Generalized Perturbation Theory for Weakly Deformed Microdisk Cavities

A perturbation treatment is derived for weakly deformed microdisk cavities in a more generalized scenario, where both resonant and non‐resonant surface scattering processes coexist. It is proved that the originally developed nondegenerate and degenerate perturbation theories are simplified forms of the generalized theory in special cases. The validity of the perturbation theory is verified by the comparison with a full numerical method for two generic classes of deformed microdisk cavities: locally notched microdisks and smooth globally deformed microdisks. The simulation of these two classes of microdisk cavities using the perturbation theory unveils the distinct characteristics of optical modes from the ones predicted by the original nondegenerate and degenerate perturbation theories in the generalized scenario. Furthermore, a hybrid‐scattering‐based exceptional point is introduced for deformed microdisk cavities in the generalized scenario by using the perturbation theory. The exceptional point exhibits higher nonorthogonality over the originally proposed resonant‐scattering‐based exceptional point, as non‐resonant scattering processes provide an additional degree of freedom. Full numerical simulations demonstrate that the exceptional point is more robust against random surface roughness at cavity boundary than the resonant‐scattering‐based exceptional point.


Introduction
11][12] The microdisk cavities with deformed boundary enable ultralow-threshold microcavity lasers. [11,13,14]Breaking the rotation symmetry of a perfect circle microdisk, highly directional free-space light output with small angular spread can be achieved by the deformed microdisk cavities with various boundary shapes, [11][12][13] e.g., the limaçon cavity, [15] the notched cavity, [16] the spiral cavity, [17,18] the face cavity, [19] the short-egg cavity, [20] etc.Compared with isotropic emission from a perfect circle microdisk, the directionality of light from deformed microdisk cavities facilitates more highly efficient optical pumping [21,22] and light emission [23][24][25][26][27] from microcavity lasers.][30][31] Additionally, deformed microdisk cavities can channel light wave to attain efficient broadband transmission between a microcavity and a waveguide. [32,33]rom the perspective of fundamental physics, intriguing phenomena in quantum chaos theory including dynamical tunneling [32][33][34][35][36] and dynamical localization [37][38][39] can be observed and studied in deformed microdisk cavities, since more morphological modes such as whispering-gallery modes (WGMs), chaotic modes, and scar modes arise from diverse ray dynamics depending on type and degree of cavity shape deformation.On the other hand, many peculiar non-Hermitian effects such as exceptional points (EPs), [40][41][42][43][44] high-order EPs, [44] optical chirality, [17,40,41,45] local vortices of light, [42,43] and generic avoided resonance crossing (ARC) [26,[46][47][48][49][50] are demonstrated in deformed microdisk cavities, as the microcavity is an open multimode system.In the multimode system, the morphological modes can be coupled with each other due to cavity shape deformation.In addition to the non-Hermitian effects, the coupling of modes leads to dramatic modifications for characteristics of microcavity modes in terms of Q factors, [46,47,49] intracavity intensity distribution, [50,51] and far-field pattern. [26,52]or the deformed microdisk cavities much larger than wavelength in vacuum, the formation and characteristics of the morphological modes can be explained by ray dynamics in the geometrical optics regime. [12]However, the ray optics models fail once the microcavity size is comparable to or even smaller than wavelength in vacuum.For a more accurate explanation and analysis, the elaborate numerical methods developed based on wave optics such as the boundary element method (BEM) [53] are necessary.The numerical methods of wave optics are too tedious to provide intuitive physical pictures.Therefore, efficient and transparent approximate treatments of wave optics are desirable for deformed microdisk cavities.
To circumvent the numerical methods, a perturbation theory was proposed by R. Dubertrand et al. [54] for weakly deformed microdisk cavities.Initially, the perturbation theory is restricted to the scenario of transverse-magnetic (TM) polarized modes in mirror-reflection symmetric microdisk cavities.Then, the theory was extended to the scenario of transverse-electric (TE) polarization in mirror-reflection symmetric microdisk cavities [55,56] and TM polarization in asymmetric microdisk cavities without mirror-reflection symmetry. [57]][58][59][60][61][62][63] In contrast to numerical methods, the perturbation theory not only allows straightforward evaluations for characteristics of modes with analytical formulas [54,58,59,61,63] but also provides a powerful tool for solving inverse design problems. [60]oreover, deep insight can be gained from the perturbation theory.For instance, the underlying mechanism of the coupling of modes with different angular momenta in a weakly deformed microdisk cavity is ascribed to the surface scattering processes of light at the microcavity sidewall, of which the strengths can be individually evaluated for different scattering paths by the perturbation theory. [64,65]In fact, the scattering processes are not limited to weakly deformed microdisk cavities, i.e., the formation and characteristics of modes in strongly deformed microdisk cavities are also governed by the scattering. [10]The scattering processes can be manipulated by imposing specific cavity shape deformation to achieve unidirectional light emission for weakly deformed microdisk cavities, according to the perturbation theory. [55,65]In general, the coupling of modes with different angular momenta via surface scattering processes is in the weak coupling regime for weakly deformed microdisk cavities, as the scattering processes are trivial non-resonant scattering. [55,64]However, the coupling of modes is radically altered in quasi-degenerate cases, where resonant scattering processes occur. [42,44,54,55]As the resonant scattering processes are nontrivial, a substantial modification is required for the perturbation theory in quasi-degenerate cases. [54]aking resonant surface scattering processes into account, a degenerate perturbation treatment is developed. [42,44,54]Using the degenerate perturbation theory, first-order resonant scattering processes and the coupling of modes via the non-trivial scattering can be described.According to the description, the resonant scat-tering processes can be delicately tailored to yield EPs through designing cavity shape deformation. [42,44]Nevertheless, only the first-order resonant scattering processes are considered within the degenerate perturbation theory.While the resonant scattering processes might be more pronounced, the non-resonant ones are more ubiquitous. [55,64]The more generalized scenario in which both the two scattering processes coexist has not been studied.Correspondingly, the generalized perturbation theory describing the scenario is rarely presented except as a treatment in the special case of a strict twofold degeneracy in asymmetric microdisk cavities. [57]However, the treatment is only restricted to the twofold degenerate regime, and cannot be applied even for the fourfold degenerate modes [44] similar to the twofold degenerate regime.
In this paper, we derive the generalized perturbation theory, in which both non-resonant and resonant surface scattering processes are incorporated into the formalism.It is demonstrated that the originally developed nondegenerate and degenerate perturbation theory are only simplified forms of the present one in special cases.The validity of the generalized perturbation theory is verified by the two generic microdisk cavities: locally notched microdisks and smooth globally deformed microdisks, where good agreement with a full numerical method is observed for the generalized theory in terms of frequencies, decay rates, intracavity distribution, and far-field patterns.The distinct physical pictures from the original nondegenerate and degenerate perturbation theories are provided by the present generalized theory in the generalized scenario, where both resonant and non-resonant surface scattering processes coexist.On one hand, quasi-degenerate modes in weakly deformed microdisk cavities can exhibit dramatic distinction from unperturbed ones or the ones described by the original nondegenerate perturbation theory, due to resonant scattering processes.On the other hand, the characteristics of quasi-degenerate modes such as complex frequencies and coupling of modes deviate from the ones described by the original degenerate perturbation theory, owing to the modification from non-resonant surface scattering processes.When both resonant and non-resonant scattering processes are exploited, a hybrid-scattering-based EP is introduced by using the present perturbation theory in the generalized scenario.In comparison to the originally proposed EP solely depending on resonant scattering processes, [42] non-resonant scattering processes offer an additional degree of freedom for the hybrid-scattering-based EP.As the consequence of non-resonant scattering processes, the EP exhibits higher nonorthogonality over the resonant-scattering-based EP.Moreover, full numerical simulations reveal the enhanced robustness of the EP against random surface roughness at the cavity boundary compared to the resonant-scattering-based EP.
The paper is organized as follows.In Section 2, the mode Equation describing optical modes in dielectric microdisk cavities and optical properties of modes in a circular microdisk cavity are briefly reviewed.Subsequently, the generalized second-order perturbation theory is derived for weakly deformed microdisk cavities.Section 3 is devoted to verifying the applicability of the perturbation theory to two generic classes of deformed microdisk cavities.Using the generalized perturbation theory, a robust EP of order 2 is proposed in Section 4, while exploiting hybrid resonant and non-resonant surface scattering processes in a deformed microdisk cavity.Finally, a conclusion and outlook are given in Section 5.

