1. Introduction

Recently, the study of non-Hermitian systems has emerged as a frontier of considerable interest. Non-Hermitian systems, characterized by their energy exchange with the environment, diverge from the traditional closed (or Hermitian) systems that have long defined in quantum mechanics [1]. Non-Hermitian systems [2–7] are distinguished by their potential to undergo eigenstate collapse at specific parameter values, giving rise to phenomena known as Exceptional Points (EPs) [8–17]. EPs, where eigenvalues and their corresponding eigenmodes coalesce, have been identified as a fertile ground for a host of innovative applications, including but not limited to precision sensing [11–13], chiral transmission [14,15], vortex emission [16] and independent control of the polarization [17].

The intrigue surrounding EPs deepens when considering systems with more than two states, where higher-order EPs [18] appear, accompanied by the vanishing of more than one state, which provides new ideas for enhanced sensing [13,19] and emission [20]. In-depth studies of complex systems reveal the interaction can lead to the emergence, coalescence, and topological properties of multiple EPs [21–26]. These complex systems have been shown to exhibit a rich tapestry of physical phenomena, each with its own potential for groundbreaking applications that transcend those found in simpler two-state systems. Although multiple EPs have been successfully demonstrated in the framework of acoustic loss-modulated resonator systems [21], their exploration in electromagnetic fields has been comparatively limited. Yet, it is within these electromagnetic systems that a wider array of novel and potentially transformative phenomena, such as enhanced sensing [27,28], mode switching [29], and dynamic modulations [30], are predicted to exist.

Surface plasmonics [31–34] provides an especially promising platform for such explorations. The field has already made significant strides, notably with the experimental realization of anti-Parity-Time (APT) symmetry. This symmetry, remarkable for its radiative properties, has been observed to offer new modalities for wave manipulation [35]. The integration of far-field radiation with the intricate landscape of multiple EPs stands at the precipice of discovery, with the promise of a myriad of applications that could revolutionize sensing technologies, light-matter interactions, and the development of new metamaterials [36,37].

In this article, we investigate the coalescence of multiple EPs within a designer surface plasmonic (DSP) quadrumer system. Beginning with a theoretical analyze based on coupled mode model, we dissect the parameters that dictate the emergence and coalescence of multiple EPs within these quadrumer structures and trace the ensuing eigenstate evolution. Advancing beyond theoretical constructs, we utilize the fine-tunable coupling parameters characteristic of DSP modes to design innovative metasurfaces. These surfaces are engineered to enable the selective excitation of distinct eigenstates, achieved through precise manipulation of the incident direction and angle of far-field plane waves. The capability to control these parameters opens up unprecedented possibilities in the field of wavefront shaping and device functionality. This investigation accomplishes the first electromagnetic realization of multiple EPs, incorporating their unique features into the domain of microwave far-field radiation [38].

2. Coupled mode model

We initiate our exploration with the coupled mode model (i.e., the tight-binding model) of a two-state system, depicted as the dashed box in Fig. 1(a). This system comprises two modes (denoted as A and B) with eigenenergies ${E_0}\textrm{ + }\delta$ and ${E_0}\textrm{ - }\delta $, respectively, where $\delta $ is the detuning. Each mode has the intrinsic loss ${\gamma _0}$. The coupling factor between the two modes (i.e., the intra-unit coupling) is a pure imaginary number $i{\chi _1}$. The eigenvalues are given by

 

Fig. 1. Multiple EPs in quadrumer system, the parameters corresponding to the eigenspectrum are selected as E₀ = 0, γ₀ = 0, χ₁ = 1 and χ₂ = 0.5. (a) a schematic drawing of a coupled quadrumer system. The gray frame represents a two-state dimer unit. (b) The eigenspectra of the two-state unit, where the stars indicate the EP. (c) and (d) are eigenspectra (upper) and eigenstates at the coalescences (bottom) with t = 1 and t = −1, respectively. The color of the curves and arrows distinguishes different states; the grey lines in the eigenspectra are projections of the real and imaginary parts of the eigenvalues, and the stars and triangles mark the spectral coalescences.

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The eigenspectra, i.e., the evolution of the real part $\textrm{Re} ({E_i})$ and imagery part ${\mathop{\rm Im}\nolimits} ({E_i})$ of the eigenvalues in $\delta /{\chi _1}$ parameter space, are illustrated in Fig. 1(b). In the APT symmetric (APT-S) phase ($\delta /{\chi _1} > 1$), the $\textrm{Re} ({E_i})$ of the two states coincide, while the ${\mathop{\rm Im}\nolimits} ({E_i})$ differs; at the EP ($\delta /{\chi _1} = 1$), the eigenvalues coalesce; in APT broken (APT-B) phase (i.e., $\delta /{\chi _1} < 1$), the real part splits while the imaginary part remains constant.

