Directional Coupling of Surface Plasmon Polaritons at Exceptional Points in the Visible Spectrum

Abstract

Robust control over the coupling and propagation of surface plasmon polaritons (SPPs) is essential for advancing various plasmonic applications. Traditional planar structures, commonly used to design SPP directional couplers, face limitations such as low extinction ratios and design complexities. These issues frequently hinder the dense integration and miniaturisation of photonic systems. Recently, exceptional points (EPs)—unique degeneracies within the parameter space of non-Hermitian systems—have garnered significant attention for enabling a range of counterintuitive phenomena in non-conservative photonic systems, including the non-trivial control of light propagation. In this work, we develop a rigorous temporal coupled-mode theory (TCMT) description of a non-Hermitian metagrating composed of alternating silicon–germanium nanostrips and use it to explore the unidirectional excitation of SPPs at EPs in the visible spectrum. Within this framework, EPs, typically associated with the coalescence of eigenvalues and eigenstates, are leveraged to manipulate light propagation in nonconservative photonic systems, facilitating the refined control of SPPs. By spatially modulating the permittivity profile at a dielectric–metal interface, we induce a passive parity–time ($\mathcal{PT}$)-symmetry, which allows for refined tuning of the SPPs’ directional propagation by optimising the structure to operate at EPs. At these EPs, a unidirectional excitation of SPPs with a directional intensity extinction ratio as high as 40 dB between the left and right excited SPP modes can be reached, with potential applications in integrated optical circuits, visible communication technologies, and optical routing, where robust and flexible control of light at the nanoscale is crucial.

1. Introduction

The quantum mechanics formalism postulates that the Hamiltonian governing a system’s state evolution in a complex Hilbert space must be Hermitian, ensuring its eigenvalues are real [1]. This criterion guarantees that the system observables—energy measurements in this case—corresponding to these eigenvalues are real and yield physically meaningful and measurable results. This notion however has been recently challenged by Bender et al. [2], who proved that systems with non-Hermitian Hamiltonians may indeed have a real spectrum provided that such systems obey the $\mathcal{PT}$-symmetry condition despite being open systems. Remarkably, these systems under non-Hermiticity parameter variation can endure spontaneous symmetry breaking and restoration upon real to complex spectral transitions at singularities called exceptional points (EPs) [3]. At exceptional points, one or more eigenstates and their corresponding eigenvalues simultaneously coalesce and become degenerate, which is notably different from the degeneracy in Hermitian systems referred to as diabolic points (DPs) [4], where only the eigenvalues coalesce. In contrast, the eigenstates remain orthogonal in the Hermitian parameter space. Hence, non-Hermitian systems exhibit reduced dimensionality at EPs [5].

Although the theoretical concept of exceptional points (EPs) originated in quantum mechanics, it sparked extensive attention in nonconservative photonic systems [6,7,8,9] and was first experimentally observed in optics and photonics [10,11]. This was due to the widespread presence of non-hermiticity in optical systems, primarily caused by intrinsic losses due to absorption and radiation leakage. Additionally, the ability to locally control the gain through methods like stimulated emission or parametric processes [12] enables easier control of non-Hermitian Hamiltonians and facilitates the exploration of exceptional points in these systems.

EPs have paved the way for fundamentally revisiting the photonic systems with losses no longer viewed as undesirable attributes but rather a useful way to manipulate the nonconservative photonic systems for novel functionalities in lasing, sensing, and light propagation control that were unattainable otherwise. It has been demonstrated that the distinctive behaviour near an EP significantly influences the spectral response and threshold of lasing cavities [13], leading to a unique reversal in pump dependence marked by a decrease in emitted light intensity as the pump power increases in coupled microdisk quantum cascade lasers [14], directional lasing in a pair of silica microcavities [15] orbital angular momentum vortex lasing [16], and single-mode lasing [17,18].

Remarkably, nanophotonic systems are increasingly recognised as versatile platforms for advanced sensing applications due to their miniaturisation capabilities and high sensitivity to small input changes [19]. In particular, it has been shown that systems of higher order EPs are subjected to eigenstates split proportional to the nth root of the input perturbation, where n is the EP order; hence, sensors operating at EPs are envisaged to have a superior response to perturbations compared to those at conventional degeneracies exhibiting a linear response [3,20,21], promising enhanced performance in applications ranging from biochemistry to the Internet of Things. In this context, Chen et al. [22] demonstrated nano-object detection by perturbing a micro-toroid cavity tuned to an exceptional point with two scatterers. Furthermore, photonic systems operating at an EP have been proposed for nanoparticle detection [23,24,25,26], angular rate and phase sensing [27,28], and refractive index sensing [29,30,31,32].

