Exceptional points (EPs), representing unique non-Hermitian degeneracies in open systems, have been the focus of significant research due to their remarkable phenomena and applications across various fields. While second-order EPs are extensively studied, higher-order EPs—requiring the degeneracy of multiple eigenvalues and corresponding eigenvectors—remain less explored. In this work, we propose a generic approach for realizing, 2Nth-order EP in an N-element photonic molecule (PM). Differing from a common PM consisting of identical microresonators, here a spiral-shaped microresonator is incorporated to form an asymmetric PM, offering unidirectional coupling between eigenmodes and transferring the chirality throughout the whole PM. Moreover, the hierarchical construction of 2Nth-order EPs can be achieved in gain–loss asymmetric PMs, as well as in all-passive ones. In the prototype demonstration based on numerical simulations, the obtained fourth-order EP reveals its superior sensitivity with a three-fold enhancement compared with that of a second-order EP to external perturbations at the single-molecule level. We envisage that the proposed asymmetric PM, circumventing the need for external components or precise control of gain–loss profiles, will pave the way for constructing higher-order EPs in integrated photonic devices and enabling applications in both linear and nonlinear regimes.

Topics
Numerical methods, Finite-element analysis, Molecular sensors, Optical microresonators, Microoptics, Nanophotonics, Chirality, Open quantum systems

In nature, open systems exhibit complex eigenenergies, rendering their Hamiltonians non-Hermitian.1,2 Exceptional points (EPs) are unique degeneracies in the parameter space of non-Hermitian systems, where at least two eigenvalues and their corresponding eigenvectors coalesce.3 In optics, whispering-gallery-mode (WGM) microresonators such as microdisks, microtoroids, and microrings have emerged as an ideal platform among a plethora of photonic and plasmonic structures for exploring EPs, owing to their high Q-factors as well as tunable gain and dissipation characteristics.4,5 The distinctive behavior of EPs constructed in WGM microresonators has led to significant advancements in integrated photonics, such as control of light emission properties,6–10 nonreciprocal transmission,11 coherent perfect absorption,12,13 chiral light generation2 and ultra-sensitive detection.14–18 More recently, attention has been extended toward the construction of higher-order EPs involving the coalescence of three or more eigenvalues and their corresponding eigenvectors.19 In this context, the system’s response scales as ϵ1/N upon external perturbation with strength ϵ.16,20 Realizing higher-order EPs becomes highly desirable for unleashing the potential in sensing toward single-molecule levels,14,21–23 optical gyroscopes,19,24,25 wireless energy transfer,26 and enhanced light–matter interactions.27,28

For the realization of higher-order EPs, typical N (N > 2) microresonators with identical geometric parameters are coupled in either a parallel arrangement (known as “photonic molecules (PM)”29 or “coupled-resonator optical waveguides”30), or in series via bus waveguides.31,32 In general, there are two primary strategies for constructing higher-order EPs in coupled WGM microresonator-based systems. The first involves judicious engineering of the gain–loss profile33 and inter-resonator coupling so that the N-element symmetric PM forms a parity-time (PT)-symmetric array with an Nth-order EP.8,14 The second way is to utilize unidirectional coupling where the clockwise (CW) and counterclockwise (CCW) propagating waves within a microresonator are coupled via asymmetric backscattering.22,34 This concept was initially utilized to realize a second-order EP35 and then further employed to construct a robust exceptional surface.34 More recently, it has been leveraged to elevate the original EPs to a new one, effectively doubling their order.32 However, this asymmetric backscattering often requires external off-chip components (e.g., mirrors13 or isolators36) or on-chip structures (e.g., reflectors37). Therefore, developing a compact and fully integrable framework of N-element PM with a maximized EP order to 2N becomes essential for advancing technologies of integrated photonics and quantum photonics.

In this work, we propose an asymmetric configuration of PM consisting of N coupled resonators. Here the approach of achieving an Nth-order EP is unified with the concept of unidirectional coupling, forming a coherent framework for constructing 2Nth-order EPs without the need for external assistance. By utilizing a spiral-shaped microring operating at a second-order chiral EP via edge-induced asymmetric backscattering as the first element of the PM, the asymmetric N-element PM follows with a transferred maximum of chirality, enabling the construction of 2Nth-order EPs with no theoretical limit on the order. In addition to such hierarchical construction on the basis of a gain–loss PM, the approach is further extended to all-passive PMs, eliminating the need for precision gain tuning. Furthermore, the theoretical framework is examined using numerical simulations demonstrating fourth-order EPs in an asymmetric dimer. Enhanced eigenvalue splitting under external perturbations offers significant advantages, particularly for applications such as single-molecule detection.

