(c)). As the parameters approach the chiral EP, the paired polarization eigenstates progressively coalesce into the LCP state located at the south pole of the Poincaré sphere [31]. The CAPC at the chiral zero-eigenvalue EP is further verified by observing that all output polarization states converge at the south pole of the Poincaré sphere for uniformly distributed input polarization states (Figure 2(d)).

2.3 Numerical verification of CAPC

To demonstrate the feasibility of gain-assisted ACPC at the chiral zero-eigenvalue EP, we designed a non-Hermitian metasurface. This metasurface consists of two coupled plasmonic nano-rod resonators placed on a gain substrate (Figure 3(a)). Each of the two plasmonic nano-rod resonators, oriented with their major axes perpendicular to each other, is designed to resonate at the same angular frequency, ω ₀, when exposed to incident light polarized along their respective major axes. By varying the geometrical parameters of the resonators and the lattice constant, we can adjust the ratio, δ, between the radiative loss rates. To achieve distinct intrinsic loss rates, two types of plasmonic nano-rods can be utilized: one made of silver and the other made of gold. Specifically, the Drude model, which describes the relative permittivity of metals, was employed in our simulations with the following parameters: a plasma frequency ω _p = 1.2052 × 10¹⁶ rad/s and a collision frequency γ = 1.3697 × 10¹⁴ rad/s for gold, and ω _p = 1.1454 × 10¹⁶ rad/s and γ = 1.1938 × 10¹⁴ rad/s for silver [35]. In the simulations conducted using a finite element method solver, the following geometrical parameters were employed (see Figure 3(a)): L = 600 nm, l ₁ = 193 nm, l ₂ = 197 nm, w ₁ = 82 nm, w ₂ = 121 nm, and d = 60 nm. The coupling between the resonators is tuned by adjusting their relative positions within the unit cell. As discussed in the previous section, by fine-tuning the gain rate and the frequency of the incident wave, we can approach the chiral zero-eigenvalue EP and thereby observe CAPC. In numerical simulations, the change in gain rate is modeled by varying the imaginary part of the substrate permittivity. The complex-valued permittivity, ɛ = ɛ′ − jɛ″, is specifically modeled after an InGaAsN/InP quantum well [36], [37], with the real part of the permittivity (ɛ′ = 9.5), held approximately constant, while the imaginary part (ɛ″) is varied [38], [39].

Figure 3:

Design of the gain-incorporated non-Hermitian metasurface for CAPC. (a) Schematic illustration of the gain-incorporated non-Hermitian metasurface, with the geometric dimensions of the unit cell determined to be L = 600 nm, l ₁ = 193 nm, l ₂ = 197 nm, w ₁ = 82 nm, w ₂ = 121 nm, and d = 60 nm. (b) Numerically extracted eigentransmission magnitude plotted in the parameter space spanned by the input frequency ( f ) and the imaginary part of substrate permittivity ( ɛ″). (c–e) Numerically extracted transmission amplitudes t L R , t R L , and t R R = t L L in a two-parameter space spanned by f and ɛ″.

To confirm the presence of a chiral zero-eigenvalue exceptional point (EP), we performed numerical calculations of the eigentransmission for the designed non-Hermitian metasurface. These calculations were conducted in a two-parameter space defined by the input frequency and the imaginary part of the permittivity (Figure 3(b)). The eigentransmission reveals the topology of a self-intersecting Riemann surface, identifying the chiral EP of a nearly zero eigenvalue at approximately f E P , ε E P ′ ′ ′ = (193.08 THz, 0.4685). Additionally, the Jones matrix elements are extracted and plotted in the two-parameter space to substantiate the fulfillment of the CAPC condition. In Figure 3(c–e), it is evident that the cross-polarized transmission t _RL and the co-polarized transmissions t _RR and t _LL sharply approach zero at the chiral zero-eigenvalue EP, whereas the other cross-polarized transmission t _LR remains nearly constant. It is worth noting that at the chiral zero-eigenvalue EP, the intensity conversion efficiency of the proposed metasurface, initially estimated to be approximately t L R 2 ≈ 4 % , can be significantly enhanced to up to 25 % by optimizing the geometrical parameters of the unit cell. In addition to examining the transmission amplitude, understanding the system behavior near the chiral EP requires a thorough analysis of the transmission phase spectra. The phase spectra offer crucial insights into the underlying dynamics, particularly in the vicinity of EPs, where rapid phase variations occur. These variations reflect the topological characteristics of the EP, manifesting as pronounced phase changes in its proximity. To further illustrate this phenomenon, the transmission phase spectra and their winding behavior across the two-parameter space are presented in Figure S2 of the Supplementary Information.

2.4 Maximal CDT at chiral zero-eigenvalue EPs

The proposed metasurface interacts with waves in a direction-dependent manner, effectively altering the polarization of light differently based on whether it is incident from the front or the back. This phenomenon is attributed to the inherent planar chirality, arising from the broken in-plane mirror symmetry of the unit cell. In particular, CAPC occurs when an LCP wave is incident on the front surface of the metasurface. Conversely, CAPC is observed with an RCP wave when it is incident from the rear. This indicates that direction-dependent CAPC can be achieved at the chiral zero-eigenvalue EP (Figure 4(a)). To demonstrate how the metasurface converts circularly polarized waves based on the incident direction, we performed simulations exposing the metasurface to all possible states of polarized waves at the chiral zero-eigenvalue EP (Figure 4(b)). In the case of forward incidence, the output polarization states converge at the south pole (LCP), whereas for backward incidence, they converge at the north pole (RCP). The output polarization states are tightly clustered around the poles, revealing an extreme case of non-unitary polarization transformation at the chiral zero-eigenvalue EP.

Figure 4:

Direction-dependent CAPC and maximal CDT at the chiral zero-eigenvalue EP. (a) Schematic illustration of direction-dependent CAPC in gain-incorporated non-Hermitian metasurfaces. (b) For incidence of the homogenous input state mapped on Poincaré sphere, the LCP and RCP components are mostly transmitted for forward and backward propagation, respectively. Uniformly distributed input polarization states mapped on Poincaré sphere (left). All input polarization states are converted to the LCP state at the output. (c) Numerically-extracted CDT in a two-parameter space spanned by f and ɛ″. CDT forward (blue surface) and CDT backward (red surface) for forward and backward propagation, respectively.

Notably, CAPC maximizes circular dichroism in transmission (CDT), defined as the normalized difference in transmission between left-handed and right-handed circularly polarized light [40], [41],

For the simulated case, all transmission amplitudes except for t _LR approach nearly zero at the chiral zero-eigenvalue EP for forward incidence. Consequently, CDT approaches a value of −1, indicating its maximum magnitude (Figure 4(c)). For backward incidence, the value of CDT is inverted, approaching +1 at the chiral zero-eigenvalue EP. Through numerical simulations, we achieve blocking of RCP at approximately 50 dB while maximizing the CDT by simultaneously suppressing co-polarized transmission. This enhanced level of polarization control is not simply a result of parameter optimization; it arises from mathematically rigorous conditions.