$\mathcal{PT}$-symmetric photonic structures have also been employed to achieve non-trivial control over light propagation at exceptional points (EPs), including unidirectional reflectionless light propagation in integrated silicon waveguide meta gratings through modulation of the dielectric permittivity [33]. Additionally, the suppression of backscattering has been successfully demonstrated in a silicon microring by integrating asymmetric Mie scatterers, which steer the system toward an EP [34], as well as through the use of external nano cylinder scatterers [35].

The interest in manipulating light transcends propagation controllability to topological phenomena at optical interfaces [36] and extends to surface plasmon polaritons (SPPs), which are evanescent electromagnetic waves coupled to charge oscillation at the metal–dielectric interfaces [37]. SPPs hold the promise of transcending the diffraction limit and enhancing light–matter interactions [38] and are envisioned to enable key applications in sensing [39,40,41], nanolasers [42,43,44], microscopy [45], spectroscopy [46,47], data transmission [48], and integrated optical communication [49,50]. Therefore, considerable efforts have been devoted to achieving robust and controllable manipulation of SPPs. Although conventional techniques such as nanoslits [51], catenary apertures [52], nanoantennas [53,54,55,56,57], and metasurfaces [58,59,60] have been explored to facilitate the robust and directional control of SPPs, these methods often suffer from considerable scattering losses, low extinction ratios, and complex manufacturing processes. Alternatively, non-Hermitian $\mathcal{PT}$-symmetric photonic systems have recently emerged as a compelling alternative for manipulating SPPs at EPs. For instance, recent theoretical studies have shown that the modulation of $\mathcal{PT}$-symmetric refractive index in a dielectric–metallic waveguide can enable unidirectional clocking for SPPs at EP [61] and the adjustment of Fermi energy and inter-ribbon distances in double-layer graphene nanoribbons to achieve unidirectional reflectionless SPP excitation in the mid-infrared range [62]. Moreover, metagratings have shown the potential for the unidirectional excitation of SPPs at EPs [63] and have been employed to achieve unidirectional SPP excitation at EPs in infrared [64], offering a novel and flexible approach for SPP manipulation.

Despite these advances, the potential for unidirectional SPP excitation at visible frequencies using non-Hermitian $\mathcal{PT}$-symmetric metagratings remains unexplored. Such an approach could leverage symmetric complex optical potentials without relying on metallic materials in the metagrating design, simplifying the fabrication and integration into practical applications. In this work, we formulate a rigorous TCMT description of non-Hermitian metagratings that maps the complex permittivity modulation onto an effective non-Hermitian two-mode Hamiltonian, allowing us to analytically identify the exceptional points and their associated unidirectional SPP eigenstates (see Appendix A). Building on this framework, we demonstrate the unidirectional excitation of SPPs in the visible spectrum through permittivity modulation using a metal-free non-Hermitian metagrating composed of alternating Si–Ge nanostrips. This design benefits from a simpler fabrication process and leverages passive $\mathcal{PT}$-symmetry, enabling the robust and energy-efficient control of SPPs in the visible spectrum, with potential applications in integrated photonic circuits, visible-band optical communications, and on-chip optical routing.

2. Materials and Methods

2.1. Theoretical Model

In this study, we investigate the unidirectional excitation of surface plasmon polaritons (SPPs) in the visible spectrum at exceptional points (EPs) under passive $\mathcal{PT}$-symmetry. This is achieved by meticulously engineering the spatial distribution of the permittivity profile at a dielectric–metal interface using a non-Hermitian metallic-free metagrating that is driven to operate at EPs at the optimised spatial distribution of both the real (${\epsilon }_r$) and imaginary (${\epsilon }_i$) parts of the permittivity.