Our discussion begins with a conventional two-element symmetric PM (i.e., a photonic dimer), consisting of two identical microrings in close proximity interacting through the evanescent fields [see Fig. 1(a)]. By adopting a traveling-wave basis with the state vector aCCWaCWbCCWbCWT⁠, where aCCW, aCW, bCCW, and bCW denote the lightwave components in the two rings, respectively, the effective Hamiltonian can be represented by a 4 × 4 matrix:
(1)
where ω0 is the original real-valued resonance frequency, γ1 and γ2 denote the gain/loss coefficients in each ring, respectively, and κ represents the inter-resonator coupling coefficient. Here, the coupling between the CW and CCW components is assumed to be negligible. For γ1 = −γ2, the Hamiltonian is PT-symmetric3 and can be expressed as [H, PT] = 0, where P and T are the parity and time-reversal operators,38 respectively. The eigenvalues ωn (n = 1, 2, 3, 4) are given by
(2)
At the critical point γ1=γ2=κ⁠, the eigenvalues become fourfold degenerate (i.e., ω1 = ω2 = ω3 = ω4). However, there are two complex (not normalized) eigenvectors, namely, 0−i10T,−i001T. The coexistence of two second-order EPs is attributed to the preserved orthogonality between the original CW and CCW modes.
FIG. 1.
(a) Schematic of a two-element symmetric PM operating at second-order EPs. (b) Schematic of spiral microring-based asymmetric PM, including gain–loss (left) and all-passive (right) configurations. The spiral geometry can be described by a spiral function ρ(ϕ)=R+T2πϕ, where T is the lateral dimension of the outer and inner spiral edges and ϕ is the azimuthal angle. The winding angle θ between 0 and 2π quantifies the relative azimuthal position of two spiral edges. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In this modeling, we set A=252+252i GHz and ω0 = 5 × 105 GHz. (d) Calculated surfaces of the imaginary part of eigenvalue in the Δω − γ parameter space for the all-passive system.
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(a) Schematic of a two-element symmetric PM operating at second-order EPs. (b) Schematic of spiral microring-based asymmetric PM, including gain–loss (left) and all-passive (right) configurations. The spiral geometry can be described by a spiral function ρ(ϕ)=R+T2πϕ⁠, where T is the lateral dimension of the outer and inner spiral edges and ϕ is the azimuthal angle. The winding angle θ between 0 and 2π quantifies the relative azimuthal position of two spiral edges. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In this modeling, we set A=252+252i GHz and ω0 = 5 × 105 GHz. (d) Calculated surfaces of the imaginary part of eigenvalue in the Δω − γ parameter space for the all-passive system.

FIG. 1.
(a) Schematic of a two-element symmetric PM operating at second-order EPs. (b) Schematic of spiral microring-based asymmetric PM, including gain–loss (left) and all-passive (right) configurations. The spiral geometry can be described by a spiral function ρ(ϕ)=R+T2πϕ, where T is the lateral dimension of the outer and inner spiral edges and ϕ is the azimuthal angle. The winding angle θ between 0 and 2π quantifies the relative azimuthal position of two spiral edges. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In this modeling, we set A=252+252i GHz and ω0 = 5 × 105 GHz. (d) Calculated surfaces of the imaginary part of eigenvalue in the Δω − γ parameter space for the all-passive system.
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(a) Schematic of a two-element symmetric PM operating at second-order EPs. (b) Schematic of spiral microring-based asymmetric PM, including gain–loss (left) and all-passive (right) configurations. The spiral geometry can be described by a spiral function ρ(ϕ)=R+T2πϕ⁠, where T is the lateral dimension of the outer and inner spiral edges and ϕ is the azimuthal angle. The winding angle θ between 0 and 2π quantifies the relative azimuthal position of two spiral edges. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In this modeling, we set A=252+252i GHz and ω0 = 5 × 105 GHz. (d) Calculated surfaces of the imaginary part of eigenvalue in the Δω − γ parameter space for the all-passive system.