Generalized Perturbation Theory
In the section, a generalized second-order perturbation theory is presented for optical modes in weakly deformed microdisk cavities.In Sections 2.1 and 2.2, the mode Equation determining optical modes in dielectric microdisk cavities and its solutions in a circular microdisk cavity are reviewed, respectively.Next, the framework of the perturbation theory is set up for weakly deformed microdisk cavities in Section 2.3.According to the framework, the first-and second-order perturbation theories are derived in Sections 2.4 and 2.5, respectively.In Section 2.6, a criterion function is given to distinguish between degenerate and nondegenerate regimes.

Mode Equation
Optical modes defined as a time-harmonic solution of Maxwell's equations are crucial for the study of optics in microdisk cavities.To solve optical modes, microdisk cavities can be first treated as a quasi-2D system to simplify Maxwell's equations and introduce mode equations.The reason for this treatment is twofold.The height of the cavity is negligible compared to its transverse size.Also, the height can be separated out using an effective index approximation.
For 2D microdisk cavities, Maxwell's equations reduce to a 2D scalar Helmholtz-type mode equation where n(r, ) is the effective index in the polar coordinates (r, ), k = /c is the wave number of light,  is the angular frequency of light, and c is the velocity of light in vacuum.The real part of the wavefunction  represents a magnetic field H z for transverse electric (TE) polarization ( E z = 0) or electric field E z for transverse magnetic (TM) polarization ( H z = 0) perpendicular to the plane of the microdisk cavity (along the height direction ' z of the cavity).With the solution  of the mode Equation (1), the other electric and magnetic field components can be calculated. [10]n addition to the mode Equation ( 1), the solution  has to fulfill boundary conditions.Along the interface r b of the cavity, the boundary conditions of the solution  and its normal derivative subject to the continuity relations of electromagnetic fields can be written by and Here the subscripts "in" and "out" represent the value inside and outside the cavity, respectively.At far positions with large r, there exists a Sommerfeld's outgoing wave condition which has to be satisfied for the solutions of the mode Equation (1).Due to the boundary condition ( 4), the solutions of the mode Equation ( 1) are quasi-bound states with discrete complexvalued frequency .The real and imaginary parts of  are respectively related to resonant frequency and decay rate of quasi-bound modes.Correspondingly, the quality (Q) factor of modes is

Optical Modes in a Circular Microdisk Cavity
Only in some special cases, can the mode Equation (1) be solved analytically, e.g., a circular microdisk cavity.When a circular microdisk cavity with the effective index n in = n > 1 and a radius of r b () = R is surrounded by air with n out = 1, the inner and outer general solutions of the mode Equation ( 1) can be given by Here J m is the Bessel functions of m order, and H m is the Hankel functions of the first kind and m order.Due to the mirrorreflection symmetry of circular cavities with respect to  = 0, the modes of m ≠ 0 are twofold degenerate.In addition to these even-parity solutions (5), odd-parity ones can be obtained by substituting sine for cosine in Equation (5a) and (5b).Putting the solutions (5) into the boundary conditions (2) and (3), the Equation (quantization condition) for discrete frequency  or dimensionless frequency x (0) = kR = R/c is achieved as Here the prime (′) denotes the first derivative of functions with respect to argument.According to Equation ( 6), the dimensionless frequency x (0) and the corresponding mode for a circular microdisk cavity are labeled by (m, l), where m ∈ N and l ∈ N + are the azimuthal mode number (angular momentum) and the radial mode number respectively related to the number of wavefunction (mode) nodes in azimuthal and radial directions.The real and imaginary parts of the dimensionless frequency x (0) = kR describe resonant frequency and decay rate of modes in a circular microdisk cavity, respectively.Thus, the quality (Q) factor of the mode can be expressed by Q = Re(x (0) )/[ − 2Im(x (0) )].To distinguish modes in deformed cavities, the modes in a circular microdisk cavity are called unperturbed modes in this following.For unperturbed modes, the dimensionless frequency is represented by x (0) .In this paper, unperturbed modes are restricted to WGMs with high Q factors or satisfying |Im(x (0) )| ≪ |Re(x (0) )|, i.e., the external modes with low Q factors and almost zero intensity in cavities are excluded. [66]

Framework of Perturbation Theory
To derive the perturbation theory applicable for weakly deformed microdisk cavities, the boundary function of the deformed microcavities is treated as a sum of a circle and a perturbation r b () = R + f () (7)   in polar coordinates (r, ).Here R is the radius of an unperturbed circular microdisk. is a formal perturbation parameter, which is used for arranging perturbation series.f() is a singlevalued deformation function with the period of 2, describing the boundary deformation of the microcavity.The cavity shape that we focus on is mirror-reflection symmetric, i.e., f () = f( − ), as a large number of deformed microcavities possess the symmetry. [24,25,34,65]Thus, modes in the symmetric microcavities are either even-or odd-parity with respect to the symmetry axis  = 0.The perturbation theory that we consider here is restricted to even-parity modes in TM polarization.
For weakly deformed microdisk cavities, f() has to satisfy |f()| ≪ R as a general condition of weak deformation.Specifically, the criterion of weakly deformed microcavities, for which the perturbation theory is applicable, has been given by [54] with Here a is the area, where the refraction index n of the deformed cavity differs from the corresponding unperturbed circular cavity with the radius of R.
In these weakly deformed microdisk cavities, the wavefunctions inside and outside of the cavity as the solution to mode Equation (1) are expanded as the unperturbed modes (5) of circular cavities in the angular momentum representation by the following ansatz where  in (r,) and  out (r,) represent the inner and outer wavefunctions of modes, respectively.Here the subscript q denotes the azimuthal mode number (angular momentum), while a q (a q + b q ) is the corresponding angular momentum component of the wavefunction inside (outside) the microcavity.The distribution of the angular momentum component versus the angular momentum is referred to as angular momentum spectra, which can be used to analyze strengths of surface scattering processes in microdisk cavities. [65]The ansatz (10) represents even-parity modes, whereas odd-parity modes can be obtained by replacing cosine functions with sine functions.Apparently, the wavefunctions (10a) and (10b) are also solutions of the mode Equation ( 1), satisfying the boundary condition (4) due to the Hankel function's asymptotic behavior.The unknown coefficients a q , b q , and the perturbed dimensionless frequency x = kR are determined by the boundary conditions ( 2) and (3a) for TM modes (( 2) and (3b) for TE modes).Note that the expansion method (10) relies on the Rayleigh hypothesis. [10]o determine the unknown quantities, the boundary conditions ( 2) and (3a) are expanded at the interface of the corresponding unperturbed circular disk cavity as the following series, Inserting the ansatz (10) into Equation (11a) and (11b), the terms J ′ q ∕J q , H ′ q ∕H q , J ′′ q ∕J q , H ′′ q ∕H q , J ′′′ q ∕J q , and H ′′′ q ∕H q emerge.Using the recurrence relations of Bessel functions and Hankel functions, the high-order derivatives of Bessel and Hankel functions are expressed by the low-order ones as follows, where Z q denotes J q or H q .With Equation (12a) and (12b), the terms J ′′ q ∕J q , H ′′ q ∕H q , J ′′′ q ∕J q , and H ′′′ q ∕H q in Equation (11a) and (11b) can be simplified and substituted by J ′ q ∕J q and H ′ q ∕H q .Then, replacing J ′ q ∕J q by the function S q in the left-hand side of Equation (6a) and H ′ q ∕H q , we get Here we omit the arguments of S q , H ′ q , and H q , which are all the functions of the perturbed dimensionless frequency x = kR.
In general, optical modes in deformed microdisk cavities cannot be labeled by a well-defined angular momentum and radial mode number like unperturbed modes in a circular microdisk cavity.However, perturbed modes in weakly deformed microdisk cavities can be labeled by a dominant angular momentum m 0 and a dominant radial mode number l 0 in the nondegenerate regime, as revealed by the nondegenerate perturbation theory. [54]Correspondingly, the mode is dominated by an unperturbed mode labelled by (m 0 ,l 0 ), showing the noticeable similarity to the unperturbed mode (m 0 ,l 0 ). [55,59,64,65]Rigorously, the modes are expanded as a series of mixed angular momenta consisting of the dominant angular momentum m 0 and infinite minor ones created via non-resonant surface scattering. [64,65]evertheless, the expansion rule fails when quasi-degeneracy appears.Quasi-degeneracy is a frequently encountered phenomenon for microdisk cavities.For circular microdisk cavities, there exists a wealth of quasi-degeneracies due to the Poisson sequences of the distribution of WGMs. [67]For quasi-degenerate cases, the nondegenerate perturbation theory and the corresponding expansion rule need to be modified. [54,57]Between quasi-degenerate modes, resonant surface scattering processes can arise instead of the non-resonant ones in the nondegenerate regime. [55,64]Compared to non-resonant surface scattering processes, the resonant ones are nontrivial.Along with the resonant scattering processes, a number of trivial non-resonant scattering processes occur in other scattering paths like the nondegenerate regime.To take the two scattering processes into account, the n dfold quasi-degenerate perturbed modes in weakly deformed microdisk cavities can be assumed to be dominated by n d -fold quasidegenerate unperturbed modes (m i ,l i ) (i = 1, 2, …n d , where n d is the degree of degeneracy) with angular momenta mixed by n d dominant ones and infinite minor ones.Note that the nondegenerate regime is incorporated by the assumption, i.e., the case of n d = 1.Consequently, the unknown coefficients a q , b q , and the perturbed frequency x = kR in Equation (10) are expanded as the following power series of (2) where m i (i = 1, 2, …n d ) and p are the dominant and minor angular momenta for the modes, respectively; a m i and a p are the corresponding dominant and minor angular momentum components of the wavefunction inside the microcavity, respectively.In Equation (14), a m i is only corrected up to the  first-power term, as the zero-power one  m i ∼  0 needs to be determined within the theory.In contrast to the expansion Equation of a p , a m i is generally dominant, reflecting that a m i and a p are respectively the dominant and minor angular momentum components.The vanishing  first-power terms of b m i and b p can be deduced from inserting Equation (11b) into Equation (11a).The multiple expansions of x at slightly different quasi-degenerate unper-turbed frequencies x (0) m i (labeled by (m i ,l i ), i = 1, 2, …n d ) are exploited for the present perturbation theory, replacing a single expansion at a fixed frequency in the originally developed first-order treatment. [42,44,54,57]With the expansion schemes of x, the function S m i (x) that emerges in Equation ( 13) can be conveniently calculated by the following expanded series at x + Δx (1) where Here Equation (16a) is obtained, since x (0) m i as the  zero-power term of x is a solution of S m i (x) = 0. Using Equation (12a) and (12b), one can readily get Equation (16b) and (16c).