Progressing from the two-state systems as fundamental units, we construct a composite four-state system as illustrated in Fig. 1(a). The detuning of the upper unit cell is $\delta $, and that of the lower unit cell is $t\delta $. The coupling factors between different unit cells (i.e., the inter-unit coupling) is also a pure imaginary number $i{\chi _2}$. The Hamiltonian of the four-states system can be articulated as $H = {E_0} + i{\gamma _0} + H^{\prime}$, where ${E_0}$ and $i{\gamma _0}$ signify the center energy and intrinsic loss of the system, respectively. The interaction term $H^{\prime}$, can be expressed as

It is noteworthy that each unit individually adheres to APT symmetry [35], the overall system lacks APT symmetry unless $t ={\pm} 1$. The analytic solution for the eigenvalues of $H^{\prime}$ takes the form

The two plus-minus signs in Eq. (3) identify four eigenstates, and the zeroes of the term after the plus-minus signs indicates the coalescence of states. A further discussion of the zeros allows a glimpse of different classes of states coalescence. There exists three classes of the coalescence of states: (Class-I) $X \pm \sqrt Y = 0$ with $X \ne 0$, $Y \ne 0$, where one state vanishes, corresponding to a normal EP; (Class-II) $X \ne 0$, $Y = 0$, where two states vanish; (Class-III) $X = Y = 0$, where three states vanish and only one four-order EP exists. Without loss of generality, we consider a scenario to visually demonstrate these coalescences with intra-unit coupling surpasses inter-unit coupling

When t = 1, the system reveals the coalescence of Class-II EPs, as indicated by star markers in the upper portion of Fig. 1(c). These stars denote the pairing of eigenmodes at these EPs, signifying the co-occurrence of two EPs at an identical parameter value, hereby designated as EP-II. In contrast, the presence of a triangle symbol signals the absence of eigenmode coalescence, suggesting an accidental degeneracy rather than a true EP. The situation inverses when t = −1, as illustrated in Fig. 1(d). There appears two Class-I EPs, also marked by stars, where the eigenvalues coalesce and the corresponding eigenmodes substantiate the EPs, labeled as EP-I₁ and EP-I₂. Notably, one of these eigenmodes ceases to exist. Although the eigenspectra echoes the dual structure of Fig. 1(b), it is now displaced within the parameter space.

Furthermore, this illustration demonstrates that the sign of the inter-unit detuning coefficient t can switch the classes of EPs. It effectively manipulates the relative positioning of multiple EPs within the parameter space, thus driving distinct phase transition phenomena. Leveraging this characteristic, the topological features of the multiple EPs can switch through simple adjustments in realizations. Additionally, the quadrumer can also be used as a unit cell to form a chain, which contains more EPs with complex interactions, as detailed in Fig. S1 of Supplement 1.

3. Designer-plasmonic realization of multiple EPs

Here, we realize the four-state non-Hermitian system owning different classes of EPs with DSP quadrumer. The DSP quadrumer system shown in Fig. 2(a) comprises three layers: The top layer is four designer surface plasmonic resonators (DSPRs) made up of groove textured ultrathin copper disks, where the inner radius is r₀, outer radius is R₀, the groove number is N = 60, the filling ratio is FR = 50%, and the distances between intra-unit and inter-unit resonators are d₁ and d₂, respectively; the middle layer is a $h = 2$ mm thin dielectric substrate with relative permittivity ${\mathrm{\epsilon }_r} = 2.2$; and the bottom layer is a complete copper plate attached to the substrate. The sandwich structure can support leaky modes [35]. The in-plane component of the leakage establishes indirect coupling channels between modes on different DSPRs, as well as the out-of-plane leakage ensures the coupling to the far field. To satisfy the relationship between the coupling factors as Eq. (6) describe, here we select the dipolar localized surface plasmonic modes with the geometric parameters of resonators as ${d_1} = {d_2} = 18$ mm, ${r_0} = 6$ mm, ${R_0} = 12$ mm. It is worth noting that well-confined modes on the DSPRs are also directly coupled though evanescent filed, while the direct coupling can be ignored because the distances between

 

Fig. 2. Structure and parameters of the DSP quadrumer system with R₀ = 12 mm, r₀ = 6 mm, d₁ = d₂ = 18 mm. (a) top and side views of the structure; (b) resonant frequencies (circles) with different Δr and the linear fitted line; (c) the absolute value of indirect coupling factors of two parities with different d, the insets show the field distributions of the two parities marked as odd and even mode.