Close modal
The orthogonality can be broken by introducing controllable backscattering inside a resonator. Here a spiral ring is employed to form an asymmetric two-element PM [see Fig. 1(b)]. For an individual spiral microring, the effective Hamiltonian is expressed as39,
(3)
where the complex-valued A (B) describes the coupling strength from the CW (CCW) to the CCW (CW) components. The spiral ring geometry naturally possesses inner and outer edges acting as effective Rayleigh scatterers.40,41 The inner edge primarily facilitates backscattering and coupling from CCW to CW components (i.e., the term B), and the outer edge accounts for coupling from CW to CCW components (i.e., the term A). In our recent study, unidirectional coupling can be achieved (i.e., either A = 0, or B = 0) via tuning the angular offset between two spiral edges θ and the dimensions of two edges T, thereby steering the system toward a second-order chiral EP without the need for external components.41 
Assuming a unidirectional coupling in a spiral ring (A ≠ 0, and B = 0), the Hamiltonian of an asymmetric PM can be expressed as
(4)
where Δω is the frequency detuning between the resonances of the spiral ring and the regular ring. The corresponding eigenvalues are given by
(5)
where δ = γ1 − γ2. The corresponding eigenvectors are
(6)
Here, v− and v+ contain non-zero components only in aCCW and bCW, respectively, indicating a traveling-wave nature with maximized mode chirality. By utilizing a lossy spiral ring and a regular ring with matched γ = γ1 = −γ2, the original PT-symmetry is broken due to the existence of unidirectional coupling [see the left panel of Fig. 1(b)]. Figure 1(c) summarizes the numerically modeled eigenvalue surfaces in the Δω − γ parameter space using Eq. (5). Each Riemann sheet suggests the coalescence of both eigenvalues and eigenvectors, allowing each sheet to be classified as an exceptional surface (ES) with second-order degeneracy.34 When Δω = 0 and γ = κ, the system reaches a fourfold degeneracy, in which v=−2iγ002κT suggests a fourth-order degeneracy.

Moreover, the concept can be extended to an all-passive configuration. As illustrated in the right panel of Fig. 1(b), the all-passive system consists of a spiral ring and a regular ring with distinct loss coefficients. Figure 1(d) depicts the calculated evolution of the imaginary part of the eigenvalues. By tuning the loss contrast to reach δ=2κ⁠, the system approaches a fourfold degeneracy of both eigenvalues and eigenvectors.

The framework can be extended to EPs of even higher orders, with no theoretical limit. For a general N-element symmetric PM, the Hamiltonian can be expressed as
(7)
where Hn=ω0−iγnAnBnω0−iγn represents the Hamiltonian of the nth resonator (n = 1, 2, …, N), and Kn=0κnκn0 denotes the coupling between the nth and (n + 1)th resonators (n = 1, 2, …, N − 1). For an ideal N-element symmetric PM,22 the coupling between the CCW and CW components in any resonator is negligible (An = Bn = 0). By setting γn = (2n − N − 1)γ and κn=n(N−n)γ⁠, where γ is an adjustable parameter, the PT-symmetric system exhibits two Nth-order EPs with distinct eigenvectors. A straightforward way to double the order of the EP to 2N is to replace the first regular microring with a spiral ring [see Fig. 2(a)] so that the original PT-symmetry becomes broken. Upon adjusted coupling in the spiral ring (A1 ≠ 0, B1 = 0), unidirectional coupling from CW to CCW components and the resulting maximized mode chirality are sequentially transferred throughout the resonator chain. Both the eigenvalues and eigenvectors of the system exhibit 2N-fold degeneracy (i.e., ω0 and v=1001⋯1001T⁠. In addition, the framework can be extended to general all-passive systems. When the loss coefficients in each element satisfy the condition γn = 2nγ and keep κn unchanged from the previous case, the eigenvalues reach a 2Nth-fold degeneracy ωn = ω0 − iγ(N + 1) at the 2Nth-order EP.
FIG. 2.
(a) Schematic illustration of an N-element asymmetric PM incorporating one spiral ring and N − 1 regular rings for the realization of a 2Nth-order EP. (b) The gain/loss coefficients (top) and coupling coefficients (bottom) adopted in the calculations at an eighth-order EP in a gain–loss system and an all-passive system with N = 4. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In the modeling, we set ω0 = 5 × 105 GHz and A = 50 GHz. (d) Calculated surfaces of the imaginary part of the eigenvalues in the Δω − γ parameter space for the all-passive system.
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(a) Schematic illustration of an N-element asymmetric PM incorporating one spiral ring and N − 1 regular rings for the realization of a 2Nth-order EP. (b) The gain/loss coefficients (top) and coupling coefficients (bottom) adopted in the calculations at an eighth-order EP in a gain–loss system and an all-passive system with N = 4. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In the modeling, we set ω0 = 5 × 105 GHz and A = 50 GHz. (d) Calculated surfaces of the imaginary part of the eigenvalues in the Δω − γ parameter space for the all-passive system.

