1. Introduction

Recently, the non-Hermitian features of optical systems have attracted significant attention [1–5]. One interesting feature is the emergence of exceptional points (EPs) corresponding to the degeneracies of a non-Hermitian Hamiltonian where the complex eigenvalues are coalesced and the eigenvectors become non-diagonalizable [6,7]. Many novel phenomena can happen in the vicinity of an EP, such as loss (pump) induced revival (depressing) of lasing [8,9], unidirectional invisibility [10,11], broadening of lasing/abosorption linewidth [12,13], enhancement of nonreciprocity [14,15], etc. Due to the Riemann-surface structure of the eigenvalues, interesting physics can emerge when encircling an EP [16,17]. In particular, dynamic effect associated with encircling EPs can give rise to chiral mode conversion [18,19], polarization state conversion [20], and topological energy transfer [21]. Compared to Hermitian degeneracies, non-Hermitian EPs manifest enhanced sensitivity to external small perturbations with a response proportional to $\epsilon ^{1/N}$ with $N$ being the order of the EPs, which have potential applications in optical sensing [22,23]. Therefore, significant efforts have been devoted to the realization of higher order exceptional points [24–30]. However, achieving higher order EPs usually requires tuning a higher dimensional parameter space, which inevitably leads to systems vulnerable to fabrication imperfections or other undesired perturbations. Interestingly, robust EPs can be achieved by employing unidirectional coupling of modes [26,31,32], where the EPs live on a surface (or hypersurface) of parameters and system imperfections only move the EP across the surface but do not destroy it. Such a surface is referred to as ES, and every point on the ES is an EP [31–33]. It emerges when an EP is protected by certain mechanism against variation of multiple system parameters. The order of the ES is equal to the order of the constituting EPs. Higher-order ESs can enable the realization of the above-mentioned intriguing properties within a large parameter space.

A simple mechanism to induce unidirectional coupling of resonance modes is the photonic spin-orbit interaction (SOI) associated with guided/surface waves [34–37], which has been applied to realize arbitrary order EPs/ESs in coupled spherical resonators [26,38]. However, these systems rely on either rotating excitation fields (i.e., circularly polarized fields) or rotating structures to break the symmetry. Here, we employ the unidirectional coupling induced by photonic SOI in coupled chiral particles sitting on a waveguide to realize an arbitrary order ES, under the excitation of a linearly polarized plane wave. Recently, an interesting mechanism to achieve light funneling has been demonstrated by employing the non-Hermitian skin effect in a setup involving time-dependent modulations [39]. We show that our system can also be applied to realize light funneling. Meanwhile, our proposed mechanism and system here are simpler and easier to implement experimentally. We demonstrate the unidirectional excitation of guided waves and the asymmetric response of the chiral particles due to the ES. In addition, we show that a mirror-symmetric configuration of the proposed structure can funnel light towards the center of the system where light energy can be harvested by a ring resonator.

The paper is organized as follows. In Section 2, we explain the physical mechanism underlying the realization of arbitrary order ES. We then demonstrate the second-order and fourth-order EP, and discuss the asymmetric response of the chiral particles and the unidirectional transport of guided light. In Section 3, we design a mirror-symmetric system that can serve as a light funnel, collecting light to the center of the system under different excitations. The conclusion is drawn in Section 4.

2. Arbitrary order exceptional surface

2.1 Unidirectional coupling induced by photonic spin-orbit interaction

We consider a silver helix particle (i.e., chiral particle) sitting on a silicon ($\varepsilon _r=12$) waveguide, as shown in Fig. 1(a). The helix has major radius $R=92$ nm and minor radius $r=11$ nm and pitch $p=50$ nm. The material properties of silver is characterized by the Drude model $\varepsilon _{\text {Ag}}=1-\omega _{\text {p}}^2/(\omega ^2+i\omega \gamma )$, where $\omega _{\text {p}}=1.37\times 10^{16}$ rad/s and $\gamma =2.73\times 10^{13}$ rad/s [40]. The waveguide has a cross section of $w \times h = 310$ nm $\times$ 640 nm. The distance between the center of the chiral particle and the upper surface of the waveguide is $g=126$ nm. The system is under the illumination of an electromagnetic plane wave with the electric field linearly polarized in $-yoz$-plane and the wavevector $\mathbf {k}$ forms an angle $\theta =20$ degrees with $-y$ direction, as shown in Fig. 1(a). In the considered frequency regime, the chiral particle is deep subwavelegnth and can be treated approximately as an electric dipole.

 

Fig. 1. (a) Chiral particle couples with a silicon waveguide. The helix has pitch $p=50$ nm, major radius $R=92$ nm, and minor radius $r=11$ nm. The incident wave is linearly polarized in the $-yoz$-plane and the wavevector $\mathbf {k}$ forms an angle of $\theta$ with the $-y$ direction. (b) Relative amplitude and phase of two dipole components $p_x$ and $p_z$. (c) Unidirectional excitation of guided wave due to spin-orbit interaction. (d) Unidirectionality as a function of frequency and damping of the chiral particle.