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DSPRs is much larger than the decay length of the evanescent filed (Fig. S2 and Fig. S3 in Supplement 1), which has been experimentally proved [35].

The coupling factors in the coupled mode models can be directly mapped to the structural parameters of the DSPR quadrumer as Figs. 2(b-c) shows. To maintain constant inherent loss, we introduce the same detuning to the inner and outer radii, ensuring uniform groove depth for each DSPR. Figure 2(b) shows the simulated resonant frequencies of the single DSPR with different $\Delta r$, revealing a linear correlation. Though the frequency splitting in a two-state system in APT-B phase, the absolute value of indirect coupling factor $|\chi |$ can be extracted from Eq. (1). It is noteworthy that $|\chi |$ is related to the parity of the mode on DSPRs. For the dipolar mode under consideration, two parities exist: odd mode and even mode, as depicted in Fig. 2(c). These correspond to intra-unit and inter-unit coupling in the four-state system. It is worth noting that although the sign of the coupling factor is related to parity, in our system, the coupling factors appears squared in the eigenvalues, therefore, the sign does not affect the result. The extracted $|\chi |$ for various resonator spacing d shows the following characteristics: (i)The indirect coupling between odd dipolar modes (i.e., $|{{\chi_1}} |$) is larger than that between even dipolar modes (i.e., $|{{\chi_2}} |$), which is due to that the maximum-to-maximum pattern of the odd mode has high coupling efficiency; (ii) The couplings are stronger at smaller distance d because the out-of-plane component of the leaky wave radiates into the air while the in-plane component is attenuated (Fig. S4 in Supplement 1). In our configuration, the strength of coupling factors can be expressed as $|{{\chi_1}} |\textrm{ = 3}\textrm{.75} \times {10^{ - 3}}c/{r_0}$ and $|{{\chi_2}} |\textrm{ = 1}\textrm{.069} \times {10^{ - 3}}c/{r_0}$, where c is the velocity of light. Hence, the frequency detuning $\delta $ can be realized by adjusting the external radii ${R_0} \pm \Delta r$ of the DSPRs, where $\Delta r$ represents the radius detuning. Additionally, the sign of t can be switched through the sign of $\Delta r$ in different units.

4. Selective excitation of the non-Hermitian eigenstates

As Figs. 1(c-d) shows, the eigenmodes at different classes of EPs distribute differently at each resonator. Due to the far-field response of the DSPRs, the eigenstates at different phases can be selectively excited by the far-filed incident plane waves. Figure 3(a) illustrates the configuration for far-field excitation, where a plane wave is obliquely incident on the sample, the angles with the x and y axis are respectively ${\theta _x}$ and ${\theta _y}$, introducing phase differences ${\phi _{x,y}} = \frac{{\omega (d + 2{R_0})}}{c}\cos {\theta _{x,y}}$ between DPSRs to efficiently excite the corresponding states. Therefore, phase differences can be introduced in the x and y directions to efficiently excite the corresponding states. Two probes, labeled as P₁ and P₂, are put at the maxima side of DSP dipolar modes. However, not all the phase-distribution of eigenmodes satisfy the far-filed phase condition [39], i.e., eigenmodes marked as red vectors in Fig. 1(c) and the green vectors in Fig. 1(d) can only be excited though near-field configurations. Figure 3(b) shows the near-field excitation configuration, featuring dipolar sources labeled S₁ and S₂, with detection and excitation locations determined by the intensity distribution. After extracting the parameters mentioned above, we can predict the eigenvalue es and eigenstates of the system with various $\Delta r$. By adjusting the amplitude and phase among the source array, eigenmodes with different amplitude and phase distributions can be near-filed excited.

 

Fig. 3. The simulation characterizations of the Class-II plasmonic multiple EPs. (a) Far-field excitation configuration with an oblique plane wave incident upon the sample, where P₁ and P₂ denote probe positions. (b) Near-field excitation configuration with S₁ and S₂ indicating source locations and P₁ and P₂ marking probe positions. (c-d) Comparison of theoretical calculations (solid curves) and simulation results (circles) for the real (c) and imaginary (d) parts of the eigenfrequencies as functions of the detuning Δr₁ = Δr₂ =Δr. (e-f) The electric field intensity distributions (E_z and |E_z|) of various states at different Δr. Blue and red outlines indicate far-field and near-field excitation, respectively, and the white arrows represent the theoretically predicted eigenvectors.