FIG. 2.
(a) Schematic illustration of an N-element asymmetric PM incorporating one spiral ring and N − 1 regular rings for the realization of a 2Nth-order EP. (b) The gain/loss coefficients (top) and coupling coefficients (bottom) adopted in the calculations at an eighth-order EP in a gain–loss system and an all-passive system with N = 4. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In the modeling, we set ω0 = 5 × 105 GHz and A = 50 GHz. (d) Calculated surfaces of the imaginary part of the eigenvalues in the Δω − γ parameter space for the all-passive system.
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(a) Schematic illustration of an N-element asymmetric PM incorporating one spiral ring and N − 1 regular rings for the realization of a 2Nth-order EP. (b) The gain/loss coefficients (top) and coupling coefficients (bottom) adopted in the calculations at an eighth-order EP in a gain–loss system and an all-passive system with N = 4. (c) Calculated eigenvalue surfaces in the Δω − γ parameter space, including the real (left) and imaginary (right) parts for the gain–loss system. In the modeling, we set ω0 = 5 × 105 GHz and A = 50 GHz. (d) Calculated surfaces of the imaginary part of the eigenvalues in the Δω − γ parameter space for the all-passive system.

Close modal

To illustrate the concept, numerical modeling results for N = 4 are presented as a prototype example. Figure 2(b) outlines the system parameters, including γn and κn following PT-symmetric settings. The evolution of eigenvalues is presented in the Δω − γ parameter space in Fig. 2(c). Each eigenvalue surface corresponds to a doubly degenerate ES. By tuning γ = κ and Δω = 0, the eighth-order EP can be achieved, where both the real and imaginary parts of the eigenvalues become degenerate. The numerical simulation results of an all-passive PM with N = 4 in Fig. 2(d) reveal that all eigenvalues are purely lossy (negative imaginary parts), and the eighth-order EP is reached under the same conditions as in Fig. 2(c).

The enhancement of nonlinear responses at EPs to local perturbations is examined. As an example, we consider a two-element PM subjected to a local perturbation (e.g., a nanoparticle or analyte) with a strength of ϵ applied near the last ring [see Fig. 3(a)]. Such local perturbations can induce backscattering that couples the CW and CCW components and can be modeled using the two-mode approximation.39 The local perturbation, acting as the perturbative Hamiltonian, causes backscattering and coupling between the CW and CCW components. Thus, the corresponding perturbation Hamiltonian can be mapped onto the traveling-wave basis and expressed as
(8)
where Hp−1=εεeiϕεe−iϕε and Hp−2=εεeiϕεe−iϕε denote the nanoparticle-induced perturbation on the spiral ring and the regular ring, ϵ is the perturbation strength, and ϕ is the phase related to the azimuthal angle and azimuthal mode number,39 respectively. For simplicity, we assume that ϕ = 0 (i.e., exp(iϕ) = 1).
FIG. 3.
(a) Schematics of enhanced responses at a second-order EP (top) and a chiral fourth-order EP (bottom) toward single-analyte-induced localized perturbation. [(b)–(d)] Calculated eigenvalue splitting upon varying perturbation strength for EPs of an order of 2 (b), 4 (c), and 8 (d). The shaded areas denote the regime of single molecule sensing with ϵ < 0.3 GHz. (e) Summary of the maximum splitting Δω upon varying ϵ across DPs and EPs of orders 2, 4, 6, and 8, on a logarithmic scale. Inset: Responses of different EP orders to ϵ of 0.07 GHz.
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(a) Schematics of enhanced responses at a second-order EP (top) and a chiral fourth-order EP (bottom) toward single-analyte-induced localized perturbation. [(b)–(d)] Calculated eigenvalue splitting upon varying perturbation strength for EPs of an order of 2 (b), 4 (c), and 8 (d). The shaded areas denote the regime of single molecule sensing with ϵ < 0.3 GHz. (e) Summary of the maximum splitting Δω upon varying ϵ across DPs and EPs of orders 2, 4, 6, and 8, on a logarithmic scale. Inset: Responses of different EP orders to ϵ of 0.07 GHz.

FIG. 3.
(a) Schematics of enhanced responses at a second-order EP (top) and a chiral fourth-order EP (bottom) toward single-analyte-induced localized perturbation. [(b)–(d)] Calculated eigenvalue splitting upon varying perturbation strength for EPs of an order of 2 (b), 4 (c), and 8 (d). The shaded areas denote the regime of single molecule sensing with ϵ < 0.3 GHz. (e) Summary of the maximum splitting Δω upon varying ϵ across DPs and EPs of orders 2, 4, 6, and 8, on a logarithmic scale. Inset: Responses of different EP orders to ϵ of 0.07 GHz.
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(a) Schematics of enhanced responses at a second-order EP (top) and a chiral fourth-order EP (bottom) toward single-analyte-induced localized perturbation. [(b)–(d)] Calculated eigenvalue splitting upon varying perturbation strength for EPs of an order of 2 (b), 4 (c), and 8 (d). The shaded areas denote the regime of single molecule sensing with ϵ < 0.3 GHz. (e) Summary of the maximum splitting Δω upon varying ϵ across DPs and EPs of orders 2, 4, 6, and 8, on a logarithmic scale. Inset: Responses of different EP orders to ϵ of 0.07 GHz.