Figure 1 illustrates a schematic of the proposed structure, which consists of a silicon–germanium (Si-Ge) metagrating placed on top of a metallic (silver) and dielectric (air) interface. This configuration introduces a non-uniform spatial distribution of the dielectric refractive index along the silver–air boundary. The perturbation of the permittivity profile $ϵ\left(x\right)$ at the metal–dielectric interface, in the presence of the metagrating, is denoted as $\Delta ϵ\left(x\right)$ and can be expressed under the assumption of a continuous passive $\mathcal{PT}$-symmetric profile as follows [33,65]:

$$\Delta ϵ\left(x\right)=\Delta ϵ\left[cos\left(k_{\mathrm{spp}}x\right)-i\delta sin(k_{\mathrm{spp}}x-\varphi )\right].$$

(1)

$\Delta ϵ$ is the perturbation strength, $\delta$, $\varphi$, and $k_{\mathrm{spp}}$ are the additional imaginary modulation, the relative phase shift between the real and imaginary parts of the permittivity perturbation, and the wavenumber of the excited SPP wave. Equation (1) can be further simplified as follows:

$$\Delta ϵ\left(x\right)=\Delta {ϵ}_r{e}^{i{k}_{\mathrm{spp}}x}+\Delta {ϵ}_i{e}^{-i{k}_{\mathrm{spp}}x},$$

(2)

where $\Delta {ϵ}_r={\displaystyle \frac{\Delta ϵ}{2}}\left(1-\delta e^{-i\varphi }\right)$, and $\Delta {ϵ}_i={\displaystyle \frac{\Delta ϵ}{2}}\left(1+\delta e^{i\varphi }\right)$. Consequently, the overall permittivity profile at the interface will be

$$ϵ\left(x\right)={ϵ}_{\mathrm{air}}+\Delta {ϵ}_r{e}^{i{k}_{\mathrm{spp}}x}+\Delta {ϵ}_i{e}^{-i{k}_{\mathrm{spp}}x}.$$

(3)

Physically, the $\Delta ϵ\left(x\right)$ term represents a complex periodic perturbation to the permittivity continuum of the Ag–Air interface. To realise the passive PT-symmetry condition $\epsilon \left(x\right)={\epsilon }^{*}(-x)$, this spatial modulation is designed such that the real permittivity component (dominated by Si) forms an even function, while the imaginary component (dominated by Ge) forms an odd function. Referring to the formal equivalence between the Schrödinger equation in quantum mechanics and the paraxial wave equation in optics, Equation (1) indicates that the $\mathcal{PT}$-symmetry condition for this optical potential, i.e., $\epsilon \left(x\right)={\epsilon }^{*}(-x)$ [66], is satisfied when $\varphi =0$ or $\pi$.

It is well-known that the interaction between the free space electromagnetic field and metal surface charges increases the SPP momentum compared to free space, resulting in a momentum mismatch between light and SPPs. This mismatch can be addressed using various configurations, such as Kretschmann and Otto prism configurations or grating couplers, which align the momentum of light with that of the SPPs according to the SPP dispersion relation [37,67].

$$k_{\mathrm{spp}}=k_0{\displaystyle \frac{{ϵ}_d{ϵ}_m}{{ϵ}_d+{ϵ}_m}}$$

(4)

Herein, ${ϵ}_d$ and ${ϵ}_m$ denote the permittivities of the dielectric, the thin metal film, $k_0$, represents the wave numbers of the incident wave in free space, and $k_{\mathrm{spp}}$ is the wavenumber of the SPP excited at the interface.

Hence, to excite SPPs along the air–silver interface with normally incident light on the metagrating, the metagrating’s period $\Lambda$ should be calibrated to address the momentum mismatch with the incident light to enable efficient coupling from the diffracted electromagnetic wave to the SPP evanescent wave propagating at the interface, and $\Lambda$ can be mathematically deduced from the SPP dispersion relation, where $k_0$ is given by $k_0={\displaystyle \frac{2\pi }{{\lambda }_0}}$, with ${\lambda }_0$ being the wavelength of the normally incident wave on the metagrating and ${\displaystyle \frac{2\pi }{\Lambda }}=Re\left(k_{\mathrm{spp}}\right)$. The coupling mechanism can be precisely controlled through the geometric and material parameters of the metagrating.

In the absence of the metagrating, the permittivity modulation $\Delta \epsilon$ in Equation (1) vanishes; so, the normally incident light cannot couple to the SPP mode, and no SPPs are excited. However, when the metagrating is introduced with the calibrated periodicity, the periodic modulation of the permittivity, as expressed in Equation (2), provides the phase-matching necessary to excite SPPs. By adjusting this perturbation, which is determined by the metagrating parameters, the coupling to the right and left of the metagrating can be effectively controlled. Additionally, unidirectional excitation can be achieved by manipulating the parameters to derive the passive $\mathcal{PT}$-symmetric system to operate at an EP, as illustrated in Figure 2.