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Figures 3 and 4 shows the eigensolutions extracted from simulated excitation configurations, which correspond to the calculated eigenvalues and eigenmodes in Figs. 1(c-d). When $\Delta {r_1} = \Delta {r_2} = \Delta r$ (i.e., the inter-unit frequency detuning $t = 1$), the Class-II EPs appears in the quadrumer. The eigenfrequencies spectra ($\textrm{Re} (\omega )$ and ${\mathop{\rm Im}\nolimits} (\omega )$) extracted from simulated excited modes (circles) is shown in Figs. 3(c-d), alongside the calculated eigenspectra from couple resonator model (dashed lines). The real part of the eigenvalue $\textrm{Re} (\omega )$ corresponds to the resonance frequencies measured at the probe, and the imaginary part ${\mathop{\rm Im}\nolimits} (\omega )$ can be extracted from the half-width of the Lorentz resonance peaks. We select both the near-field and far-field excitation based on the characteristics of the state to enhance the measurement accuracy. The eigenmodes with larger ${\mathop{\rm Im}\nolimits} (\omega )$ imply the higher radiation losses, which is more efficient to be excited by the far field, as the red circles depicted in the Fig. 3(d).

 

Fig. 4. The simulation characterizations of the Class-I plasmonic multiple EPs. (a-b) Comparison of theoretical calculations (solid curves) and simulation results (circles) for the real (a) and imaginary (b) parts of the eigenfrequencies as functions of the detuning Δr₁ = −Δr₂ =Δr. (c) The electric field intensity distributions (E_z and |E_z|) of various states with different Δr, with certain states highlighting the E-field distribution. Blue and red outlines indicate far-field and near-field excitation, respectively, and the white arrows represent the theoretically predicted eigenvectors.

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By adjusting radius detuning $\Delta r$ between DSPRs, the eigenmodes evolute in different phases and can be categorized into two groups characterized by phase differences of field on resonators B₁ and B₂, as shown in Figs. 3(e-f). The blue and red frames represent far-field and near-field excitation, respectively. Based on the perturbation theory [39], the dipolar mode on each resonator can be regarded as dipole, and the total dipole moment can be approximated as the vector sum of eigenmodes because of the subwavelength size of the spacing. The far-field configuration can only excite the eigenmodes with non-zero dipole moments, leading to the eigenmodes with zero dipole moments shown in Fig. 3(f) are more efficiently excited by the near field. Notably, some state in the blue frame of Fig. 3(f) can still be excited by the far field, because of 180° the phase mismatching between adjacent resonators from the near field inevitably. We also explore the evolution of the eigenstates in parameter space: with the increasing of $0 < \Delta r < 0.14$ mm, the phases of pattern on resonators A₁ and A₂ changes continuously; however, at $\Delta r = 0.14$ mm(i.e., at Class-II EPs), the phase becomes identical, and the eigenmodes coalesce; as $\Delta r > 0.14$ mm, the magnitude of patterns on A and B becomes inconsistent, and the four different eigenstates reappear.

When $\Delta {r_1} ={-} \Delta {r_2} = \Delta r$ (i.e., the inter-unit frequency detuning $t ={-} 1$), there exist two Class-I EPs, the corresponding simulated results are shown in Fig. 4. Figure 4(c) reveal that the eigenstates can also be divided into two groups according to the phase difference between A₁ and B₂. When $\Delta r < 0.08$ mm, the phase difference of the pattern on A₂ and B₁ decreases and the states in the two groups reach the Class-II EPs. Class-I EPs are significantly different from the Class-II shown in Fig. 3: (i) The Class-I EPs located at $\Delta r = 0.08$ mm and 0.21 mm, respectively, resulting in a new phase transition; (ii) Class-II EP located at two different imaginary frequencies ${\mathop{\rm Im}\nolimits} (\omega )$, and some patterns can hardly be excited far field incident wave; while Class-I EPs located at the same imaginary frequency as shown in Fig. 4(b), the corresponding eigenstates can all be excited by the far-field incident plane wave. These results confirm the specific far-field response of the multiple EPs.

In addition, the proposed platform supports higher-order EP, i.e., the Class-III coalescence in the theoretical analysis, and the parameters and eigenspectrum of realizing a third-order EP are shown as Fig. S5 in Supplement 1.

5. Discussion and outlook

We elucidate the properties of multiple EPs by employing DSPR quadrumer system. By employing far-field plane waves at various incident angles, we selectively excited eigenstates, elucidating their evolution near distinct EP types. This work offers a platform for probing topological properties of complex system with multiple EPs, with significant implications for wireless sensing and far-field detection technologies. Further, our findings suggest potential expansion into the terahertz and optical frequency domains, paving the way for innovative designs of highly sensitive, miniaturized terahertz devices. This advancement promises to enhance the resolution and sensitivity of future photonic sensing systems, marking a substantial leap forward in the field.