Close modal
The total Hamiltonian is H = H0 + Hp. Here, we analyze the perturbations induced by a nanoparticle on regular rings, that is, Hp−1 = 0 [see the lower panel of Fig. 3(a)]. For the gain–loss symmetric PM operating at two second-order EPs, the eigenvalues are given by
(9)
The induced eigenvalue splitting between ω3 and ω4 is presented in the numerical modeling results in Fig. 3(b). As the perturbation strength ϵ (up to 5 GHz) is significantly smaller than the coupling strength κ (∼100 GHz), the eigenvalue splitting Δω = ω3 − ω4 is proportional to ϵ1/2. In contrast, the other two eigenmodes are completely silent with Δω = 0, as the lack of chiral EPs prevents the localized perturbation from simultaneously influencing both “even” and “odd” modes.
For an asymmetric PM operating at a chiral fourth-order EP (i.e., γ = γ1 = −γ2 = κ, A ≠ 0, B = 0) [see the lower panel of Fig. 3(a)], while the perturbation strength ϵ is much smaller than the unidirectional coupling strength A, the responses of eigenvalues can be analyzed using the Newton–Puiseux series theorem:42 
(10)

For the same range of ϵ up to 5 GHz in Fig. 3(c), significantly stronger nonlinear splitting is observed for ω1 and ω4. In addition, the other two split modes are no longer silent to the external perturbation, which is in sharp contrast with those of the second-order EP in Fig. 3(b). Cascading more microring resonators allows the construction of EPs with even higher orders. Figure 3(d) presents mode splitting at an eighth-order EP using a four-element asymmetric PM. Figure 3(e) summarizes the responses of discernible spacing between all split modes for EPs with orders of 2, 4, 6, and 8, as well as the diabolic point (DP) in a single regular microring [A = B = 0, in Eq. (3)], where only eigenvalues are degenerate. At the DP, the mode splitting scales linearly with ϵ (i.e., ΔωDP = 2ϵ),43 as expected from a conventional Hermitian system. In contrast, at higher-order EPs, the splitting exhibits a nonlinear dependence on ϵ, transitioning from ϵ1/2 at the second-order EP to ϵ1/8 at the eighth-order EP. This highlights the fundamental difference between DPs and EPs in terms of their sensitivity to perturbations.

Here, we highlight the potential applications of single-molecule detection at higher-order EPs. Typically, the perturbation caused by a binding or unbinding event of individual biomolecules (e.g., antibodies,44 nucleic acids,45 and viruses46,47) within the order of 10–300 MHz is much smaller than the practically configured coefficient |A| (∼10 GHz in our previous experimental demonstration of a spiral ring40), ensuring the effectiveness of EP-enhanced sensing. The inset of Fig. 3(e) shows the response of various EP orders at ϵ of 0.07 GHz reported in previous studies of Influenza A viruses.46 At the second-order EP, splitting is enhanced ∼20-fold, increasing to ∼135-fold at the fourth-order EP, underscoring the exponential advantage of higher-order EPs in ultra-sensitive detection. The enhancement approaches saturation at the eighth-order EP.

To validate the proposed framework, numerical simulations based on the finite-element method (COMSOL Multiphysics wave-optics module) are performed. Here, a prototype demonstration with an optically passive, two-element asymmetric PM is presented. First, following our prior study,41 a second-order chiral EP is achieved by tuning the relative angular offset θ and the dimension of spiral edges T. Then, a regular microring with the same radius of 3 μm is placed next to the spiral ring with a fixed gap spacing of 200 nm, ensuring constant inter-resonator coupling strength κ. The evolution of the eigenfrequencies of the PM is monitored in the zero-detuning case (Δω = 0) by varying the imaginary part of the refractive index of one ring and changing the loss contrast. As shown in Figs. 4(a) and 4(b), instead of four independent branches, only two “exceptional branches (EBs)” with two-fold degeneracy are discernible, which aligns well with the ESs predicted by theory.

