Hybridization of different types of exceptional points

A large number of different types of second-order non-Hermitian degeneracies called exceptional points (EPs) were found in various physical systems depending on the mechanism of coupling between eigenstates. We show that these EPs can be hybridized to form higher-order EPs, which preserve the original properties of the initial EPs before hybridization. For a demonstration, we hybridize chiral and supermode second-order EPs, where the former and the latter are the results of intra-disk and inter-disk mode coupling in an optical system comprised of two Mie-scale microdisks and one Rayleigh-scale scatterer. The high sensitivity of the resulting third-order EP against external perturbations in our feasible system is emphasized.

1 INTRODUCTION

Eigenvalues (wavenumber) in open quantum (wave) systems described in terms of non-Hermitian Hamiltonian are complex, and their corresponding eigenvectors (modes) are not completely orthogonal. One fascinating issue arising in an open system is the unique point of non-Hermitian degeneracy, the so-called “exceptional point” (EP), where the eigenvalues as well as the corresponding eigenvectors coalesce simultaneously [1 3" target="_self" style="display: inline;">–3]. So far, much effort has been devoted to figuring out this issue, and EPs have been found in such extensive systems as atomic systems [4], lasers [5,6], microwave and optical systems [7 –10" target="_self" style="display: inline;">–10], acoustics [11,12], and electronic circuits [13,14].

In an earlier stage of studying EPs, the focus was on a two-level system $(N = 2)$ , which helps to understand the physical mechanism of a second-order EP (EP2). Several types of EP2s have been found in various systems, depending on their intrinsically different mechanism of mode coupling: a parity-time symmetry [15 17" target="_self" style="display: inline;">–17], asymmetric back scattering [18 –21" target="_self" style="display: inline;">–21], supermode coupling [22 –26" target="_self" style="display: inline;">–26], an internal-external mode pair [27], a concentric layered microdisk [28], and resonance-assisted tunneling [29].

After the reports of applicability of EPs, e.g., sensors [30] for nanoparticle detection [31 –35" target="_self" style="display: inline;">–35] or gyroscopes [36 –39" target="_self" style="display: inline;">–39], a growing interest has been concentrated on higher-order EPs $(N \geq 3)$ . This is because, as splitting of eigenvalues $Δ λ$ against a small perturbation $(ϵ ≪ 1)$ is proportional to $ϵ^{1 / N}$ in the vicinity of EPs, sensitivity can be enhanced by as much as the increased order $N$ . Hence, higher-order EPs have become a key agenda today. Considerable attempts have been made theoretically to obtain higher-order EPs for several systems [28,40 –50" target="_self" style="display: inline;">–50]. However, because of their fragile configurations, only a restricted number of experiments to obtain higher-order EPs were successful [45,49,50].

Each of those higher-order EPs mentioned above uses its own single coupling mechanism, and yet, no attempt has been made to hybridize two different types of EP2s for a higher-order EP. In this paper, we show that different types of EP2s can be hybridized for a higher-order EP. As a demonstration, we hybridize a chiral EP and a supermode EP, where the former and the latter are the results of intra-disk and inter-disk mode coupling in an optical microdisk system [51] comprised of two Mie-scale microdisks and one Rayleigh-scale scatterer as shown in Fig. 1.

Fig. 1. System configuration comprised of two microdisks and one scatterer. The scatterer is magnified in the dotted box.

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When an even and an odd parity mode in a single microdisk coalesce due to asymmetric backscattering, we call this type of EP2 a “chiral EP” [18 –21" target="_self" style="display: inline;">–21,36,52]. On the other hand, optical modes in separate microdisks can be coupled to form a “supermodes” pair. When they coalesce [22], we call this type of EP2 a “supermode EP.” These intra- and inter-disk mode couplings can contribute simultaneously in our system to generation of a chiral EP and a supermode EP, respectively, as well as their hybridization for third-order EPs (EP3s). The setup in Fig. 1 is experimentally feasible since the system parameters are easily controllable as in Refs. [35,52].