First-Order Theory
Finally, after extracting the terms of Equation (11a) and (11b) with the same power of  ( 1 ,  2 ) and performing Fourier transformation in both left-and right-hand sides, the equations for the unknown quantities a m i , a p , b m i , b p , and x are determined.In these equations, the cavity shape is encoded in the Fourier harmonics of the deformation function f() and its square where c i = 1 for i = 0, otherwise c i = 2 for i ≠ 0.Here A ij and B ij are referred to as the coupling integrals. [56]From the first-power of , the boundary condition (11) leads to where x (1) = x (0) m i is the dimensionless frequency within the first-order perturbation theory.
The first-order treatment as Equation (18a) ((18b)) indicates that the cavity boundary deformation described by A m i m j and A pm j can induce the conversions m j → m i (p) between different angular momenta.The conversions of angular momentum components are considered as the consequence of first-order surface scattering processes. [55,64]Conversely, the scattering processes can be well analyzed by the angular momentum components (spectra) obtained from Equation (18).When the appropriate deformation is introduced to microdisk cavities, optical properties of modes such as the complex frequency x (1) and the angular momentum components  m i ( p ) can be tuned via controlling the first-order surface scattering processes m j → m i (p).
The significant difference between Equations (18a) and (18b) manifests a substantial distinction between resonant and nonresonant scattering processes.The scattering process m i → m j that arises between the quasi-degenerate modes (m i , l i ) and (m j , l j ) belongs to the resonant scattering due to high spectral overlap or aligned frequencies, whereas the scattering processes m i → p are non-resonant due to low spectral overlap or different frequencies.Thus, the 1-order resonant and non-resonant scattering processes are described by Equations (18a) and (18b), respectively.As the key factor (1/S p ) of  p in Equation (18b) is small due to the lower spectral overlap, the non-resonant scattering is weak and insensitive to change of cavity boundary shape compared to the resonant one in general.Within the originally developed degenerate perturbation theory, the non-resonant scattering and the resulting first-order corrected component  p in Equation (18b) are not taken into account. [54]Here we retain the nonresonant scattering to derive a rigorous high-order perturbation theory incorporating both resonant and non-resonant scattering processes.However, only the impact of first-order resonant scattering processes on the dimensionless frequency and the dominant angular momentum components are considered within the first-order theory.Thus, the first-order theory primarily describes first-order resonant scattering processes.
Furthermore, Equation (18a) reveals that the quasi-degenerate modes (m i , l i ) and (m j , l j ) can be coupled by the first-order resonant scattering m i → m j .According to the treatment of the originally developed perturbation theory, [42] the coupling can be explicitly unveiled by rewriting Equation (18a) as an eigenvalue equation with an effective non-Hermitian Hamiltonian Equations ( 19) and (18a) are approximately equivalent to the ones within the originally developed degenerate perturbation theory, [42,54] provided that the dominant quasi-degenerate unperturbed modes labeled by (m i ,l i ) (i = 1, 2, …n d ) that are considered in the theory are all WGMs satisfying |Im(x In other words, the originally developed degenerate perturbation theory is only the first-order form of the generalized theory presented here.

Second-Order Theory
From the treatment of the second-power terms of  in the boundary conditions (11a) and (11b), we get the equations of the secondorder perturbation theory ) where m i is the second-order corrected dimensionless frequency x.For brevity, we respectively shorten the first-and second-order perturbation theory shown by Equations ( 18) and ( 21) to the 1PT and 2PT, in the following.
Equations ( 21) and ( 18) reveal that the multiple conversions of angular momenta such as m i → q → m j and m i → q → p (i, j = 1, 2, …n d ) can arise, as long as the appropriate deformation is introduced to cavity boundary.The conversion m i → q → m j (m i → q → p) can be regarded as the consequence of second-order surface scattering processes or equivalently two consecutive firstorder processes m i → q and q → m j (q → p). [64,65]The second-order surface scattering in turn can also be analyzed by the angular momentum components obtained from Equation ( 21), e.g,  m i in Equation (21a) can be employed for estimating the secondorder scattering contribution to the angular momentum m i .According to the 2PT, the second-order scattering processes might also be exploited and manipulated to alter optical properties of modes such as complex frequency and angular momentum components.Furthermore, the coupling between quasi-degenerate modes (m i , l i ) and (m j , l j ) may be modified by the second-order scattering processes m i → q → m j .
In addition to Equation ( 18) and ( 21) obtained from boundary conditions, a universal normalization condition is required for the unknown quantities (a m i , a p , b m i , b p , and x) and imposed on the dominant angular momentum components a m i as follows, When the perturbation theory is applicable, the different order corrections of a m i satisfy Due to the arbitrary nature of phase for modes, the argument  m 1 = arg(a m 1 ) can be specified and fixed as  m 1 = 0, i.e., arg ( Therefore, the condition ( 22) is approximately written as Combining Equation (18a) with the constraint condition (23a), x (1) and  m j are given.Here, these obtained angular momentum components  m j has to be replaced by  m j ∕exp[i m 1 ] to fulfill the specified phase, if  m 1 = arg( m 1 ) ≠ 0.Then, the  firstpower term of a p can be calculated by using x (1) and  m j in Equation (18b) within the 1PT.Subsequently, with  m j ,  p , and x (1) from the first-order treatment, x (2) and the  first-power terms of a m i are obtained by solving the linear equations consisting of Equation (21a) and the constraint condition (23b) within the 2PT.At last, the  second-power terms of a p and b p can be directly computed using Equation (21b) and (21c), respectively.
Note that we do not restrict the degree of degeneracy to n d > 1 for the perturbation theory presented here.Hence, the scenario of n d = 1 corresponding to the nondegenerate regime can be described by the 1PT and 2PT.When n d = 1, the constraint conditions (22) and ( 23) reduce to  m 1 = 1 and  m 1 = 0.In the limiting case, the 1PT and 2PT presented here can indeed be simplified as the 1-order and 2-order nondegenerate perturbation theory [54] using the relation 0]65] Frequencies, decay rates, and Q factors of modes can be given by the dimensionless frequencies x (1) and x (2) .The intracavity field distribution of modes is respectively calculated within the 1PT and 2PT by (1) r∕R ) J m i (nx (1) ) cos and In comparison to the intracavity field, far-field pattern of modes is a more crucial characteristics of modes for microcavity based lasers [11,13] and sensors. [68,69]By using the asymptotic behavior of Hankel functions, the outgoing wavefunction (10b) at large distance (r → ∞) becomes Here the profile F() describes the far-field pattern of modes depending on azimuth angle .Within the 1PT and 2PT, F() is respectively computed by and