FIG. 4.
[(a) and (b)] Extracted evolution of the real (a) and imaginary (b) parts of the eigenfrequencies as a function of the ratio between the loss contrast and the inter-resonator coupling |γ1 − γ2|/κ. In the simulation, the refractive indices of the microrings and the surrounding medium are 2 (silicon nitride) and 1 (air), respectively. T is set to 41.8 nm. The widths of two microrings are adjusted independently (Wspiral = 125 nm and Wring = 135 nm) to have a match between resonant frequencies. Insets [(i)–(iii)]: Simulated mode field distribution in the regime of strong (i) and weak coupling [(ii) and (iii)]. (c) Simulated mode field distribution at the chiral fourth-order EP. Insets [(iv) and (v)]: Zoomed-in views around the spiral edges (iv) and the corresponding region in the regular ring (v). (d) Extracted internal emerging Husimi functions for (c), including the spiral (left) and the regular (right) ring. The positive region (sin (χ) > 0) stands for CCW components whereas the negative region (sin (χ) < 0) stands for CW components. The dashed lines indicate the critical angle for total internal reflection.
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[(a) and (b)] Extracted evolution of the real (a) and imaginary (b) parts of the eigenfrequencies as a function of the ratio between the loss contrast and the inter-resonator coupling |γ1 − γ2|/κ. In the simulation, the refractive indices of the microrings and the surrounding medium are 2 (silicon nitride) and 1 (air), respectively. T is set to 41.8 nm. The widths of two microrings are adjusted independently (Wspiral = 125 nm and Wring = 135 nm) to have a match between resonant frequencies. Insets [(i)–(iii)]: Simulated mode field distribution in the regime of strong (i) and weak coupling [(ii) and (iii)]. (c) Simulated mode field distribution at the chiral fourth-order EP. Insets [(iv) and (v)]: Zoomed-in views around the spiral edges (iv) and the corresponding region in the regular ring (v). (d) Extracted internal emerging Husimi functions for (c), including the spiral (left) and the regular (right) ring. The positive region (sin (χ) > 0) stands for CCW components whereas the negative region (sin (χ) < 0) stands for CW components. The dashed lines indicate the critical angle for total internal reflection.

FIG. 4.
[(a) and (b)] Extracted evolution of the real (a) and imaginary (b) parts of the eigenfrequencies as a function of the ratio between the loss contrast and the inter-resonator coupling |γ1 − γ2|/κ. In the simulation, the refractive indices of the microrings and the surrounding medium are 2 (silicon nitride) and 1 (air), respectively. T is set to 41.8 nm. The widths of two microrings are adjusted independently (Wspiral = 125 nm and Wring = 135 nm) to have a match between resonant frequencies. Insets [(i)–(iii)]: Simulated mode field distribution in the regime of strong (i) and weak coupling [(ii) and (iii)]. (c) Simulated mode field distribution at the chiral fourth-order EP. Insets [(iv) and (v)]: Zoomed-in views around the spiral edges (iv) and the corresponding region in the regular ring (v). (d) Extracted internal emerging Husimi functions for (c), including the spiral (left) and the regular (right) ring. The positive region (sin (χ) > 0) stands for CCW components whereas the negative region (sin (χ) < 0) stands for CW components. The dashed lines indicate the critical angle for total internal reflection.
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[(a) and (b)] Extracted evolution of the real (a) and imaginary (b) parts of the eigenfrequencies as a function of the ratio between the loss contrast and the inter-resonator coupling |γ1 − γ2|/κ. In the simulation, the refractive indices of the microrings and the surrounding medium are 2 (silicon nitride) and 1 (air), respectively. T is set to 41.8 nm. The widths of two microrings are adjusted independently (Wspiral = 125 nm and Wring = 135 nm) to have a match between resonant frequencies. Insets [(i)–(iii)]: Simulated mode field distribution in the regime of strong (i) and weak coupling [(ii) and (iii)]. (c) Simulated mode field distribution at the chiral fourth-order EP. Insets [(iv) and (v)]: Zoomed-in views around the spiral edges (iv) and the corresponding region in the regular ring (v). (d) Extracted internal emerging Husimi functions for (c), including the spiral (left) and the regular (right) ring. The positive region (sin (χ) > 0) stands for CCW components whereas the negative region (sin (χ) < 0) stands for CW components. The dashed lines indicate the critical angle for total internal reflection.