The impact of this continuous perturbing modulation of the dielectric permittivity at the interface on the excitation of the SPP evanescent wave is quantified using the TCMT framework detailed in Appendix A, which maps the complex permittivity modulation onto an effective non-Hermitian two-mode Hamiltonian for the right- and left-propagating SPPs. Within this description, the steady-state solution of the coupled-mode equations shows that the field magnitude ratio of the right ($H_{+}$) and left ($H_{-}$) branches of the supported transverse magnetic (TM) mode at the interface is linearly proportional to the ratio of the first-order Fourier coefficients of the complex permittivity modulation associated with the two propagation directions,

$$\left|{\displaystyle \frac{H_{+}}{H_{-}}}\right|=\left|{\displaystyle \frac{\Delta {ϵ}_i}{\Delta {ϵ}_r}}\right|.$$

(5)

From Equations (1), (4) and (5), we discern that the asymmetric ratio $\left|{\displaystyle \frac{\Delta {ϵ}_i}{\Delta {ϵ}_r}}\right|$ in the introduced permittivity modulation underpins the momentum compensation for SPP propagation towards the right or left direction on the surface. Hereby, it is possible to finely control this directional propagation by controlling the strength of the permittivity perturbation. Although implementing a continuous permittivity perturbation is challenging in practical scenarios, it can be effectively approximated through discrete spatial modulation using metagrating nanostrips, and the perturbation strength can then be controlled by varying the geometric parameters of these nanostrips ($w_1$, $h_1$, $w_2$, $h_2$), along with their centre-to-centre separation distance D as shown in Figure 1. In practice, the effective modulation depth $\Delta \epsilon$ and the relative imaginary modulation parameter $\delta$ are determined by the complex permittivities of Si and Ge at ${\lambda }_0=633\mathrm{nm}$ and by the filling factors of the two materials within a unit cell (set by $w_1$, $w_2$, $h_1$, and $h_2$). The phase $\varphi$ arises from the relative shift of the strips inside the period $\Lambda$, which controls the relative phase of the forward and backward Bloch components and, thus, the balance between the right and left SPP channels.

Consequently, the momentum coupling from the incident free-space light to either the left or right branch of the excited SPP can be adjusted, as visually depicted in the dispersion curve in Figure 2. Within the TCMT framework (Appendix A), the complex eigenvalues associated with the two counter-propagating SPP modes are

$${\lambda }_{1,2}\approx 1\pm \sqrt{{\displaystyle \frac{\Delta {ϵ}_i\Delta {ϵ}_r}{2}}}\approx 1\pm {\displaystyle \frac{\Delta ϵ}{2\sqrt{2}}}\sqrt{1-{\delta }^2+2i\delta sin\varphi }.$$

(6)

The eigenvalues coalesce when the square-root term vanishes, i.e., when $\Delta {ϵ}_r\Delta {ϵ}_i=0$. Physically, this coalescence arises from the interplay between the coherent coupling rate (J) and the effective gain/loss contrast ($\Gamma$), as defined in the temporal coupled-mode theory description in Appendix A (see Equation (A6)). When these two competing rates are perfectly balanced ($\left|J\right|=|\Gamma |$), the system undergoes a phase transition where the two counter-propagating modes can no longer exist as distinct states, causing them to collapse into a single unidirectional eigenstate. In that case, one of the directional coupling channels is suppressed, and the two eigenmodes collapse into a single unidirectional SPP state, aligned with either $H_{+}$ or $H_{-}$ (see Equation (5)). Using the explicit expressions for $\Delta {ϵ}_r$ and $\Delta {ϵ}_i$, this degeneracy occurs at $\delta =\pm 1$ and $\varphi =0$ or $\pi$, where the system sits at an exceptional point with a real eigenvalue spectrum.

The non-Hermitian system eigenvalue evolution in the 1D and 2D parameter spaces is shown in panels A–D and E–F of Figure 3, respectively. A bifurcation occurs when the square-root term in Equation (6), $1-{\delta }^2+2i\delta sin\varphi$, vanishes, marking the topological transition from the $\mathcal{PT}$-symmetric to the $\mathcal{PT}$-broken phase, as seen in Figure 3A,C. This condition is satisfied at $(\delta ,\varphi )=(\pm 1,0)$ and $(\pm 1,\pi )$, which, in terms of the effective modulation coefficients, correspond to $\Delta {ϵ}_r=0$ or $\Delta {ϵ}_i=0$, i.e., the two EPs of the system. As illustrated by the Riemann surfaces in Figure 3E,F, these four points organise into two symmetry-related pairs: the transformation $(\delta ,\varphi )\to (-\delta ,\varphi +\pi )$ leaves $\Delta {ϵ}_r$ and $\Delta {ϵ}_i$ invariant, so that $(\delta ,\varphi )=(1,0)$ is equivalent to $(-1,\pi )$, and $(-1,0)$ is equivalent to $(1,\pi )$. Consequently, the parameter space hosts two physically distinct EPs that appear symmetrically about $\delta =0$.