Criterion Function for Quasi-Degeneracy
In the previous subsection, we derived the generalized perturbation theory, within which both the nondegenerate ( n d = 1) and degenerate (n d > 1) regimes can be described.However, a criterion condition has not be presented to distinguish between the nondegenerate ( n d = 1) and degenerate (n d > 1) regimes.Indeed, the criterion is determined while deriving the 2PT.The function S m 1 (x) is expanded up to its second-order terms as Equation (15) shows.The expansion implies that the third-order term of S m 1 (x) should be much smaller than the second-order one as follows, When the two unperturbed modes (m 1 ,l 1 ) and (m 2 ,l 2 ) are quasidegenerate, the inequality (27) m 2 are respectively the dimensionless frequencies of the two un-perturbed modes (m 1 ,l 1 ) and (m 2 ,l 2 ).Therefore, the criterion is rewritten as where f c (m 1 ,l 1 ; m 2 ,l 2 ) is the criterion function defined as Here the criterion ( 28) can be used to distinguish the nondegenerate ( n d = 1) and degenerate (n d > 1) regimes.When the inequality ( 28) is fulfilled, the modes are at least twofold quasidegenerate.When there is no another mode along with the mode (m 1 ,l 1 ) to satisfy the inequality (28), the modes are nondegenerate.For n d -fold quasi-degenerate modes, the arbitrary two dominant unperturbed modes (m i ,l i ) and (m j ,l j ) (i, For convenience, the n d -fold quasi-degenerate modes are further labelled by the dominant mode number (m 1 , l 1 ; m 2 , l 2 ; … m n d , l n d ) as the arguments of the criterion function f c in the following.In the limiting case of n d = 1, the modes are labelled by (m 1 ,l 1 ) as the convention in the nondegenerate regime.

Applications of Perturbation Theory
In this Section we apply the generalized perturbation theory to two generic classes of deformed microdisk cavities to verify its validity.In Section 3.1, we first present several examples of twofold and threefold quasi-degenerate unperturbed modes.In Section 3.2 and 3.3, we compare optical properties of these quasidegenerate modes predicted by the perturbation theory with the numerical finite element method (FEM) using Comsol Multiphysics 5.5 for the microdisk cavities with local and global boundary deformation, respectively.

Quasi-Degenerate Unperturbed Modes in A Circular Disk Cavity
To verify the perturbation theory presented here, we apply the theory to the more generalized scenario, where resonant and nonresonant surface scattering processes coexist.Resonant surface scattering processes occur, when quasi-degeneracy emerges for microdisk cavities.Thus, quasi-degenerate unperturbed modes in a circular microdisk cavity first need to be determined.To search for twofold quasi-degenerate unperturbed modes, one can employ the function of degeneracy defined as, [43] Δ Here x (0) m 2 as solutions of Equation (6a) are the dimensionless frequencies of unperturbed modes respectively labelled by (m 1 ,l 1 ) and (m 2 ,l 2 ).The function of degeneracy depends on the effective refraction index of cavities, the azimuthal mode number (m 1 and m 2 ), and the radial mode number (l 1 and l 2 ).Peaks of Δ −1 indicate quasi-degeneracy.

Microdisk Cavity with Local Boundary Deformation
To demonstrate the validity of the perturbation theory, we apply the theory to optical properties of quasi-degenerate modes in microdisk cavities with local boundary deformation.Among the class of microcavities, locally notched cavities are widely explored for directional light emission, [16,70] lifting degeneracies, [70] light outcoupling, [71] and laser mode selection. [31]The boundary of the microdisk cavity with local notches can be defined as [57,58] where N  ,   ,   , and   represent the number, the azimuthal angle, the depth, and the width of notches, respectively.Here  ∈ Z ensures the 2-periodicity of the boundary function r b ().For a local deformation   ≪ 1, the sum from  = − 1 to  = 1 is a enough sufficient approximation of the one ( − ∞ ≤  ≤ ∞) in Equation (35) for practical numerical computation. [57]To guarantee the mirror-reflection symmetry of the boundary function, the microdisk cavities notched at symmetric azimuthal angles with identical sizes are considered here.The quasi-degenerate modes simulated here are the ones with the dominant mode number (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) shown in Equation ( 32), (33), and (34), respectively.For the modes, both the 1PT and 2PT are utilized for simulation.Since the 1PT is approximately equivalent to the originally developed degenerate perturbation theory, the comparison of the 1PT and 2PT can unfold the distinction between the original and generalized theories.Figure 2 and 3 show the dependence of real and imaginary parts of the perturbed dimensionless frequency x on notches for the foregoing quasi-degenerate modes, when two notches as perturbations are introduced to the boundary of a microdisk cavity.The notches scatter unperturbed modes and alter optical properties of the modes.As the width or the depth of the notches increases, the real and imaginary parts of x deviate from the unperturbed ones.The general trend of the dimensionless frequency is captured by both the 1PT and 2PT.However, better agreement with the numerical method is observed for the 2PT compared to the 1PT.Note that the boundary functions of the deformed microdisk cavities calculated in Figures 2 and 3 (c,d), and (e,f), the dominant mode numbers of modes are (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) as shown in Equation ( 32), (33), and (34), respectively.The boundary of the microdisk cavity is defined by Equation ( 35  As shown in Figure 4 and 5, the intracavity intensity distribution and the far-field pattern of the aforementioned modes are radically different from unperturbed ones or the perturbed ones in the nondegenerate regime described by the original nondegenerate perturbation theory, despite the weak local boundary deformation of the microdisk cavities satisfying the inequation (8) as s n a(nRe(k)) 2 /8 < 10 −4 .As illustrated in Figure 6, the angular momentum spectra of the perturbed modes indicate that the modes are hybridized by multiple unperturbed modes rather than dominated by an individual one in the nondegener-ate regime or the unperturbed case.The hybridization implies the efficient coupling of dominate quasi-degenerate unperturbed modes labelled by (m i ,l i ) and (m j ,l j ).Since the dimensionless frequency in Figure 2 and 3, the field distributions in Figures 4  and 5, as well as the angular momentum spectra in Figure 6 are approximately reproduced by the 1PT, the differences between perturbed modes and unperturbed modes (or the modes described by the original nondegenerate perturbation theory) are primarily attributed to the first-order resonant scattering processes m i → m j in the quasi-degenerate regime introduced by the notches described by the 1PT.Nevertheless, the 2PT agrees better with the numerical results compared to the 1PT.The comparison  , (c,d), and (e,f), the dominant mode numbers of modes are (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) as shown in Equation ( 32), (33), and (34), respectively.The boundary of the microdisk cavity is defined by Equation (35), in which two notches ( N  = 2) of identical sizes (  1 =  2 ,  1 =  2 ) are introduced at the symmetric angle (  2 = −  1 ).Filled square, circle, and triangle points are the numerical results using the FEM, whereas dashed and solid curves respectively depict the results obtained from the 1PT and the 2PT.
between the 2PT and the 1PT (or equivalent the original degenerate perturbation theory) in terms of dimensionless frequency, field distributions, and angular momentum spectra reveals that high-order surface scattering processes cannot be thoroughly neglected in the degenerate regime, even though the local boundary deformation is weak.