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We conducted full-wave numerical simulation of the system by using COMSOL Multiphysics [41] and calculated the electric dipole moments induced in the chiral particle. Figure 1(b) shows the relative amplitude and phase difference between the $p_x$ and $p_z$ components of the dipole. The electric dipole resonates at the frequency $f=101.5$ THz. We notice that $|p_x/p_z|=1.17$ and Arg$(p_z)-$Arg$(p_x)=90$ degrees are achieved at $f=102.3$ THz, approximately corresponding to a circularly polarized electric dipole, i.e., a chiral dipole carrying spin angular momentum. The coupling from the chiral dipole to the waveguide is dictated by the spin-orbit interaction (SOI) associated with the evanescent wave near the waveguide surface [42,43], where the propagation direction of the guided wave is locked to the transverse spin of the evanescent wave, a property known as spin-momentum locking [44]. In this case, the chiral dipole couples to the guided wave propagating to the $+x$ direction, as shown in Fig. 1(c). We define the unidirectionality as $\alpha =S(+x)/S(-x)$, where $S(+x)$ and $S(-x)$ denotes time-average power flow inside the waveguide along $+x$ and $-x$ directions, respectively. Figure 1(d) shows the unidirectionality $\alpha$ as a function of the excitation frequency and the loss $\gamma$ of the chiral particle. As seen, a high unidirectionality of $\alpha >50$ can be achieved at the resonance frequency for a range of $\gamma$ values, indicating the robustness of the unidirectional coupling against variation of loss. This robustness is essential to achieving the ES in the next section.

2.2 Arbitrary order exceptional surface in coupled chiral particles

We now consider two same chiral particles $p_1$ and $p_2$ sitting on the waveguide and separated by a distantance $d$, as shown in Fig. 2(a). Under the excitation of the same incident plane wave as in Fig. 1(a), the scattering fields of the chiral particles are both coupled to the guided wave propagating in $+x$ direction. This induces unidirectional coupling from $p_1$ to $p_2$, and the two chiral dipole modes can be described by the coupled mode equations:

In the above equations, $a_1(a_2)$ denotes the dipole mode amplitude of $p_1$ ($p_2$); $\Gamma =\gamma +\gamma _{\text {c}}$ is the total loss with $\gamma$ being the material loss and $\gamma _c$ being the radiation loss of the particles; $a_\text {in}$ denotes the excitation; $\kappa _{21}=-\text {i}\kappa _0 e^{i k_\text {wg} d}$ is the coefficient that characterizes the coupling from $p_1$ to $p_2$, where $\kappa _0$ is the positive real-valued coupling strength and $k_\text {wg}$ is the propagation constant of the guided wave; $\omega _0$ is the dipole resonance frequency of isolated chiral particle. The effective Hamiltonian $H$ has two degenerate eigenvalues $\omega _{1,2}=\omega _{0}-\frac {\mathrm {i}}{2}\Gamma$, corresponding to a second order EP. Importantly, the EP can be achieved for a range of loss value $\gamma$ (corresponding to the result in Fig. 1(d)) and for arbitrary coupling distance $d$, as the unidirectionality is protected by the SOI of the guided wave. Thus, the considered system gives rise to an ES defined on the parameter space of $(\gamma, d)$.

 

Fig. 2. (a) Two chiral particles couple via a silicon waveguide. The incident wave is linearly polarized in the $-yoz$-plane and the wavevector $\mathbf {k}$ forms an angle of $\theta$ with the $-y$ direction. The induced electric dipole amplitude of (b) the first and (c) the second chiral particle as a function of frequency and the coupling distance $d$. (d) Unidirectionality as a function of the frequency and coupling distance. (e) The magnetic field $|H_y|$ (denoted by the color and height) inside the waveguide for $d=2.1\lambda$ and $f=103.2$ THz, showing the power flows in $+x$ direction. The black lines denote the frame of the waveguide. The $p_1$ and $p_2$ label the two chiral particles.

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Substituting $a_{\text {in }}=A_{\text {in }} e^{-\mathrm {i} \omega t}$ and $a_i=A_i e^{-\mathrm {i} \omega t}$ into Eq. (1), we obtain the complex amplitudes of the chiral dipoles