2.2. Numerical Simulation

We have performed a full-wave modelling for the structure depicted in Figure 1 using COMSOL Multiphysics commercial software (version: Comsol 6.0) to verify the theoretical model. In the simulation, a Gaussian beam profile was implemented as the source with a waist radius covering the metagrating. The incident light was set at the visible wavelength of ${\lambda }_0=633$ nm, with the polarisation oriented along the x-direction, the scattering boundary condition at the top, and perfectly matched layers (PML) of $\lambda$ thickness at the left, right, and bottom, and the domain was meshed using the physics-controlled `extremely fine’ setting, with a maximum element size of $\lambda /60$ and $\lambda /30$ localised at the nanostrips and the grating Ag-Air interface, respectively. The metagrating period $\Lambda$ was set at 616 nm to meet the phase matching condition for SPP excitation as per the dispersion relation in Equation (4). After that, the metagrating nanostrips’ geometric parameters, their separation distance, and their position along the grating unit cell are carefully optimised to approximate the theoretical continuous $\mathcal{PT}$-symmetric modulated permittivity profile in Equation (2) to further control the coupling strengths of the excited SPPs to the left and right directions of the metagrating opting to maximise or minimise the ratio $\left|{\displaystyle \frac{\Delta {ϵ}_i}{\Delta {ϵ}_r}}\right|$, where the system EPs occur, and the SPP modes coalesce, as inferred from Equations (5) and (6).

In a unit cell of the metagrating, the nanostrips are made of silicon (Si), which has nearly real permittivity at the operating wavelength of 633 nm [68]; hence, it contributes to the real modulation of the permittivity. On the other hand, the second nanostrip is made of germanium (Ge), which has relatively high real and imaginary permittivity at the chosen wavelength [69] and contributes to both the real and imaginary modulation of the permittivity profile at the interface. The permittivity of the underlying Ag film was taken from Johnson and Christy experimental data [70]. Here, we note that while the inclusion of absorptive Ge nanostrips introduces ohmic losses, these are strictly localised to the metagrating region. Consequently, these losses primarily affect the coupling efficiency (the ratio of incident photon flux to launched SPP power) but do not degrade the propagation length or confinement of the excited SPP. Once launched, the SPP propagates along the bare Ag–Air interface, where its decay characteristics are governed solely by the intrinsic losses of the silver film at the operating wavelength.

3. Results and Discussion

The intensity of the transverse magnetic field $|H_y{|}^2$ of the SPP TM mode excited by a metagrating consisting of seven-unit cells in total with alternating Si-Ge nanostrips is plotted in Figure 4. At the optimised geometric parameters of $w_1=36$ nm and $h_1=30$ nm for the Si nanostrip, $w_2=64$ nm and $h_2=30$ nm for the Ge nanostrip and nanostrips, and a centre-to-centre separation distance $D=162$ nm, a unidirectional excitation of SPP can be observed along the $+x$ direction to the right of the metagrating as shown in Figure 4B. Hence, a separation distance equivalent to approximately a quarter grating period between the Si-Ge nanostrips in a unit cell of the metagrating at their optimised geometric parameters is required to tune the imaginary modulation of the metagrating $\delta$ to a value of 1 necessary to break the $\mathcal{PT}$-symmetry and drive the system into the first exceptional point at which a unidirectional excitation of SPP to the right direction occurs.

As shown in Figure 5, the field contrast of the left and right SPP modes at the optimised geometric parameters is controlled by adjusting the separation distance D between the nanostrips in the grating unit cell. This adjustment controls the strength of the imaginary modulation $\delta$ of the permittivity perturbation, thereby altering the ratio $\left|{\displaystyle \frac{\Delta {ϵ}_i}{\Delta {ϵ}_r}}\right|$ (refer to Equation (3)) and facilitating the control of SPP excitation (refer to Equation (5)). As illustrated in Figure 4A, at a separation distance of 100 nm, the field contrast $\left|{\displaystyle \frac{H_{+}}{H_{-}}}\right|$ was ≈4.11 dB. However, it increased dramatically to ≈20.9 dB at a separation distance of 162 nm (Figure 4B) corresponding to $\delta \approx 1$ and the first exceptional point of the non-Hermitian metagrating. Notably, Figure 5 demonstrates that a high field contrast (>10 dB) is maintained over a separation range of approximately $D=162\pm 10$ nm, indicating that the unidirectional excitation is robust against typical fabrication tolerances in electron beam lithography.

