In contrast to closed Hermitian systems, open non-Hermitian systems can feature an interesting type of degeneracies where not only the eigenvalues or frequencies but simultaneously also the corresponding eigenstates or modes of the Hamiltonian coalesce [1–3]. This phenomenon typically occurs at specific points in the parameter space, which are therefore called exceptional points (EPs). In the past years, the research on EPs has been growing enormously especially but not exclusively in optics and photonics [4–7]. A plethora of interesting phenomena and applications arise around EPs, some of which are EP-based sensing [8–15], loss-induced revival of lasing [16], orbital angular momentum lasers [17], single-mode lasing [18], topological defect engineering [19], topologically curved microcavities [20], time-symmetry breaking in cavity quantum electrodynamics systems [21], electromagnetically induced transparency [22], synthetic dimensions with anti-PT symmetry [23], or optical amplifiers [24].

A key challenge is the generation of higher-order EPs where n>2 eigenstates and eigenvectors coalesce. Such EPs of order n (EPn) have a characteristic nth root topology of their complex eigenvalues in parameter space, which makes them exceedingly suited for sensing applications as a small perturbation generically results in a sizable response of the frequencies due to the steep slope of the nth root at the EP [25].

On the other hand, the extreme sensitivity makes the realization of a system at an EPn challenging and often includes a delicate fine tuning of parameters. For example, Hodaei et al. [11] realized an enhanced frequency splitting at an EP3 with three optical ring resonators with fine-tuned gain and loss. In Ref. [26], the authors proposed an optical microcavity with a precisely tuned boundary shape to generate an EP4. Other systems with high-order EPs suggested are optomechanical systems [27,28], resonators with spin–orbit interaction [29], parity–time symmetric cavities [11,30,31], waveguide (WG)-coupled refractive index tuned cavities [32], WG-coupled cavities with optical isolators [33], and recently coupled cavities with tuned asymmetric auxiliary WG elements [34].

A solution for the robust fabrication of an EP2 has been proposed in terms of exceptional surfaces [35–37], whereby fabrication tolerances represent non-generic perturbations that do not drive the system away from the EP whereas, e.g., a test particle (TP) does drive the system away from the EP and, therefore, leads to an EP-enhanced frequency splitting. Zhong et al. [35] proposed an exceptional surface in an elegant way by coupling a microring cavity to a WG with a mirror at the end. Therefore, clockwise (CW) and counterclockwise (CCW) propagating waves in the microring are coupled via asymmetric backscattering [38] such that an EP2 is generated. The existence of the EP2 is independent from fabrication parameters like the precise edge-to-edge distance between the WG and microring but shows an EP-enhanced sensing of a TP.

Furthermore, Wang et al. [32] showed that already two (three) microring cavities coupled by a single WG form two EP2s (EP3s). They additionally increase the order of the EP to four (six) via a sophisticated fine tuning of the refractive index profile of the individual microrings.

In this paper, we build on the approaches by Zhong et al. [35] and Wang et al. [32] and develop them further. In contrast to their works, we construct higher-order EPs with an easy-to-realize and robust fabrication scheme where a fine tuning of parameters is not necessary. Our proposed setup is supposed to work at room temperature and is scalable in size. In particular, we investigate two setups of N WG-coupled microring cavities that form an EP of order 2N by placing a mirror at one end of a WG. One of these setups with N=3 cavities and the corresponding eigenmodes of an EP6 is illustrated in the front of Fig. 1. The analog setup without a mirror is visualized in the background. In contrast to the proposed scheme, the mirrorless setup is at two EP3s with similar eigenmodes. We show that systems consisting of WG-coupled microring cavities with mirror-induced asymmetric backscattering enable an EP-enhanced particle sensing with an interesting and potentially useful cavity-selective scaling of the sensitivity.

Within coupled-mode theory, an effective Hamiltonian as a 2N×2N matrix is constructed, which allows for an intuitive understanding of the EP formation. In addition, the effective Hamiltonian description has an excellent agreement with the numerical finite-element method (FEM) simulations of the system including the complex eigenfrequencies and the reflection spectra of the system perturbed by a TP.