Microdisk Cavity with Global Boundary Deformation
In this subsection, the present perturbation theory is applied to microdisk cavities with smooth global deformation for further testing its validity.In contrast with the previous microcavity with local deformations such as subwavelength scaled notches, the globally deformed microcavity can relax the fabrication accuracy of cavity boundary.Therefore, the microdisk cavity with smooth global deformation are more intensively explored. [12]enerally, the boundary of the microcavity is defined by a Fourier series with finite terms in the polar coordinates as where R is the unperturbed radius of the microdisk cavity, and   is the dimensionless strength of the th harmonic boundary deformation.As a sum of cosine functions, the cavity bound- ary function (36) satisfies the mirror-reflection symmetry with respect to  = 0.The limaçon [15] and quadrupole [23] cavities respectively defined as r b ( belong to this class of microcavities. To test the performance of the perturbation theory in a more generalized scenario, we consider the modes with different angular momenta coupled via hybrid resonant and non-resonant surface scattering processes in the microcavity with smooth global deformation.Thus, the quasi-degenerate unperturbed modes shown in Equation ( 32), ( 33), (34) in globally deformed microdisk cavities are studied by using both the 1PT and 2PT.According to the 1PT, the Hamiltonian (20) shows that two quasi-degenerate unperturbed modes with angular momenta m i and m j can be coupled via the first-order resonant surface scattering m i → m j by introducing a harmonic modulation  m i −m j cos[(m i − m j )] to the cavity boundary.Meanwhile, Equations ( 18) and (21) show that the second-order processes that scatter light from m i to m j through an intermediate p 0 such as m i → p 0 → m j can form a different coupling path with the harmonic boundary perturbations The second-order processes are non-resonant, provided that the frequency of the intermediate WGM with p 0 is not aligned to the quasi-degenerate one with m i or m j .Therefore, to create the coupling paths between quasi-degenerate unperturbed modes via the hybrid resonant and non-resonant surface scattering processes, the deformation function of the cavity can be designed as where m 1 and m 2 (m 1 , m 2 , and m 3 ) are the dominant angular momenta of twofold (threefold) quasi-degenerate modes; p 0 is an intermediate angular momentum.Specifically, p 0 = 11, 8, and 40 are respectively adopted for the quasi-degenerate unperturbed modes shown in Equations ( 32), (33), and (34), to avoid the alignment of frequency.
As shown by the angular momentum spectra in Figure 7, the perturbed degenerate modes are decomposed into multiple angular momentum components with leading amplitudes in the degenerate regime rather than an individual dominant one in a nondegenerate or unperturbed case, when the global deformation function (37) or (38) of small amplitudes |  | ≪ 1 is imposed to a circular microdisk cavity.As the angular momentum spectra can be approximately captured by the 1PT, the efficient coupling  , (c,d), and (e,g), the dominant mode numbers of modes are (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) as shown in Equation ( 32), (33), and (34), respectively.In (a-c), the boundary of the microcavity is defined by Equation (35) with two notches ( N  = 2) of identical size (  1 =  2 ,  1 =  2 ) at the symmetric angle (  2 = −  1 ), respectively corresponding to the cases marked by black dotted lines in Figure 2a,c,e.
of modes with different angular momenta is naturally explained by nontrivial first-order resonant surface scattering processes described by the 1PT in the degenerate regime.Due to the resonant scattering processes, the dimensionless frequencies of the perturbed modes depart from the unperturbed values shown in Equations ( 32), (33), and ( 34), e.g., the imaginary parts of the dimensionless frequencies shift beyond one order of magnitude for some modes.Furthermore, as a consequence of the resonant scattering, a significant difference is manifested in the intracavity intensity in terms of positions and shapes of nodes compared to the unperturbed modes, as shown in Figure 8.
To demonstrate the distinct effects of resonant and nonresonant surface scattering processes on the coupling of modes with different angular momenta, the dependence of the dominant angular momentum components a m i on the harmonic strengths of the microcavity boundary deformations (37) and ( 38) is calculated using the present perturbation theory, as shown in Figures 9-11.When the harmonic strength   of the cavity deformation functions is varied, Figures 9-11 illustrate that finite and non-vanishing amplitude ratios |a m i ∕a m j | of the dominant angular momentum components are roughly captured by the 1PT, indicating that first-order resonant scattering processes are more pronounced than high-order ones including non-resonant scattering processes for the perturbed modes in the microcavity with small deformation.Nevertheless, the 2PT is in better agreement with the numerical method in comparison to the 1PT (or equivalent the original degenerate perturbation theory) with modest deviation.The better agreement between the 2PT and numerical results pinpoints that the coupling of modes with different angular momenta originates from both first-and second-order surface scattering processes described by the 2PT, demonstrating the validity of the theory.The difference between the 2PT and 1PT (or equivalent the original degenerate perturbation theory)  (a-c), the dominant mode numbers of modes are (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) as shown in Equation ( 32), (33), and (34), respectively.The boundary functions of the microcavities in (a-c) correspond to the ones in Figure 7a-g, respectively.
results can be attributed as the second-order surface scattering processes, since the 1PT only describes the first-order one.
Figures 9-11 show that the contribution from the second-order surface scattering processes cannot be ignored for the weakly deformed microcavities, even though the first-order resonant scattering is dominant due to quasi-degeneracy.Particularly, the difference between the 1PT and 2PT results unveil that the coupling between m i and m j via the second-order surface scattering processes depends on the harmonic strengths  m i −p 0 and  p 0 −m j , implying the existence of the second-order non-resonant scattering m i → p 0 → m j as expected.Therefore, the coupling between modes with different angular momenta is accomplished via hybrid resonant and non-resonant surface scattering processes by introducing the appropriate harmonic perturbations to microdisk boundary, in the degenerate regime.The dependence on the harmonic strengths  m i −p 0 and  p 0 −m j cannot be ex-plained by individual first-order resonant scattering processes, as its trend cannot be reproduced by the 1PT (or equivalent the original degenerate perturbation theory).Compared to the coupling depending on the harmonic strengths  m i −p 0 and  p 0 −m j , the dependence of the coupling via resonant scattering processes on  m i −m j shown in Figures 9a, 10a, and 11a is captured by the 1PT in terms of its trend at least, even though the 2PT agrees with the numerical results better.In contrast to the dependence of the coupling via resonant scattering on  m i −m j , the coupling via the second-order non-resonant scattering is less sensitive to the changes of  m i −p 0 and  p 0 −m j .
Apart from the preceding angular momentum spectra and the coupling of modes, complex frequencies of quasi-degenerate modes in a weakly deformed microdisk cavity are dependent on both resonant and non-resonant scattering processes, according to the present perturbation theory.Within the 1PT, Figure 9. Simulated amplitude ratio and phase difference between two dominant angular momentum components of modes versus harmonic strengths   of cavity boundary for twofold quasi-degenerate modes in a globally deformed microdisk cavity.The dominant mode number of modes is (12,1;6,3) as shown in Equation (32).The boundary of the deformed cavity is by the deformation function (37) with m 1 = 12, m 2 = 6, and p 0 = 11, i.e., f()/R =  1 cos () +  5 cos (5) +  6 cos (6).The effective index of the cavity is n = 3.4.Filled (square or circle) points are the numerical results obtained by using the FEM, whereas solid and dashed curves respectively depict the results calculated by the 2PT and the 1PT.
the diagonal elements of the Hamiltonian (20) indicate that the first-order corrected complex frequency x (1) as the eigenvalue of the Hamiltonian can be directly controlled by the firstorder resonant scattering process m i ↔ − m i , while imposing the harmonic perturbations  2m i cos(2m i ) to microdisk boundary, where m i is one of the dominant angular momenta.Considering second-order scattering processes within the 2PT, Equation (21) shows that the second-order scattering process m i → p 0 → m i assisted by an intermediate angular momentum p 0 can exist and consequently might modify complex frequencies of quasidegenerate modes for a microdisk cavity with the harmonic perturbation  p 0 −m i cos[(p 0 − m i )].The second-order processes are non-resonant, when the frequency of the intermediate WGM with p 0 is not aligned to the one with m i .Thus, to confirm the effects of the two generic scattering processes on complex fre-quencies of quasi-degenerate modes, the deformation functions of microdisk cavities with the specified harmonic perturbations is and where (m 1 , l 1 ;m 2 , l 2 ) as well as (m 1 , l 1 ;m 2 , l 2 ;m 3 , l 3 ) in Equations (39) and ( 40) are the dominant mode numbers of twofold and threefold degenerate modes, respectively.Specifically, p 0 = 11, 8, and 40 are respectively chosen for the quasi-degenerate unperturbed modes shown in Equations ( 32), (33), and (34), to circumvent the alignment of frequency.Note that the secondorder non-resonant scattering m j → p 0 → m i (j ≠ i) is inhibited for the deformation functions ( 39) and ( 40) due to the absence of the boundary perturbation  m j −p 0 cos[(m j − p 0 )], compared to the deformation functions (37) and (38).
Figure 12 shows the dependence of the dimensionless frequency x of quasi-degenerate modes in the microdisk cavity with the deformation functions ( 39) and ( 40) on the harmonic strengths of cavity boundary.The numerical results of complex frequencies can be accurately reproduced by the 2PT, verifying the validity of the perturbation theory.In comparison to the 2PT, the 1PT (or equivalent the original degenerate perturbation theory) can only predict approximate results, in particular the dependence of the complex frequency on  2m i resulting from the first-order resonant scattering process m i ↔ − m i .The deviation of the 1PT (or equivalent the original degenerate perturbation theory) from the numerical method is visible.Owe to the outperformance of the 2PT, it can be inferred that the deviation mainly originates from second-order scattering processes described by the 2PT.As illustrated in Figure 12, the difference between the 2PT and 1PT (or equivalent the original degenerate perturbation theory) depends on the harmonic strengths  p 0 −m 2 and  p 0 −m 3 of deformation functions, implying the occurrence of second-order non-resonant scattering processes (e.g., m i → p 0 → m i ).The dependence unveils that the complex frequencies of quasi-Figure 11.Simulated amplitude ratio and phase difference between three dominant angular momentum components of modes versus harmonic strengths of cavity boundary for threefold quasi-degenerate modes in a globally deformed microdisk cavity.The dominant mode number of modes is (46, 1; 41, 2; 37, 3) as shown in Equation (34).The boundary of the deformed cavity is defined by the deformation function (38) with m 1 = 46, m 2 = 41, m 3 = 37, and p 0 = 40, i.e., f()/R =  1 cos () +  3 cos (3) +  4 cos (4) +  5 cos (5) +  9 cos (9).The effective index of the cavity is n = 1.5.For the three quasi-degenerate perturbed modes in the microcavity, the one with an intermediate Q factor labelled as Mode 2 is shown here.Filled (square or circle) points are the numerical results obtained by using the FEM, whereas solid and dashed curves respectively depict the results calculated by the 2PT and the 1PT.
degenerate modes in the microcavity can be tuned through controlling high-order non-resonant scattering processes by tailoring relevant harmonic perturbations of cavity boundary, in addition to exploiting first-order resonant scattering.