The emergence of the ES is accompanied by a "chirality" of the system: the light energy tend to localize on one side of the system, analogous to the non-Hermitian skin effect [39,45,46]. Here, the "chirality" can be characterized by the unidirectionality $\alpha$ measuring the asymmetry of power output at the two ports of the waveguide. Figure 2(d) shows the value of $\alpha$ as a function of frequency and coupling distance $d$. Interestingly, $\alpha$ is larger than 50 for a wide range of $d$ value except near $d/\lambda =2.5$. This can be understood as follows. When $d$ equals half-integer wavelength, the second particle undergoes constructive interference, giving rise to amplification of $p_2$ and enhanced dipole radiation [26]. In contrast, the guided waves excited by the two particles undergo destructive interference, leading to suppression of the guided wave propagating in $+x$ direction. Therefore, the unidirectionality $\alpha$ is reduced at this coupling distance. The maximum unidirectionality is achieved at $d=n\lambda$ with $n$ being a positive integer. At this coupling distance, the chiral dipoles interference destructively, suppressing the radiation of the second particle. In contrast, the guided waves excited by the dipoles interfere constructively, giving rise to amplified field on the right side of $p_2$, as shown in Fig. 2(e). This indicates that the incident light can be coupled to the waveguide and redirected to the $+x$ direction. We note that the phenomenon here is reciprocal, and the unidirectional propagation of the guided light is attributed to the asymmetric excitation by the circularly polarized dipole of the helix particle.

The mechanism can be extended to the higher-order ES straightforwardly. Consider four chiral particles locating on the waveguide and are equally separated with a distance $d$, as shown in Fig. 3(a), where the particles are labeled by $p_1,p_2,p_3,p_4$. Under the excitation of the same linerly polarized plane wave, the coupling of the four chiral particles can be described by the following effective Hamiltonian

 

Fig. 3. (a) Four chiral particles (labeled as $p_1, p_2, p_3, p_4$) couple with each other via photonic spin-orbit interaction. The surface color and height show the magnetic field amplitude $|H_y|$ inside the waveguide for $d=3\lambda$ and $f=103$ THz. (b) The electric dipole amplitudes of the four chiral particles for $d=3\lambda$. (c) Unidirectionality as a function of frequency and coupling distance $d$.

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3. Light funneling in a mirror-symmetric system

The structure in Fig. 3(a) can collect and transport light energy to the right side of the waveguide. To realize a complete light funnel, we consider the design in Fig. 4(a), where eight chiral particles locate on the waveguide, among which four are left-handed and four are right-handed, forming a mirror-symmetric system. The separation of two sets of chiral particles is $D=8\lambda$. In addition, a silicon ring resonator with inner radius $L$ and outer radius $L+h$ locates below the waveguide. We set the radius $L=1800$ nm so that a whispering gallery mode appear at the resonance frequency of the dipoles. This system can enable light harvest in the ring resonator by the eight spin-orbit-coupled chiral particles. We conducted full-wave simulation of this system under three different excitations to verify its light funneling functionality.

 

Fig. 4. (a) A light funnel consists of four left-handed particles and four right-handed particles sitting on a silicon waveguide. The coupling distances are $d=2.8\lambda$ and $D=8\lambda$. A ring resonator locates right below the waveguide. (b) Magnetic field amplitude inside the waveguide when the right half of the system is excited by the incident light. (c) Magnetic field amplitude inside the waveguide when the left half of the system is excited by the incident light. (d) Magnetic field amplitude inside the waveguide when the whole system is excited.

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In Fig. 4(b), only the right half of the system with four left-handed chiral particles are excited by the incident wave. The color and height of the pattern denote the amplitude of the magnetic field in the center of the waveguide. We notice that light is collected and transport to the center of the system where the guided wave is further coupled to the ring resonator. The light is then damped in the ring resonator and can not couple back to the straight waveguide (in practical applications, the ring resonator can be replaced by other devices that consume the light energy). Figure 4(c) shows the field pattern when the left half of the system with four right-handed chiral particles are excited. As expected, light coupled via the chiral particles are right-transported to the center of the system, i.e., corresponding to a mirror-symmetric scenario of the case in Fig. 4(b). Figure 4(d) shows the field pattern when the full system is excited by the incident plane wave. Interestingly, light is funneled to the center from both sides of the system. The phenomena in Fig. 4(b)-(d) indicates that the proposed system can realize robust funneling of lights illuminating different parts of the system.

4. Conclusion

We propose a mechanism to achieve light funneling by employing spin-orbit-coupled chiral particles on an arbitrary order ES. We demonstrate the mechanism for the cases of a second order and fourth order ESs. On the ES, the chiral particles convert the incident light to guided light that propagate unidrectionally, and the direction depends on the spin of the chiral dipoles induced in the particles. When the separation of chiral particles equals integer multiples of the wavelength inside the waveguide, the guided waves undergo constructive interference, giving rise to the amplification of the guided wave in one direction. By employing a mirror-symmetric system consisting of both left-handed and right-handed chiral particles, we demonstrate the light funneling phenomenon robust against variation of excitation positions. While various two-dimensional metasurfaces have been proposed for light harvest/absorption in free space [47–49], the considered structure can serve as a one-dimensional metasurface suitable for on-chip harvest of light energy. It may also find applications in on-chip conversion of guided waves and in designing novel optical switches.