The paper is organized as follows. In Section 2, the system of two WG-coupled cavities is revised. In Section 3, the two schemes for the higher-order EP construction are discussed. A summary is given in Section 4.

First, some preliminary considerations using a matrix model are in order. A single microring cavity is assumed, which is coupled to a WG. This situation is very well described by an effective two-mode Hamiltonian H0 in the basis of CCW and CW propagating waves as H0=(ΩDDΩ),(1)where Ω is the complex frequency of the CCW and CW waves. The complex off-diagonal elements D reflect the small symmetric coupling between CW and CCW waves due to the weak backreflections at the WG. Adding a small TP to the cavity is typically described by a perturbation matrix. An approach for the design of such a perturbation matrix for an isolated microdisk perturbed by two TPs is given in Ref. [39]. Here, we interpret the WG already as one of the TPs leading to the matrix in Eq. (1). Thus, adding a TP to the WG-perturbed cavity adds a perturbation matrix of the following form: HTP=(V+U(V−U)e−i2mϕTP(V−U)ei2mϕTPV+U),(2)where m is the azimuthal mode number of the mode to be perturbed and ϕTP is the angle between the WG and the TP. The complex parameters V and U include the perturbation strength induced by the TP. Usually |U| is much smaller than |V|. For small perturbations, i.e., small radii rTP of the TP, the parameter U is negligible. Therefore, HTP(ϵ)≈ϵ(1e−i2mϕTPei2mϕTP1),(3)where the perturbation strength ϵ≡V scales with rTP2. This statement is in agreement with Zhong et al. [30] and further reviewed in Appendix A.

Hence, the eigenvalues Ωi of the perturbed system H(ϵ)=H0+HTP(ϵ) have a splitting, ΔΩ=|Ω1−Ω2|=2|D2+2Dϵcos(2mϕTP)+ϵ2|,(4)which increases linearly in |ϵ| for |D|≪|ϵ|. Note that this is the typical observation as D is often negligibly small.

The extension of the matrix model to a system with two identical microring cavities coupled via a WG is straightforward; see Fig. 2(a) and Ref. [32]. In the basis of CCW/CW waves of the left and CCW/CW waves of the right microring, the effective Hamiltonian reads H0=(ΩDA0DΩ0000ΩD0ADΩ).(5)Due to the spatial symmetry of the WG-microring system, the complex coupling parameter A is the same for CCW waves from the right to the left cavity and for CW waves from the left to the right cavity. It is assumed that the symmetric internal backscattering D at the WG is small compared to the directional coupling from one cavity to the other implying |A|≫|D| holds. In particular, this condition is fulfilled close to the critical coupling regime. For the idealized case D=0, the effective Hamiltonian [Eq. (5)] has two EP2s with eigenvalue Ω and associated eigenstates (1,0,0,±1)T. Thus, two microring cavities coupled via a WG realize in a good approximation an EP; see Ref. [32].

If a perturbation of a higher-order EPN is considered, the N frequencies Ωi diverge along the Riemann sheets of an Nth-order root. Therefore, the generalization of the splitting ΔΩ from Eq. (4) to more than two frequencies is not straightforward. In this paper, we use the convenient definition, ΔΩ=maxi,j|Ωi−Ωj|,(6)which is the maximal distance in the complex plane between two frequencies emerging from the EP for a given perturbation; see Fig. 3 for an illustration. The definition is consistent with Eq. (4) and captures the characteristic behavior of the frequencies around an EPN.