Robust Exceptional Points
In this section, we propose a scheme of exceptional points (EPs) of order 2, which are generated in a generalized scenario where resonant and non-resonant surface scattering processes are both exploited in a deformed microdisk cavity, in contrast to the EPs solely via first-order resonant scattering. [42,44]In Section 4.1, the EP via first-order resonant scattering is briefly reviewed.Then, the EPs via hybrid resonant and non-resonant scattering processes are introduced by using the present 2PT in Section 4.2.Finally, the effects of surface roughness on the two classes of EPs are compared in Section 4.3.
In general, EPs are created by coupled modes in optical microcavity systems, where complex frequencies match with coupling strength of modes. [12,72]For twofold quasi-degenerate modes (m 1 ,l 1 ; m 2 ,l 2 ) in a weakly deformed microdisk cavity, the complex frequencies and coupling of modes are respectively described by the diagonal and off-diagonal elements in the Hamiltonian (20), when only first-order resonant surface scattering processes are considered.Emerging in the diagonal and off-diagonal elements in the Hamiltonian (20), A m i m i and A m i m j (i ≠ j) depend on the har-Figure 12. Dimensionless complex frequency x of quasi-degenerate modes versus boundary harmonic strengths  v of a globally deformed microdisk cavity.In (a-d), (e-h), and (i-n), the dominant mode numbers of modes are (12,1;6,3), (9,1;6,2), and (46,1;41,2;37,3) as shown in Equation ( 32), (33), and (34), respectively.The deformation functions of the deformed cavities defined by Equation ( 39) and ( 40) are (a-d) f()/R =  5 cos (5) +  6 cos (6) +  24 cos (24), (e-h) f()/R =  2 cos (2) +  3 cos (3) +  18 cos (18), and (i-n) f()/R =  3 cos (3) +  4 cos (4) +  5 cos (5) +  9 cos (9) +  74 cos (74), respectively.Filled square and circle points are the numerical results using the FEM, whereas solid and dashed curves respectively depict the results from the 2PT and the 1PT.monic perturbations  2m i cos(2m i ) and  m i −m j cos[(m i − m j )] introduced to boundary shapes of microdisk cavities, respectively.Consequently, the complex frequencies and coupling of modes can be respectively controlled through tuning the first-order resonant surface scattering processes m i ↔ − m i and m i ↔m j by designing the strengths  2m i and  m i −m j of the harmonic perturbations.Moreover, as revealed in the previous section, the complex frequencies and coupling of quasi-degenerate modes can be manipulated via the second-order non-resonant scattering processes m i → p 0 → m i and m i → p 0 → m j intermediated by another angular momentum p 0 provided that the frequency of the WGMs with p 0 is not aligned with the one with m i or m j , when the harmonic perturbations  m i −p 0 cos[(m i − p 0 )] and  p 0 −m j cos[(p 0 − m j )] are imposed.In the case of twofold quasi-degenerate modes (m 1 ,l 1 ; m 2 ,l 2 )(m 1 > m 2 ), to adjust the complex frequencies and coupling of modes for generating the EP of order 2 via the hybrid resonant and non-resonant surface scattering processes, we synthesize the mentioned harmonic perturbations and specify a family of microdisk cavities with the deformation function Here p 0 = 11 and p 0 = 8 are chosen to avoid the alignment of frequency for the twofold quasi-degenerate unperturbed modes (12, 1; 6, 3) and (9, 1; 6, 2) shown in Equation ( 32) and (33), respectively.
According to the 1PT, the Hamiltonian (20) for the deformation function (41) reads provided that the special cases of m 1 = 3m 2 , m 1 = p 0 + 2m 2 , or p 0 = 3m 2 are excluded.Here x (0) m 2 are the dimensionless frequencies of the unperturbed modes (m 1 ,l 1 ) and (m 2 ,l 2 ), respectively.The diagonal and off-diagonal elements in the non-Hermitian Hamiltonian Equation ( 42) confirm that complex frequencies and coupling of modes can be controlled via harnessing the first-order resonant surface scattering processes m 1 ↔ − m 1 and m 1 ↔m 2 by designing  2m 1 and  m 1 −m 2 , respectively.Inserting Equation ( 42) into the characteristic Equation det( Ĥ − x (1) Î) = 0, one gets the 1-order corrected dimensionless frequency x (1) as the eigenvalue of the Hamiltonian ( 42) where m 2 , and x . According to the 1PT, the 1-order corrected dominant angular momentum compo-nents  m 1 and  m 2 are given as the eigenvector of the Hamiltonian (42) where c n ± is the normalization constant.For the microdisk cavity with the deformation function (41), the impacts of the nonresonant scattering processes m i → p 0 → m i and m i → p 0 → m j (i, j = 1, 2) on the frequencies and angular momentum components of the quasi-degenerate modes are not taken into account within the 1PT.To precisely calculate the modes under the coexistence of resonant and non-resonant scattering processes, the 2PT has to be used by substituting the eigenvalue (43) and eigenvector (44) into Equation (21a), (21b), and (21c).