2 NON-HYBRIDIZED EXCEPTIONAL POINTS

Optical modes in our system are obtained by solving Maxwell’s equations reduced to a two-dimensional wave equation $- \nabla^{2} ψ = n^{2} (\vec{r}) k^{2} ψ$ , where $ψ$ , $n \in R$ , and $k \in C$ are the $z$ component of electric field $E_{z}$ (i.e., TM polarization), the refractive index, and the vacuum wavenumber, respectively. We set $n = 2$ inside of the microdisks and the scatterer and $n = 1$ outside of them. In numerical computations, we use the boundary element method [53], and the modes obtained are expressed in terms of dimensionless wavenumbers $k R$ .

Figure 2 shows the $Re (k R)$ of modes as a function of $r_{1} / R$ in coupled microdisks without a scatterer for $d_{1} / R \approx 0.5121$ . The horizontal and oblique curves represent modes localized on the left and the right microdisk, respectively. The highlighted thick curves originate from the $(l, m) = (1, 6)$ and the (1, 5) mode, which are to be used as four basis modes in the following discussions. Here $l$ and $m$ are the radial and the azimuthal mode number, respectively, in each microdisk. The four basis modes are defined as follows: inset I ( $ϕ_{1}$ ) is the “left-even” mode, II ( $ϕ_{2}$ ) the “left-odd”, III ( $ϕ_{3}$ ) the “right-even”, and IV ( $ϕ_{4}$ ) the “right-odd.” Here, “right” and “left” indicate the location of modes on the microdisks, and “even” and “odd” indicate the parity with respect to the axis of mirror-reflection symmetry ( $x$ axis). We focus on the region marked by the blue dashed circle, where the four modes interact with each other. The pairs $ϕ_{2, 4}$ and $ϕ_{1, 3}$ are in the weak and the strong coupling, respectively [29]. This is because the real and the imaginary parts (not shown) of the wavenumbers of the pair $ϕ_{2, 4}$ cross and repulse, respectively, while those of the pair $ϕ_{1, 3}$ exhibit an avoided crossing and a crossing, respectively. Note that the even (solid curves) and odd (dashed curves) parity modes in Fig. 2 are orthogonal because of the mirror-reflection symmetry of the system, even though the system is non-Hermitian. It is known that the orthogonality of eigenvectors in a non-Hermitian system can be characterized by the definition of the phase rigidity [54,55].

Re(kR) of supermodes as a function of r1/R and their basis modes in coupled microdisks without a scatterer. The solid and the dashed curves are the even and odd parity modes, respectively. In the region marked by a blue dashed circle, the four modes originating from (l,m)=(1,6) and (1, 5) are highlighted by the thick solid and dotted curves. The insets I, II, III, and IV are the four basis modes defined as ϕ1, ϕ2, ϕ3, and ϕ4, respectively.

Fig. 2. $Re (k R)$ of supermodes as a function of $r_{1} / R$ and their basis modes in coupled microdisks without a scatterer. The solid and the dashed curves are the even and odd parity modes, respectively. In the region marked by a blue dashed circle, the four modes originating from $(l, m) = (1, 6)$ and (1, 5) are highlighted by the thick solid and dotted curves. The insets I, II, III, and IV are the four basis modes defined as $ϕ_{1}$ , $ϕ_{2}$ , $ϕ_{3}$ , and $ϕ_{4}$ , respectively.

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In the marked region, we add a scatterer and observe the wavenumber variations of the four modes depending on $θ$ at $r_{1} / R \approx 0.8587$ , $d_{1} / R \approx 0.5121$ , $r_{2} / R \approx 0.1140$ , and $d_{2} / R \approx 0.0065$ . Figures 3(a) and 3(b) show $Re (k R)$ and $Im (k R)$ , respectively. By introducing the scatterer, the even and odd parity modes are now highly nonorthogonal because of the asymmetric backscattering induced by the scatterer. This additional non-Hermiticity, caused by the scatterer, can induce couplings among all the basis modes, I, II, III, and IV, in Fig. 2, no matter what their parity symmetries were before. Therefore, the wave functions corresponding to the curves in Figs. 3(a) and 3(b) exhibit the superpositions of the basis modes. To quantify how close the modes are to the EPs, we obtain wavenumber differences $| Δ k R |$ for all combinations of the four modes as shown in Fig. 3(c). The figure exhibits three sharp local minima at (A) $θ \approx 0.669$ , (B) 0.909, and (C) 1.969. According to the locations and parities of the modes, we can infer that A and B are the proximity of a chiral EP and a supermode EP because the modes there originate from $ϕ_{3, 4}$ and $ϕ_{2, 4}$ , respectively. C is an EP3 of three coalescent modes originating from $ϕ_{2, 3, 4}$ . In this case, since the $ϕ_{3, 4}$ combination implies a chiral EP and the $ϕ_{2, 4}$ one a supermode EP, this EP3 is a hybridization of the two different types of EP2. Hence, accessing points A, B, and C, we can switch the three different EPs by controlling the scatterer in our system.