Close modal

When γ1−γ2 is smaller than κ, the real part of the eigenfrequencies splits, while the imaginary part remains degenerate, indicating that the system operates in the strong coupling regime. The simulated mode field distribution in the inset (i) reveals balanced intensities in two resonators, corresponding to the PT-symmetric regime in a gain–loss system. Upon further increasing γ1−γ2 beyond the transition point, the real part of the eigenfrequencies becomes degenerate, while the imaginary part splits, indicating that the system enters the weak coupling regime. Insets (ii) and (iii) display the distinct mode field distributions for two split modes. There exists a fourth-order EP at the transition point between strong and weak coupling, where γ1−γ2=2κ⁠. Using Eq. (5), one can accurately fit the simulation results [see the gray curve in Figs. 4(a) and 4(b)] to extract the values.

Here, we employ the Husimi projection to correlate the field distribution at the cavity boundaries with the leakage in phase space and to visualize the CCW and CW components.39,48 The detailed procedure has been well explained in previous studies.39 The calculated value H reflects the relative mode field intensity at a given leakage angle sin χ and boundary position ϕ. In Fig. 4(d), the projection for the spiral ring is primarily distributed in the CCW part (sin (χ) > 0) above the leaky region, which contrasts sharply with the projection for the regular ring. For each resonator, the local chirality α can be expressed as39,48
(11)
where ∑CCWHusimi and ∑CWHusimi are the total distribution probabilities of the CCW and CW components, respectively. One should note this particular definition only reflects the weights between two components and does not allow for negative values, which differs from some of the other alternatives.41 The extracted α reaches 0.9901 for the spiral ring and 0.9896 for the regular ring, highlighting the chiral nature of this four-fold degeneracy. Notably, the high mode chirality is maintained, with a calculated α > 0.98 even when the system is away from the fourth-order EP. This further corroborates that the chiral nature in such asymmetric PM is robust to the changes in the system loss or coupling,35 consistent with the theoretical analysis in Eq. (6) in which only aCCW and bCW are non-zero.

To simulate the EP-enhanced sensitivity, a nanoparticle with a radius of 10 nm and a refractive index of 1.4 is introduced as a local perturbation near the regular ring. Adjusting the particle-ring spacing h between 0 and 120 nm leads to a variation of the perturbation strength ϵ between ∼0.06 and ∼3 GHz. The tracked mode splitting on a logarithmic scale in Fig. 5(a) demonstrates excellent agreement with the theoretical prediction, namely, a slope of 1/2 for the symmetric PM and a slope of 1/4 for the asymmetric PM. The perturbation is visualized in the simulated mode field distribution in Fig. 5(b). In contrast to the clear traveling wave characteristics in Fig. 4(c), distinct nodes and antinodes along the resonating path emerge in both microrings, reflecting the additional backscattering due to the nanoparticle. In this case, the extracted Husimi projection in the spiral ring still retains high chirality (∼0.931)⁠, with a clear dominance by the CCW component. In contrast, for the regular microring, the Husimi projection reveals a comparable contribution between CW and CCW components, resulting in a very low chirality of 0.104.

FIG. 5.
(a) Extracted mode splitting ΔωEP as a function of ϵ at second-order EP, fourth-order EP, and DP. Dots: simulation results. Lines: fits based on theory. (b) Simulated mode field distribution for the chiral fourth-order EP with a perturbation by a 10 nm-sized nanoparticle with spacing h = 20 nm. Insets [(i) and (ii)]: Zoomed-in views around the spiral edges (i) and the region where the nanoparticle interacts with the evanescent field (ii). (c) Extracted internal emerging Husimi functions for (b), including the spiral (left) and the regular (right) ring. (d) The extracted enhancement ratio ΔωEP/ΔωDP with a lower bound of ϵ at the level of single analyte binding.
View largeDownload slide

(a) Extracted mode splitting ΔωEP as a function of ϵ at second-order EP, fourth-order EP, and DP. Dots: simulation results. Lines: fits based on theory. (b) Simulated mode field distribution for the chiral fourth-order EP with a perturbation by a 10 nm-sized nanoparticle with spacing h = 20 nm. Insets [(i) and (ii)]: Zoomed-in views around the spiral edges (i) and the region where the nanoparticle interacts with the evanescent field (ii). (c) Extracted internal emerging Husimi functions for (b), including the spiral (left) and the regular (right) ring. (d) The extracted enhancement ratio ΔωEP/ΔωDP with a lower bound of ϵ at the level of single analyte binding.