Resonant-Scattering-Based EPs
For the microdisk cavity with the deformation function ( 41), the eigenvalue (43) and eigenvector (44) signify that EPs can be achieved solely through the first-order resonant surface scattering processes m 1 ↔ − m 1 and m 1 ↔m 2 by tuning the relevant deformation strengths  2m 1 and  m 1 −m 2 .To avoid the non-resonant scattering processes m i → p 0 → m i and m i → p 0 → m j intermediated by the angular momentum p 0 , we impose  m 1 −p 0 = 0 and  p 0 −m 2 = 0.When  m 1 −p 0 = 0 and  p 0 −m 2 = 0, the deformation function ( 41) reduces to Here Equation ( 45) is a simplified form of the deformation function (41).For the microdisk cavity with the deformation function (45), one can infer from Equation ( 43) and ( 44) that an EP emerges when the square root term x (0) sq is vanishing, as not only the eigenvalues but also the eigenvectors of the Hamiltonian (42) coalesce.To attain x (0) sq = 0, the deformation strengths  2m 1 and  m 1 −m 2 of the cavity can be tuned to where ] are the Q factors of the dominant unperturbed modes (m 1 ,l 1 ) and (m 2 ,l 2 ), respectively.For the unperturbed mode pairs (32)  Here the EP of  6,EP > 0 ( 3,EP > 0) in the configuration (48a) ((48b)) is marked as the "EP+", whereas the opposing case is marked as the "EP-".With the value of  2m 1 , EP and  m 1 −m 2 ,EP , the eigenvector ( m 1 ,  m 2 ) T at the EP is where sgn(x) represents the sign function.5][76][77][78] The chirality is determined by the sign of  m 1 −m 2 ,EP .As the EP is formed via first-order resonant scattering processes within the 1PT, it is referred to as resonantscattering-based EPs (RSEPs) in the following.The EP has been introduced. [42]n the vicinity of  2m 1 , EP and  m 1 −m 2 ,EP , as the deformation parameters  2m 1 or  m 1 −m 2 is individually varied, the eigenvalue x (1) ± respectively exhibits the square root topology as follows, x (1) x (1) where the arguments Near the EP, avoided resonance crossing [12] (level repulsion) [79][80][81] can also arise via the resonant surface scattering processes.Generally, EPs are considered as a critical point between weak and strong coupling regimes of avoided resonance crossing. [12]Since the off-diagonal elements in the Hamiltonian (42) describing the coupling of modes is propor- imply the weak and strong coupling regimes, respectively.As illustrated in Figure 13, the two regimes of avoided resonance crossing are observed as expected, i.e., a crossing and an anticrossing respectively occur in the real (imaginary) and imaginary (real) parts of the eigenvalue x Accompanied by the crossing and anticrossing of the complex frequencies, Figure 14 shows that the eigenvectors become the superpositions of two modes with different angular momenta, i.e., the hybridization of modes.The phases of angular momentum components swap in the weak coupling regime, whereas the amplitudes exchange in the strong coupling regime.
Although the numerical results of dimensionless frequency and angular momentum components in Figure 13 and 14 can be captured by the 1PT with only slight deviation, the 2PT are in better agreement in comparison to the 1PT.The difference between the 1PT and 2PT results from multiple second-order scattering processes, which are not taken into account by the 1PT.For concreteness, the deformation function (45) with only two harmonic perturbations is simplified and designed to avoid the second-order non-resonant scattering processes m i → p 0 → m i and m i → p 0 → m j (i, j ∈ {1, 2}; i ≠ j) via the harmonic perturbations  m i −p 0 cos[(m i − p 0 )] and  p 0 −m j cos[(p 0 − m j )] with respect to the deformation function (41).Nevertheless, the second-order scattering processes such as and − m 2 → −m 1 → m 1 can occur via the only two harmonic perturbations  m 1 −m 2 cos[(m 1 − m 2 )] and  2m 1 cos(2m 1 ) of the deformation function (45), according to the 2PT.The second-order scattering processes and their impacts are described within the framework of the 2PT rather than the 1PT, i.e., the second-order corrected frequency and angular momentum components (the difference between the 1PT and 2PT) are determined by secondorder scattering processes within the 2PT.Therefore, the difference between the 1PT and 2PT in Figures 13 and 14 originates from the second-order scattering processes induced by the two harmonic perturbations of the deformation function (45).As the second-order scattering processes are dependent on the two harmonic perturbations  m 1 −m 2 cos[(m 1 − m 2 )] and  2m 1 cos(2m 1 ), the difference between the 1PT and 2PT varies along with  2m 1 , as shown in Figures 13 and 14.