(a) and (b) are the θ-dependent Re(kR) and Im(kR) of the four modes originating from the modes in the blue dashed circle in Fig. 2, respectively; (c) shows the absolute values of wavenumber differences for all combinations of the four modes. The black-solid curve in (c) is for the black-solid and black-dashed curves in (a) and (b). The orange-dashed curve in (c) is for the black-solid and orange-solid curves in (a) and (b). The blue-dotted curve in (c) is for the orange-solid and black-dashed curves in (a) and (b). Vertical lines at A, B, and C mark the local minima of these three curves in (c).

Fig. 3. (a) and (b) are the $θ$ -dependent $Re (k R)$ and $Im (k R)$ of the four modes originating from the modes in the blue dashed circle in Fig. 2, respectively; (c) shows the absolute values of wavenumber differences for all combinations of the four modes. The black-solid curve in (c) is for the black-solid and black-dashed curves in (a) and (b). The orange-dashed curve in (c) is for the black-solid and orange-solid curves in (a) and (b). The blue-dotted curve in (c) is for the orange-solid and black-dashed curves in (a) and (b). Vertical lines at A, B, and C mark the local minima of these three curves in (c).

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We first analyze the supermode EP at B. By means of “downhill simplex method” [56], we obtain more accurate parameters of the EP2. Figures 4(a) and 4(b) show Riemann surfaces of $Re (k R)$ and $Im (k R)$ around the supermode EP at $θ \approx 0.9115$ and $r_{2} / R \approx 0.1163$ . The coalescent modes shown in Fig. 4(c) are not completely symmetric to the $x$ axis, unlike the ones in Ref. [22], which are obtained by controlling the size of one microdisk and the distance between two microdisks. In our system, just by controlling the size and position of the scatterer, we can achieve a supermode EP. Note that the other modes in Figs. 4(d) and 4(e) are not symmetric as well.

(a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the supermode EP kEPR≈4.219−i0.0430. The dashed curves are the branch cuts, and the large dots are the EPs. Here Δr2=r2−rEP and Δθ=θ−θEP. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

Fig. 4. (a) and (b) Riemann surfaces of $Re (k R)$ and $Im (k R)$ , respectively, around the supermode EP $k_{EP} R \approx 4.219 - i 0.0430$ . The dashed curves are the branch cuts, and the large dots are the EPs. Here $Δ r_{2} = r_{2} - r_{EP}$ and $Δ θ = θ - θ_{EP}$ . (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

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In the case of the chiral EP at A, Riemann surfaces of $Re (k R)$ and $Im (k R)$ around the EP at $θ \approx 0.6654$ and $r_{2} / R \approx 0.1094$ are shown in Figs. 5(a) and 5(b). The coalescent mode at this EP is given in Fig. 5(c), and the other two are in Figs. 5(d) and 5(e). Although we tune the scatterer belonging to the left microdisk, the chiral EP mode in the right one becomes strongly chiral [i.e., traveling wave; see Fig. 5(c)]. This implies that we can remotely control the traveling waves in the target microdisk (i.e., the right one) without a direct perturbation.

(a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the chiral EP kEPR≈4.232−i0.0438. The dashed curves, the large dots, Δr2, and Δθ are the same as in Fig. 4. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

Fig. 5. (a) and (b) Riemann surfaces of $Re (k R)$ and $Im (k R)$ , respectively, around the chiral EP $k_{EP} R \approx 4.232 - i 0.0438$ . The dashed curves, the large dots, $Δ r_{2}$ , and $Δ θ$ are the same as in Fig. 4. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