For the purpose of TP-sensing, it needs to be distinguished at which cavity a TP is placed. The perturbation matrix for a TP at the left or at the right cavity can be constructed via HTP(1)(ϵ)=HTP(ϵ)⊕02,2 or HTP(2)(ϵ)=02,2⊕HTP(ϵ), respectively. Thus, the effective Hamiltonian of the system with one TP at the left cavity reads H(1)(ϵ)=H0+HTP(1)(ϵ),(7)with H0 from Eq. (5). In contrast, placing two identical TPs at each cavity leads to the effective Hamiltonian, H(1+2)(ϵ)=H0+HTP(1)(ϵ)+HTP(2)(ϵ).(8)These two scenarios lead to a different behavior of the splitting ΔΩ as shown in Fig. 2. For this idealized case with D=0 [see Fig. 2(b)], the system shows the expected square-root scaling of the splitting ΔΩ at an EP2 if it is perturbed by two identical TPs, one at each cavity. A closer look at the analytic solution of the eigenvalue problem with the Hamiltonian H(1+2) for the exemplary parameters (D,m,ϕTP)=(0,20,π/2) reveals that for |ϵ/A|≪1 the eigenstates are ∼(1,0,0,±1)T with a dominant square-root scaling of the corresponding eigenvalues. If |ϵ/A|≪1 is not fulfilled, the linear scaling of the eigenvalues is dominant. The transition from a square-root to a linear scaling of the splitting is only present for symmetric perturbations of both cavities. Placing a single TP solely at the left cavity represents a non-generic perturbation [25] within which only a linear scaling of the frequency splitting can be observed for D=0 over the whole ϵ range. To understand this behavior, the solution of the eigenvalue problem with H(1) for parameters (D,m,ϕTP)=(0,20,π/2) in Eq. (7) is useful. It shows that, through the perturbation, an EP3 with eigenvalue Ω and eigenstate (0,0,0,1)T independent from ϵ is formed. This means that the perturbation forms an EP3 and moves the system along a corresponding exceptional surface [35,36] only changing the remaining eigenvalue Ω+2ϵ and eigenstate (2ϵ/A,2ϵ/A,0,1)T. Remarkably, if 2ϵ/A is chosen small but still finite to break the symmetry, the eigenstates approximately coalesce to ∼(0,0,0,1)T, which indicates that the system is close to an EP4 in parameter space.

For small but finite D≠0, interesting differences can be observed as shown in Fig. 2(c). One of the differences is the quartic-root scaling of the splitting ΔΩ by placing a TP at the left cavity. In order to get an understanding for the observation, the matrix related to the parameter D can be apprehended as another perturbation. The combination of the ϵ and D perturbation leads the system being near an EP4 even though the system itself without TP only supports two EP2s. The finite ϵ and D in parallel excite the quartic-root scaling the most for a parameter range around ϵ≈10−9,…,10−6. Note that this quartic-root, however, does not lead to an increased sensitivity as splitting for H(1) is smaller than the splitting for H(1+2); see Fig. 2(c).

As mentioned before, if 2ϵ/A≪1 is not fulfilled, the system is not close enough to an EP4 in the parameter space; therefore, the linear scaling of the splitting becomes dominant. For ϵ≪|D|, the TP-induced perturbation is negligible compared to the small symmetric coupling D. Therefore, the observed saturation in Fig. 2(c) for small perturbation strength ϵ occurs. It is remarkable that the saturation value in ΔΩ is orders of magnitudes larger than |D| itself, whereas in the single-cavity case it is of the order of |D|. Here, the analytic solution of the eigenvalue problem with H0 in Eq. (5) can provide an explanation: the eigenvalues are Ω±D2+DA and Ω±D2−DA. Thus, for small |D|≪|A|, the splitting ΔΩ is of the order |D|, i.e., the square root leads to the differences in magnitudes for the saturation values. For the case with two identical TPs on each cavity, the difference between Figs. 3(b) and 3(c) is the saturation effect for negligibly small TPs.

As will be discussed in the next section, already an infinitesimal small, asymmetric coupling between CCW and CW waves in the form of a single TP at one cavity could lead to the formation of a higher-order EP. However, the scaling of the frequency splitting ΔΩ of the system is strongly related to the perturbation. In the example above, it is illustrated how the interaction of two perturbations, namely ϵ and D, can lead to an interesting scaling behavior of ΔΩ.