FIG. 5.
(a) Extracted mode splitting ΔωEP as a function of ϵ at second-order EP, fourth-order EP, and DP. Dots: simulation results. Lines: fits based on theory. (b) Simulated mode field distribution for the chiral fourth-order EP with a perturbation by a 10 nm-sized nanoparticle with spacing h = 20 nm. Insets [(i) and (ii)]: Zoomed-in views around the spiral edges (i) and the region where the nanoparticle interacts with the evanescent field (ii). (c) Extracted internal emerging Husimi functions for (b), including the spiral (left) and the regular (right) ring. (d) The extracted enhancement ratio ΔωEP/ΔωDP with a lower bound of ϵ at the level of single analyte binding.
View largeDownload slide

(a) Extracted mode splitting ΔωEP as a function of ϵ at second-order EP, fourth-order EP, and DP. Dots: simulation results. Lines: fits based on theory. (b) Simulated mode field distribution for the chiral fourth-order EP with a perturbation by a 10 nm-sized nanoparticle with spacing h = 20 nm. Insets [(i) and (ii)]: Zoomed-in views around the spiral edges (i) and the region where the nanoparticle interacts with the evanescent field (ii). (c) Extracted internal emerging Husimi functions for (b), including the spiral (left) and the regular (right) ring. (d) The extracted enhancement ratio ΔωEP/ΔωDP with a lower bound of ϵ at the level of single analyte binding.

Close modal

The scenario of single binding events causing an ultra-weak perturbation is further studied. As the infinite length in the z-axis in our 2D simulation actually exaggerates the perturbation effect, here the radius of a directly bound nanoparticle (h = 0) in the 2D simulation is set to be smaller than the size of any real molecules (e.g., immunoglobulin G molecule with a mass of 14.4 kDa and a size of ∼8×4×15nm3⁠) as 1 nm (limited by the meshing condition), yielding the lower bound of ϵ to be ∼0.08 GHz. The response at the chiral fourth-order EP is enhanced by 40 times compared with that at a DP and is also three times higher than that obtained at a second-order EP [see Fig. 5(b)]. In reality, by monitoring splitting-induced intensity change at a fixed probe wavelength, it is possible to track in real-time the transient processes, such as the binding and unbinding of single molecules, thereby enabling the study of molecular affinity.23 

In this work, we have reported a generic approach to constructing higher-order EPs in a PM. By utilizing a chiral microresonator as the primary element and standard microring resonators as the remaining components, the order of non-Hermitian degeneracy in an N-element PM can be increased to 2N. This hierarchical approach is applicable to gain–loss systems, as well as all-passive systems, thus eliminating the need for dedicated gain engineering. Through the adoption of a spiral-shaped microring with maximized mode chirality, we have numerically confirmed the existence of a fourth-order chiral EP in an asymmetric photonic dimer. Moreover, the EP-enhanced sensitivity to localized perturbations has been explored through numerical simulations. Both theoretical analysis and simulation results confirm that the proposed asymmetric PM system enhances system response at a higher-order EP. The increased sensitivity to weak perturbations, facilitated by higher-order EPs, presents new opportunities for detections of biomolecules and airborne contaminants.49,50 In addition to frequency shifts, the response to perturbations may also lead to enhanced frequency conversion in a nonlinear system,51 which could be further explored for sensing applications.

In general, the choice of chiral microresonators can be extended to other types, such as those mediated by the “two-scatterers” scheme,15,52 Taiji resonators,53 and others with deformed shapes.54 Given our recent experimental demonstration of spiral microring resonators using the foundry process,40 the concept of all-passive PM can be readily realized in silicon photonics platforms40 as well as polymer material platforms.55 Meanwhile, the concept of gain–loss PM systems can be realized in III–V semiconductor56 and rare-earth57 material platforms. Overall, the versatility provides a compact and integrable solution for realizing higher-order EPs across different platforms, paving the way for their widespread adoption in diverse photonic applications such as quantum light sources and nonreciprocal transmission.

The authors acknowledge the support from the National Natural Science Foundation of China under Grant Nos. (62422503), (12474375), (62305093), (U22A2093), and (22371059), the Guangdong Basic and Applied Basic Research Foundation Regional Joint Fund under Grant Nos. (2023A1515011944) and (2022A1515220158), and the Science and Technology Innovation Commission of Shenzhen under Grant Nos. (JCYJ20220531095604009), (JCYJ20240813104819027), and (RCYX20221008092907027).

The authors have no conflicts to disclose.

Jin Li: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Kexun Wu: Formal analysis (supporting); Investigation (supporting); Software (equal); Validation (equal); Writing – review & editing (supporting). Jian Ding: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). Mingquan Deng: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). Xiujie Dou: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). Xingyi Ma: Investigation (supporting); Writing – review & editing (supporting). Xiaochuan Xu: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Jianan Duan: Funding acquisition (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Jiawei Wang: Conceptualization (lead); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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