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The chirality of modes can be quantified by employing the Husimi function [57] of $H (\vec{γ})$ where $\vec{γ} \equiv (q, p)$ , which encodes a probability of propagation directions in the phase space as $p > 0$ for the counter-clockwise direction and $p < 0$ for the clockwise direction, respectively. Here, $q$ and $χ$ represent the position of the impact point and the incident angle of the ray with respect to the outward normal direction on the microdisk boundary. The chirality is given as $α \equiv 1 - \frac{\min {\int_{p < 0} H (\vec{γ}) d q d p, \int_{p > 0} H (\vec{γ}) d q d p}}{\max {\int_{p < 0} H (\vec{γ}) d q d p, \int_{p > 0} H (\vec{γ}) d q d p}} .$ (1)The chirality of the modes in the left and right microdisk in Fig. 5(c) is $(α_{L}, α_{R}) \approx (0.906, 0.977)$ . These values that are slightly less than 1 are counterintuitive when we consider the previous works on a chiral EP $(α = 1)$ in a different system [58]. Note that the chiralities of all the remaining modes in Figs. 5(d) and 5(e) as well as of the supermode EPs in Figs. 4(c)–4(e) are found to have non-zero values less than 1. It is also noted that the mode properties of our supermode EP2s and chiral EP2s are not the same as the ones found in typical systems consisting of only two microdisks or a single microdisk with two scatterers.

3 HYBRIDIZATION OF EXCEPTIONAL POINTS

Eventually, we turn our attention to the EP3 at C in Fig. 3. After carrying out tedious but definite procedures of numerical scanning in a multi-valued parameter space, EP3s are obtained around $θ \approx 1.969$ and $r_{2} / R \approx 0.1140$ . Riemann surfaces for $Re (k R)$ and $Im (k R)$ , shown in Figs. 6(a) and 6(b), clearly exhibit the typical structure of an EP3 (see, e.g., Ref. [59]). We find that three nearly identical $k R$ of the coalescent modes of $Ψ_{1}$ , $Ψ_{2}$ , and $Ψ_{3}$ are $k_{1} R \approx 4.2245 - i 0.0478$ , $k_{2} R \approx 4.2244 - i 0.0478$ , and $k_{3} R \approx 4.2243 - i 0.0478$ , respectively. The wavenumber of the remaining one, $Ψ_{4}$ , is $k_{4} R \approx 4.1726 - i 0.0418$ . The chirality of EP3 modes in each microdisk is $(α_{L}, α_{R}) \approx (0.79, 0.96)$ as shown in Fig. 7(a), and that of the remaining mode is (0.35,0.10). Again, we can see that the chirality is neither 1 nor 0, in spite of the fact that the chirality of the mode in the left microdisk should be $α_{L} = 0$ if our EP3 is a pure combination of three basis modes excluding $ϕ_{1}$ .

Riemann surfaces and modes of the EP3 by hybridizing a supermode and a chiral EP; (a) and (b) are Re(kR) and Im(kR), respectively. The dashed curves are the branch cuts and the large dots are the EPs. Here Δr2=r2−rEP and Δθ=θ−θEP. (c) and (d) The coalescent and the remaining modes at this EP parameter.

Fig. 6. Riemann surfaces and modes of the EP3 by hybridizing a supermode and a chiral EP; (a) and (b) are $Re (k R)$ and $Im (k R)$ , respectively. The dashed curves are the branch cuts and the large dots are the EPs. Here $Δ r_{2} = r_{2} - r_{EP}$ and $Δ θ = θ - θ_{EP}$ . (c) and (d) The coalescent and the remaining modes at this EP parameter.

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(a) Accumulated Husimi function of the coalescent mode as a function of p. The black-solid and the red-dashed curves are for the left and right microdisk, respectively. (b) Overlaps of all combinations of the four modes at EP3. The subscripts 1, 2, and 3 are the modes forming EP3, and the subscript 4 is the remaining mode.

Fig. 7. (a) Accumulated Husimi function of the coalescent mode as a function of $p$ . The black-solid and the red-dashed curves are for the left and right microdisk, respectively. (b) Overlaps of all combinations of the four modes at EP3. The subscripts 1, 2, and 3 are the modes forming EP3, and the subscript 4 is the remaining mode.