Next, a numerical simulation to verify the matrix model of the WG-coupled cavities is in order. Therefore, the FEM solver COMSOL Multiphysics [40] is used to find solutions of the mode equation, (Δ+n2k2)ψ(x,y)=0,(9)for the quasi-two-dimensional geometry of the WG-coupled microrings; Ω=kR is the dimensionless complex frequency, k is the complex wavenumber, and n is the effective refractive index. In the simulation, we specify transverse magnetic polarization where ψ represents the z component of the electric field. The outer radius of the mircorings is set to R=10μm, and the width of the microrings is Rw=0.13R, the same as the width of the WG hwg. The edge-to-edge space between the microring and WG is s=0.133R, and the edge-to-edge distance between the two rings is d=1.6R. The refractive index of the microrings and WG is n=3.1, and in the surrounding n is set to unity. For this setup, four quasi-degenerate modes with Ω1–4≈8.3126−3.5×10−4i can be calculated numerically. The perturbation is simulated via TPs with variable radii rTP and fixed refractive index nTP=1.5. The TPs are placed with an angle ϕTP=0.4π and a fixed edge-to-edge distance dTP=0.01R to the microring. From now on, the perturbation strength is defined as ϵ≡(rTP/R)2. As shown in Fig. 4, the results from the FEM simulation are in very good agreement with the matrix model. In particular, the correct saturation for ϵ→0 is observed as well as the scaling in the intermediate regime with ϵ1/4 or ϵ1/2. For larger ϵ≳10−3, deviations from the matrix model arise. This is, however, expected as such a relatively large TP is no longer a localized perturbation to a microring, which is assumed in the perturbation matrix HTP in Eq. (2).

For the WG-coupled cavities without TP, one of the four mode patterns |ψ| is shown in Fig. 5. Note that the computational domain is adjusted to the geometry for all the simulated structures in the manuscript, which reduces the FEM grid by 10 to 25 percent compared to a comprising rectangular domain. In the mode pattern in Fig. 5, a finite D≠0 does not lead to a deviation from the idealized case D=0 that can be observed by eyes. Thus, all four mode patterns look almost identical and consist of a CCW wave in the left ring and a CW wave in the right ring, which is consistently described by the two eigenstates (1,0,0,±1) of the Hamiltonian H0 from Eq. (5) for D=0.

It should be noted that the mechanisms leading to a finite backscattering D can be diverse. One is the already mentioned backscattering at the WG. In a realistic experimental implementation, the backscattering might also arise from fabrication tolerances and sidewall roughness. But even in the full numerical simulations, a finite simulation domain, a finite FEM mesh, or even computations with machine precision could lead to a finite D [31]. The latter seems negligibly small as the machine precision is of the order 10−16. However, a naïve estimate shows that at an EP4 this could lead to an induced splitting, ΔΩ∼10−164=10−4.

Previous works in Ref. [35] have demonstrated that a single cavity can be tuned into an EP2 by coupling it to a WG with a mirror on one end as such a mirror leads to a fully asymmetric coupling between CW and CCW waves in the microring. In this section, we construct systems of N microring cavities that exhibit an EP of order 2N. In particular, we discuss two implementations: (i) N cavities that are all coupled to the same WG [32], which has a mirror at one side, and (ii) N cavities that are coupled via N+1 WGs [41], where only one WG has a mirror at the end. Both implementations have in common that the WG introduces a directional coupling between the cavities as, e.g., a CCW mode in a cavity only couples to the CW or the CCW mode of the next cavity but not to both CW and CCW. Then the mirror at the WG end introduces an asymmetric coupling between CCW and CW waves.