As the separation distance further increased beyond 162 nm, the imaginary modulation index $\delta$ became less than 1, which is physically translated to more light being coupled to the left SPP mode, but it still couples more efficiently to the right SPP mode, and the field contrast decreased accordingly.

Remarkably, the imaginary modulation $\delta$ reaches approximately 0, at which the field contrast is ≈0 dB when the separation distance reaches a value of $D=296$ nm. In this case, the effective dielectric constant $ϵ\left(x\right)$ is real, and the permittivity profile is effectively Hermitian in agreement with what is expected from scattering of light from subwavelength apertures [71] and as inferred from Equations (2) and (5) when $\Delta {ϵ}_i=\Delta {ϵ}_r$; this is evident by the symmetrically equal excitation of SPP to the left and right of the metagrating as seen in Figure 4D. Beyond this point, the light tends to couple more efficiently to the left SPP mode, and the field contrast ratio plummeted dramatically to ≈19.1 dB at $D=456$ nm, corresponding to $\delta =-1$ and the second exceptional point of the system, with SPP steering towards unidirectional excitation in the $-x$ direction as presented in Figure 4F. The previous results demonstrate that the permittivity profile of the Si-Ge metagrating conforms to a passive $\mathcal{PT}$-symmetric model, as delineated in Equation (1), that expressed the introduced permittivity perturbation at the interface as a periodic perturbing variation in the real and imaginary parts of the permittivity, expanded in terms of Fourier series of backward and forward Bloch modes. The induced phase shift $\varphi$ in the model can be adjusted while preserving the $\mathcal{PT}$-symmetry by altering the positioning of the nanostrips within the unit cells of the metagrating. The effect of swapping the positions of the Si and Ge nanostrips on the SPP excitation is elucidated in Figure 6, which shows that exchanging the Si and Ge positions in the unit cell yielded a complete reversal of the SPP pattern, confirming an induced phase shift of $\varphi =\pi$ in line with the mirror symmetry in the parameter space as seen in Figure 3.

Figure 5 elaborates the dependence of SPP excitation and the right and left SPP modes extinction ratio on the nanostrips’ separation distance, where, at EPs, a field amplitude extinction ratio $\approx 20$ dB equivalent to ≈40 dB optical power extinction ratio can be achieved in the visible frequencies, outperforming those achieved in conventional planar structures and apertures, as summarised in

Table 1. Comparison of unidirectional SPP excitation techniques in the visible/near-visible range.

Technique	Mechanism	Extinction Ratio (dB)	Footprint	Ref.
Si-Ge Metagrating	EP Coalescence	≈40 (Sim)	∼4.3 $\mu$m	This Work
Asymmetric Nanoslit	FP Cavity Interference	∼16 (Sim)	∼0.4 $\mu$m	[51]
Catenary Apertures	Geometric Phase	∼27 (Sim)	∼0.6 $\mu$m	[52]
Crossed Nanoantennas	Dipole Interference	∼17.6 (Sim)	∼1.0 $\mu$m	[57]
Plasmonic Apertures	Polarisation Interference	∼14 (Exp)	Multi-column	[60]
Metasurface	Phase Discontinuity	∼14 (Sim)	∼17 $\mu$m	[59]

.

While conventional couplers rely on precise interference effects accumulated over a specific interaction length—where the extinction ratio is essentially limited by phase matching constraints—the proposed EP-based device leverages a topological singularity. At the EP, the eigenstates coalesce, leading to the mathematical collapse of the backward mode. This renders the unidirectionality an intrinsic property of the system’s eigenstates rather than a result of distributed interference, fundamentally enabling superior extinction ratios.

4. Conclusions

In this study, we successfully demonstrated a robust unidirectional excitation of SPPs at EPs in the visible spectrum with an intensity extinction ratio of about 40 dB that outperforms those achieved in conventional planar structures and apertures. This was solely achieved by engineering the parameters of a non-metallic Si-Ge metagrating designed to approximate a passive $\mathcal{PT}$-symmetric optical potential, optimised to operate at EPs. These findings highlight the potential of non-Hermitian metagratings for advanced applications in optical integrated circuits, optical routing, and communication systems, where robust and flexible control of light at the nanoscale is crucial.