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Though all three coalescent modes at the EP3 seem to be identical like the modes in Fig. 6(c), it is necessary to verify whether they are really identical or not. To this end, as pointed out in Ref. [60], it is useful to compute the overlaps of the modes at the EPs by using the following equation: $S_{μ ν} \equiv \frac{| \int d x d y Ψ_{μ}^{*} Ψ_{ν} |}{\sqrt{\int d x d y Ψ_{μ}^{*} Ψ_{μ}} \sqrt{\int d x d y Ψ_{ν}^{*} Ψ_{ν}}},$ (2)where $μ$ and $ν$ imply two different modes. The overlaps of all the combinations of the three coalescent modes at the EP3 are above 0.9999 as shown by the red symbols (square, circle, and downward triangle) in Fig. 7(b), while the overlaps between the three coalescent modes $Ψ_{μ}$ ( $μ = 1, 2, 3$ ) and $Ψ_{4}$ are about 0.065 as shown by the black symbols (upward triangle, diamond, and star) in Fig. 7(b). The overlap of 0.065 implies that $Ψ_{1}$ and $Ψ_{2, 3, 4}$ are not orthogonal and that all four basis modes contribute to the EP3.

To estimate the contribution of the four basis modes to the EP3, we obtain overlaps $c_{μ ν}$ between $Ψ_{μ}$ and the four basis modes $ϕ_{ν}$ by using the definition $c_{μ ν} \equiv ⟨ ϕ_{ν}^{*} | Ψ_{μ} ⟩$ [40,55,61,62]. Then the ratio of the overlaps among the four basis modes are $| c_{11} | ∶ | c_{12} | ∶ | c_{13} | ∶ | c_{14} | = 0.525 ∶ 0.829 ∶ 0.881 ∶ 1.000$ . Although the EP3 is a coalescence of three modes originating from $ϕ_{2, 3, 4}$ , the non-negligible contribution of $ϕ_{1}$ is observed as the value of $| c_{11} |$ . Hence, we describe our EP3 by using a $4 \times 4$ Hamiltonian as follows: $H = (\begin{matrix} \begin{matrix} k_{L, E} R & a \\ a & k_{L, O} R \end{matrix} & \begin{matrix} b & ϵ_{1} \\ ϵ_{2} & c \end{matrix} \\ \begin{matrix} b & ϵ_{2} \\ ϵ_{1} & c \end{matrix} & \begin{matrix} k_{R, E} R & δ \\ δ & k_{R, O} R \end{matrix} \end{matrix}),$ (3)where the subscripts $L$ $(R)$ and $E$ $(O)$ denote the left (right) microdisk and the even (odd) parity, respectively. In the Hamiltonian, $a$ is the intra-disk coupling of $ϕ_{1, 2}$ in the left microdisk due to a scatterer, and $b$ $(c)$ is the inter-disk coupling of even (odd) parity modes between the left and the right microdisks. The left-upper and the right-lower dashed boxes represent the left and right microdisk, respectively. For the Hamiltonian, we conjecture: (i) all off-diagonal elements for a supermode (not EP) without a scatterer are 0, except $b$ and $c$ , because a supermode can be formed by the same parity modes; and (ii) when a scatterer is added, $ε_{1, 2}$ and $δ$ are smaller than $a$ , $b$ , and $c$ , but are not exactly zero because the scatterer globally perturbs all basis modes in the formation of an EP3.