Note that a recent publication by Liu et al. [34] proposed a setup to implement a higher-order exceptional surface with microring cavities that, at first glance, looks similar to our proposed schemes. However, our schemes offer two main advantages. (i) The asymmetric backscattering is induced globally as it comes from a single mirror at one WG, whereas in Ref. [34] at each cavity an auxiliary WG element for the asymmetric backscattering is placed. (ii) In Ref. [34], the position of the auxiliary WG elements needs to be tuned for the backscattering to be fully asymmetric. In contrast, our setups are rather robust against the exact position of the mirror. Further note that the authors of Ref. [34] focused on the lasing properties of the system, whereas the goal of this section is the frequency splitting and characteristic behavior of the mode pattern induced by an external perturbation to the system.

The setup for N microring cavities horizontally coupled via a single WG with a mirror is shown in Fig. 6(a). For the simulation of the system, we use the same parameters as in Section 2. The mirror is realized with a slit of gold with refractive index n=0.5+10i. The slit has a width w=hwg/2 and is placed l=1.2R right to the center of the most right cavity and introduces a backscattering coefficient Rm. By simulating a TP at the most left cavity in the respective N-cavity setup, the characteristic 2Nth root scaling of the frequency splitting at an EP2N can be observed in Fig. 6.

Exemplary, the three-ring setup is considered. The effective Hamiltonian in the basis of CCW and CW waves in each microring is constructed as H0=(ΩDA000DΩ000000ΩDA00ADΩ000000ΩD+Rm000ADΩ),(10)where it is assumed that a cavity couples solely to its neighbor. The coupling to the next but one cavity can be neglected. This assumption is valid, e.g., under critical coupling. For the ideal case D=0, the effective Hamiltonian has one eigenvalue Ω and a single eigenvector (1,0,0,0,0,0)T indicating an EP6 with a pure CCW eigenstate in the leftmost microring. The role of the mirror-induced asymmetric backscattering can be seen even more obviously by reordering the traveling-wave basis such that the first three components are the CCW waves in cavities 1, 2, and 3, and the second three entries represent CW waves in cavities 3, 2, and 1; the cavities are counted from left to right. Then the effective Hamiltonian assuming D=0 reads H0=(ΩA00000ΩA00000ΩRm00000ΩA00000ΩA00000Ω),(11)which for any Rm≠0 represents an EP6. It also shows that the system without a mirror at the WG has two EP3 but is arbitrarily close to the EP6. The structure of H0 in Eq. (11) is similar to the structure of the Jordan normal form and appears also in the context of the Hatano–Nelson model of a cylindrical superconductor [42], where it can be understood as the nonperiodic, fully asymmetric limiting case [25].

The formation of the EP6 is confirmed by FEM simulations of the dielectric structure as shown in Fig. 7 where the mode pattern of one of the calculated modes is presented. A difference to the other five mode patterns cannot be observed by eyes (not shown) due to the extreme non-orthogonality.

For single-particle sensing, three cases can be distinguished. The TP can be placed from left to right at cavity 1, 2, or 3 as indicated by the symbols in Fig. 8(a). These three cases lead to a different perturbation matrix HTP(1)(ϵ)=HTP(ϵ)⊕04,4, HTP(2)(ϵ)=02,2⊕HTP(ϵ)⊕02,2, or HTP(3)(ϵ)=04,4⊕HTP(ϵ) [in the basis used in Eq. (10)]. In the parameter space, these three cases represent selected perturbation direction, which in the ideal case of D=0 results in a frequency splitting according to ∼ϵ1/6, ∼ϵ1/4, and ∼ϵ1/2, respectively. This cavity-selective scaling of the splitting is confirmed by FEM simulations of the dielectric structure as shown in Fig. 8(b).

The second setup to generate an EP of order 2N utilizes N vertically arranged cavities that are coupled via N+1 WGs; see Fig. 9(a). The most upper WG has a mirror at a specific end that induces an asymmetric backscattering between CW and CCW waves in the adjacent cavity. The most lower WG is placed to have the same WG-induced internal backscattering for each of the cavities. To verify the respective order of the EP, a TP at the most lower cavity is placed. As shown in Fig. 9(b) systematically, the expected scaling of the frequency splitting can be observed for a variation of the TP radius. In addition, a cavity-selective scaling of the frequency splitting is shown in Fig. 9(c) for a four-ring setup.

The effective Hamiltonian describing the four-ring setup in the basis of CCW and CW waves from the lowest to the upper cavity reads H0=(ΩD000000DΩA0000000ΩD0A00A0DΩ0000000AΩD000000DΩA0000000ΩD+Rm0000A0DΩ).(12)For D=0, the Hamiltonian is at an EP8 with eigenvalue Ω and eigenvector (0,1,0,0,0,0,0,0) that represents a pure CW wave in the bottom-most cavity. This can be seen well in the FEM simulations in Fig. 10. Progressing to a finite D≈10−10(−11.4803−4.3397i), Ω≈8.3132−6.9647×10−4i, and (A,Rm)≈10−4(−2.44547−6.6842i,−1.2194+6.2931i) (see Appendix B) allows us to capture the correct splitting including the cavity-selective sensing with ϵ1/8, ϵ1/6, ϵ1/4, or ϵ1/2 for a TP at cavity 1, 2, 3, or 4 (from bottom to top respectively); see Fig. 9(c).

Next, the behavior of the mode patterns for a TP perturbation at a given cavity is analyzed. Therefore, the intensity for each of the eight mode patterns g in the cavity f is calculated numerically by integrating the mode pattern. Additionally, performing an average over the eight modes gives the averaged intensity If in cavity f as If=18∑g=18∫cavity f|ψg(x,y)|2dxdy.(13)The intensities If are then normalized to give relative intensities, i.e., ∑f=14If=1. The behavior of these relative intensities is shown in Fig. 11 for a variation of the TP radius and a perturbation of each cavity. For very small TP radii, the intensities converge to the mode pattern at the EP8 (see Fig. 10), where (almost) all intensity accumulates in cavity 1. Increasing the TP radius leads to an exponential redistribution of the intensity. In cavities 2, 3, and 4, it exponentially increases with the perturbation strength while the relative intensity in cavity 1 decreases. The exponent of the exponential redistribution depends on the cavity where the TP is placed. Calculating the relative intensities from the effective Hamiltonian by analyzing its eigenvectors gives again an excellent agreement to the FEM simulations.

In practice, the position at which a TP interacts with a given cavity might not be controlled precisely. Therefore, Fig. 12 shows the eigenfrequency trajectories for a two-ring and four-ring setup if the angle ϕTP of the TP is varied. Since the azimuthal mode number is m=20, a variation of ϕTP from π/2 to π/2−π/m represents a nearly periodic perturbation leading to a characteristic cyclic rotation of the 2N eigenfrequencies in complex plane, which is referred to as chirality of an EP2N [43,44]. For larger TP radii, the eigenfrequency trajectories get deformed until eventually one or two frequencies split and form their individual cyclic behavior in the complex plane. Although such a behavior is a change in the topology of the eigenfrequency trajectories, it is very well described by the effective Hamiltonian (see Fig. 12). Consistently with the sensitivity at a higher-order EP, the separation of such an eigenfrequency trajectory happens for the four-ring setup at smaller TP radii as for the two-ring setup [cf. rTP=0.06R in Fig. 12(e) and rTP=0.12R in Fig. 12(j)].

Additionally, the effective Hamiltonian allows for a calculation of the reflection spectra if the system is excited with a (real-valued) frequency ω at a WG [26,45,46]. To do so, the effective Green’s function, G(ω)=(ω1−H)−1,(14)is calculated, where H=H0+HTP(1) is given as a 2N×2N matrix with H0 for the system of the N WG-coupled cavities [see Eq. (12)] and HTP(1) represents the perturbation by the TP at the bottom-most cavity (cf. Appendix B). Consequently, also the Green’s function is given as a matrix of the same dimensions as the effective Hamiltonian H. For the reflection spectra, we consider a monochromatic wave with frequency ω entering from the left side of the bottom-most WG. Such a wave couples to the adjacent first cavity exciting a CCW wave. In the traveling-wave basis, the excitation is, therefore, described by a vector with only one nonzero element in the first component. On the other hand, the light that leads to the reflected intensity in the most lower WG comes from the CW propagating wave in the first cavity, which is described by the second component of a vector in the traveling-wave basis. The matrix element G21 of the Green’s function connects the CCW waves to the CW waves in the most lower cavity. Thus, the reflection spectra are obtained as [26] R(ω)=|G21(ω)A|2.(15)In Fig. 13, it is demonstrated that the reflection spectra from Eq. (15) are in a very good agreement with the FEM simulations for a four-ring setup and a two-ring setup and different positions ϕTP of the TP.