Now, we numerically obtain all elements by the expansion $Ψ_{μ} = \sum_{ν = 1}^{4} c_{μ ν} ϕ_{ν}$ and the definition $| Ψ_{μ} ⟩ \equiv {(c_{μ 1}, c_{μ 2}, c_{μ 3}, c_{μ 4})}^{T}$ . For $μ = 1$ , 2, 3, and 4, we can identify all elements in Eq. (3) by setting 16 equations of $(H - k_{μ} R I) | Ψ_{μ} ⟩ = 0$ , where $I$ is the unit matrix. Conjecture (i) is confirmed, as a supermode (not EP) has $b \approx 0.033 - i 0.002$ and $c \approx 0.006 + i 0.009$ , while all the other off-diagonal elements are zero. In the case of a heterogeneous EP3, the Hamiltonian is obtained as follows: $H \approx (\begin{matrix} \begin{matrix} 4.199 - 0.041 i & \begin{array}{l} 0.013 + 0.007 i \end{array} \\ \begin{array}{l} 0.012 + 0.008 i \end{array} & \begin{array}{l} 4.212 - 0.036 i \end{array} \end{matrix} & \begin{matrix} \begin{array}{l} 0.031 - 0.006 i \end{array} & \begin{array}{l} - 0.002 - 0.003 i \end{array} \\ \begin{array}{l} 0.003 + 0.001 i \end{array} & \begin{array}{l} - 0.006 - 0.007 i \end{array} \end{matrix} \\ \begin{matrix} \begin{array}{l} 0.028 - 0.005 i \end{array} & \begin{array}{l} 0.003 + 0.001 i \end{array} \\ \begin{array}{l} - 0.002 - 0.002 i \end{array} & \begin{array}{l} - 0.007 - 0.008 i \end{array} \end{matrix} & \begin{matrix} \begin{array}{l} 4.209 - 0.052 i \end{array} & \begin{array}{l} 0.001 + 0.001 i \end{array} \\ \begin{array}{l} 0.001 + 0.001 i \end{array} & \begin{array}{l} 4.226 - 0.056 i \end{array} \end{matrix} \end{matrix}) .$ (4)We can see that the elements $a$ , $b$ , and $c$ are much bigger than non-zero $ϵ_{1, 2}$ and $δ$ as are conjectured in (ii). Moreover, it turns out that $c$ in the EP3 is similar to the one in the supermode. Based on this observation, we can conclude that a contribution of the remaining mode $ϕ_{1}$ cannot be neglected in our EP3 described in a $4 \times 4$ Hamiltonian.

Now, we test the sensitivity of our EP3 against external perturbations in comparison with that of a chiral EP and a supermode EP. We exemplify a deviation $| Δ k R |$ as a function of refractive index variation $Δ n$ outside of the microdisks. This $Δ n$ could be interpreted as the impurity concentration in a solvent, like $D_{2} O$ in $H_{2} O$ [63]. In Fig. 8, we can see that the slopes of the chiral and the supermode EPs are ruled by $\sim {(Δ n)}^{1 / 2}$ , whereas our EP3 exhibits $\sim {(Δ n)}^{1 / 3}$ . This is further evidence of an EP3. This impressive sensitivity was already predicted in previous literature, e.g., in Refs. [28,49]. To emphasize our point, our proposed system is obviously feasible since we can take advantage of the system implemented in Ref. [35].

$Wavenumber splittings depending on the outside refractive index of the microdisk in double logarithmic scale. Thick magenta, blue, and green curves correspond to an EP3, a supermode EP, and a chiral EP, respectively. Thin black-solid and black-dashed curves are the fittings of |ΔkR|∼(Δn)1/3 and ∼(Δn)1/2, respectively.$

Fig. 8. Wavenumber splittings depending on the outside refractive index of the microdisk in double logarithmic scale. Thick magenta, blue, and green curves correspond to an EP3, a supermode EP, and a chiral EP, respectively. Thin black-solid and black-dashed curves are the fittings of $| Δ k R | \sim {(Δ n)}^{1 / 3}$ and $\sim {(Δ n)}^{1 / 2}$ , respectively.

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4 CONCLUSION

To summarize, we have demonstrated hybridization of two different types of EP2. This hybridization results in a third-order EP in a system comprised of two microdisks and a scatterer. The high-order sensitivity against an external perturbation can function to resolve the imperative issue concerning real applications of EP3s to sensors. We believe that the unveiled hybridization mechanism can shed light on both improving the performance of applications and advancing the physics related to EPs. Most importantly, since this hybridization mechanism can extend to various cases of EP2s in extensive physical systems, we believe that our findings will initiate new directions of studying EPs.

5 Acknowledgment

Acknowledgment. This research was supported by Ministry of Health and Welfare, Republic of Korea (Government-wide R&D Fund project for infectious disease research, HG18C0069).

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1 INTRODUCTION

2 NON-HYBRIDIZED EXCEPTIONAL POINTS

3 HYBRIDIZATION OF EXCEPTIONAL POINTS

4 CONCLUSION

5 Acknowledgment

Jinhyeok Ryu, Sunjae Gwak, Jaewon Kim, Hyeon-Hye Yu, Ji-Hwan Kim, Ji-Won Lee, Chang-Hwan Yi, Chil-Min Kim. Hybridization of different types of exceptional points[J]. Photonics Research, 2019, 7(12): 12001473.