Depending on the TP position ϕTP, individual peaks can be seen in the four-ring setup in Fig. 13(b) compared to a broad peak in the two-ring setup in Fig. 13(a). The reason is that the four-ring setup supports narrow peaks for the modes, whereas in the two-ring setup, the peaks associated with the modes are more broad and, therefore, overlap. Note that in order to separate the peaks of individual modes, it is also possible to add gain to the microrings. This typically results in more narrow peaks of the associated modes such that the modes can be distinguished in the spectrum.

However, narrowing of the peaks in the spectrum can already be seen in the passive systems without TP as shown in Fig. 14 (cf. [24,47]). The peaks in the reflection spectra are described by powers of a Lorentz curve L(ω)∼|ω−ΩEP|−2. Each additional cavity increases the order of the EP by 2, which gives an additional term of order 2N in the Green’s function at the EP2N [1,25,48], G(ω)=∑k=12NMk−1(ω−ΩEP)k,(16)where M0=1 and Mk=(H0−ΩEP1)k for k≥1. Thus, the reflection peaks in the N-ring setup are described by R(ω)∼[L(ω)]2N. Consequently, the four-ring setup shows more narrow peaks than the two-ring setup, which are then more likely to be separated rather than overlapping for a given perturbation. Note that the exponent 2N of the Lorentzian response is the same as the order of the EP if the effective excitation |p⟩ is generic, i.e., M2N−1|p⟩≠0 [25]. The line shape might differ for non-generic excitation and different output channels [41].

In this paper, we demonstrated two schemes for a handy and robust implementation of higher-order EPs with WG-coupled microring cavities. To do so, we synergized two approaches, namely the directional coupling of cavities via a WG [32] and the induced asymmetric backscattering from a mirror at the WG [35]. Thus, in a system of N cavities, an EP of order 2N is realized without complicated fine-tuned parameters. The order of the EP is verified by TP sensing. It is remarkable that the sensitivity is cavity-selective. Therefore, the induced frequency splitting not only depends on the perturbation strength, i.e., the size of the TP, but also on the cavity at which it interacts with the system. Conclusively, the WG-coupled microcavities are ideal systems to implement and study non-generic perturbations at higher-order EPs. An effective Hamiltonian based on coupled-mode theory is used to motivate the formation of the EPs as well as the sensing properties of the systems including the reflection spectra. Therefore, the effective Hamiltonian approach has been proven as an intuitive and insightful description of such systems at an EP of second or higher order.

A topic controversy discussed in recent literature is the signal-to-noise ratio of EP-based sensors. Some works indicate no enhancement of the signal-to-noise ratio at EPs [49–52], whereas others clearly show an enhancement [15,53]. Our proposed setup might be interesting for future studies on this topic due to its simplicity and versatility to perform measurements at different WG ports.

Furthermore, recent works such as Ref. [34] show an increasing interest for the use of EP and exceptional surface physics in photonic devices. Recently employed experimental platforms range from rather conventional InGaAsP single and multiple quantum wells [18,54,55] to erbium ions [10], polymers [56], and perovskites [34], underlining the timeliness and versatility of EP photonics.

Acknowledgment. The authors acknowledge fruitful discussions with R. El-Ganainy. SK is grateful for funding support from the Deutsche Forschungsgemeinschaft. We acknowledge support for the Book Processing Charge by the Open Access Publication Fund of Magdeburg University.