Hybrid-Scattering-Based EPs
When  m 1 −p 0 ≠ 0 and  p 0 −m 2 ≠ 0 for the deformation function (41) with an appropriate p 0 avoiding the alignment of frequency, the resulting non-resonant surface scattering processes such as m i ↔p 0 need to be taken into consideration.As revealed in the foregoing section, the non-resonant scattering that can constitute second-order scattering paths has impacts on both complex frequencies and angular momentum components of quasidegenerate modes.Therefore, EPs might be achieved for quasidegenerate modes in a microdisk cavity with the deformation function (41) in the case of the coexistence of non-resonant and resonant surface scattering processes.
To attain the EPs, we first fix the contribution of resonant surface scattering processes by using the constant strengths of the harmonic perturbations  2m i cos(2m i ) and  m i −m j cos[(m i − m j )] in the deformation function (41).Specifically, for the quasidegenerate unperturbed mode pairs (32) and (33),  2m i and  m i −m j are fixed at {  24 = 1.009  24, EP = 0.00229722 , for the mode pair (32) (51a) , for the mode pair (33) (51b) which deviate from  2m 1 , EP and  m 1 −m 2 ,EP in the configuration (48)  predicted for the resonant-scattering-based EPs by using the 1PT.Note that the perturbed modes dominated by the mode pairs (32)  and (33) are merely in the strong coupling regime for the microcavity with the boundary deformation (41) and the configuration (51), even if the non-resonant surface scattering process m i ↔p 0 are ignored for vanishing  m 1 −p 0 and  p 0 −m 2 , as shown in Figure 13 and 14.
Then, the non-resonant scattering processes m i ↔p 0 (i = 1, 2) is harnessed and tuned by choosing appropriate non-vanishing  m 1 −p 0 and  p 0 −m 2 , in order to generate EPs in the parameter space  (c,d) n = 3.2.Filled square and circle points are the numerical results obtained by using the FEM, whereas dashed and solid curves respectively depict the results of x (1) and x (2) calculated by the 1PT and the 2PT.
of  m 1 −p 0 and  p 0 −m 2 for the microcavity with the boundary deformation (41) and the configuration (51).To search the location of the EP in the parameter space of  m 1 −p 0 and  p 0 −m 2 , we employ the auxiliary function measuring nonorthogonality of all normalized eigenvectors defined as [42,44] where ⃗ v i(j) is a vector composed of the dominant angular momentum components a m i (i = 1, 2, 3…n d ) for n d -fold quasi-degenerate modes The vector ( 53) is a generalized version for the eigenvector of Hamiltonian (20), i.e., the vector (53) ⃗ v i can be written as within the 1PT and 2PT, respectively.Within the perturbation theory, the vector ( 53) is normalized due to the requirement of the normalization condition (22).As modes coalesce at EPs, the generalized vectors (54a) and (54b) of all modes at EPs are collinear within the 1PT and 2PT, respectively.For the collinear vectors, g = 0. Therefore, EPs can be determined by searching the minimums of g with g ≈ 0 by using the perturbation theory.41) and the configuration (51).In (a) and (b), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation ( 32) and (33), respectively.The deformation functions of the microcavity defined by Equation ( 41) are (a) f()/R =  1 cos (1) +  5 cos (5) +  6 cos (6) +  24 cos (24) with the configuration (51a) as well as (b) f()/R =  1 cos (1) +  2 cos (2) +  3 cos (3) +  18 cos (18) with the configuration (51b).Here the auxiliary function g is calculated by using the normalized vectors (54b) within the 2PT, to take the impacts of non-resonant scattering processes on modes into consideration.The minima of the auxiliary function g that indicate hybridscattering-based EPs are marked as "HSEP±".(c) and (d) illustrate the ratio g/g 0 of the auxiliary function respectively corresponding to (a) and (b).In (c) and (d), the regions with g ≤ g 0 and g > g 0 are depicted by green and transparent color, respectively.Here g 0 is the auxiliary function value of the corresponding resonant-scattering-based EP calculated by Equation ( 44) and (54a), e.g., g 0 is respectively g 0 ≈ 10 −5.23 and g 0 ≈ 10 −5.34 for the modes in (c) and (d).
Here the cases of ( 5 > 0,  1 > 0) as well as ( 2 > 0,  1 > 0) in the configuration (56) are marked as "HSEP+" as shown in Figure 15a,b, whereas the opposing cases are marked as "HSEP-".The non-vanishing  m 1 −p 0 and  p 0 −m 2 indicate that the nonresonant scattering processes m i ↔p 0 (i = 1, 2) can indeed be exploited as an additional degree of freedom to steer the microcavity system from the strong coupling regime at the origin of Figure 15a,b to the EPs.Therefore, the EPs are generated via hybrid non-resonant and resonant scattering processes in the microcavity.In the following, the EP corresponding to the configuration (56) predicted by the 2PT is referred to as the hybrid-scattering-based EP (HSEP), in contrast to the resonantscattering-based EP at the configuration (48) predicted by the 1PT.
For comparison, the contour map of g in the vicinity of the resonant-scattering-based EP at the configuration ( 48) is also given by using the 1PT, as shown in Figure 16.The minima g 0 of g are respectively g 0 ≈ 10 −5.23 and g 0 ≈ 10 −5.34 in Figure 16a,b, respectively locating at the configurations (48a) and (48b).Compared to g 0 at the resonant-scattering-based EP, the higher nonorthogonality with g ≤ g 0 is achieved in a 2dimensional region with finite area around the hybrid-scatteringbased EP in the space of  m 1 −p 0 and  p 0 −m 2 , as shown by green in Figure 15c,d.The 2-dimensional region of higher nonorthogonality indicates that the hybrid-scattering-based EP can be more robust against the deviation of  m 1 −p 0 and  p 0 −m 2 than the corresponding resonant-scattering-based EP, according to the auxiliary function ratio shown in Figure 15c,d.The enhanced robustness of the hybrid-scattering-based EP in the parameter space is attributed to the introduction of non-resonant surface scattering processes as an additional degree of freedom with respect to the resonant-scattering-based EP.The robustness can also be intuitively explained as the weak response strength of the EP for the perturbations of  m 1 −p 0 and  p 0 −m 2 . [82]he deformation function (41) corresponding to the configuration (56) designed for the hybrid-scattering-based EP is shown  (45).In (a) and (b), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation ( 32) and (33), respectively.The deformation functions of the microcavity defined by Equation ( 45) are (a) f()/R =  6 cos (6) +  24 cos (24), and (b) f()/R =  3 cos (3) +  18 cos (18), respectively.Here the auxiliary function g is calculated by using the normalized vector (54a) within the 1PT.The minima g 0 of the auxiliary function g that indicate the resonant-scattering-based EPs at the configuration (48) are marked as "RSEP±".
in Figure 17a,b.On one hand, the weak deformations fulfill the criterion (8), confirming that the perturbation theory is applicable for predicting the EPs.On the other hand, the current fabrication limitations of 10 −4 order [83,84] can satisfy the accuracy requirement of the microcavity, as the overall dimension-less deformation f()/R in Figure 17a,b is beyond 10 −3 order.For comparison, the deformation function (45) corresponding to the configuration (48) designed for the resonant-scatteringbased EP is illustrated in Figure 17c,d.As shown in Figure 17, the two generic deformation functions for the hybrid-scattering-  To demonstrate that the configuration (56) predicted by the 2PT indeed corresponds to an EP, the stroboscopic evolution of the dimensionless frequency of the microcavity is illustrated in Figure 18, as the position at the configuration ( 56) is encircled in the parameter space of  2m 1 and  m 1 −m 2 .Both the real and imaginary parts of the dimensionless frequency calculated by the 2PT reveal that the characteristic behaviors of EPs of order 2 can be observed, i.e., one closed loop encircling the EPs can only exchange the complex frequencies whereas the two closed loops are able to recover their initial values.Note that the behavior is confirmed by the numerical FEM, which is in good agreement with the 2PT in Figure 18.
Characteristic nonorthogonality of eigenstates at EPs can be observed for the modes in the deformed microdisk cavity with the deformation function (41) and the configuration (56).As the present perturbation theory is an approximate method for solving the modes Equation (1) in weakly deformed microdisk cavities, the configuration (56) predicted by the 2PT might not exactly cor-respond to an EP within the numerical FEM.The modes at the configuration (56) can be distinguished by using the numerical FEM.As shown in Figure 19a,b, the intracavity intensity distributions of the twofold degenerate modes at the configuration (56) calculated by the numerical method are actually similar to each other but slightly different.To quantitatively evaluate the similarity or nonorthogonality between the modes, one can utilize the normalized pairwise overlap [42,44]

S
[ where  i (i = 1,2) represents one wavefunction of degenerate modes.For the degenerate mode pairs calculated by the numerical method in Figure 19a  In addition, the modes at the configuration (56) display the self-orthogonality [85] of EPs.For the modes calculated by the numerical method shown in Figure 19a,b, S[𝜓  * i ,  i ] that respectively satisfies S[ * i ,  i ] < 0.15 and S[ * i ,  i ] < 0.29 is strikingly smaller than unity.The self-orthogonality S[ * i ,  i ] → 0 of the modes also confirms that the configuration (56) predicted by the 2PT for the hybrid-scattering-based EP is close to an EP.
The local features of EPs are manifested for the modes at the configuration (56).Figure 19a,b illustrate that the nonuniform nodes and the indistinct interference pattern in the radial direc-tion arise in the intracavity intensity distribution of the modes, which are similar to the ones at an EP. [42]On the other hand, local vortex structures appear in Poynting vectors of the modes as a demonstration for the spatial local chirality of wave propagation at an EP, [42,43,76] as shown in Figure 20.The local chirality has been explored for unidirectional laser emission of deformed microcavities. [76]

Effect of Surface Roughness on EPs
In the previous subsection, a hybrid-scattering-based EP is introduced for a deformed microdisk cavity in the generalized scenario, where both resonant and non-resonant surface scattering processes are exploited.In contrast to the resonant-scatteringbased EP, the hybrid-scattering-based EP embeds in a higherdimensional parameter space, where more degrees of freedom are available due to the introduction of non-resonant scattering processes.In the additional dimensions associated with non-resonant scattering processes, higher nonorthogonality is achieved by the hybrid-scattering-based EP in the 2-dimenional region with respect to the resonant-scattering-based EP, as shown in Figure 15c,d.As a consequence, the hybrid-scattering-based EPs can be more robust against deviations compared to the resonant-scattering-based EP, provided that the deviations drive the system only along the additional dimensions.
For the hybrid-scattering-based EP in the deformed microdisk cavity with the deformation function (41), the robustness is The relations are fulfilled, even though p 0 is generalized to an integer between m 2 and m 1 rather than p 0 = m 2 + 1.Therefore, the robustness can always be attained against the deviations along the two harmonic components  m 1 −p 0 and  p 0 −m 2 with two small integers (m 1 − p 0 ) and (p 0 − m 2 ) for the hybrid-scattering-based EP in the case of the deformation function (41).
In fact, the fabrication errors of microdisk cavity boundary is primarily characterized by small surface roughness at the sidewall of microdisk cavities, [65] since the current fabrication capability of microdisk cavities can meet the requirement of overall design boundary shape up to the relative size accuracy of 10 −4 order. [83,84]88][89][90][91] The surface roughness is dependent on geometry, material, and fabrication process of microdisk cavities.
For microdisk cavities, the surface roughness can be modeled as a perturbation of cavity boundary [61,86,90,91] r b () = r 0 () + Δr () (58)   where r 0 () is the boundary function of an ideal microdisk cavity with vanishing roughness, representing the overall design boundary shape of the cavity; the 2-periodic random variable Δr as a perturbation represents the surface roughness.Assuming that the surface roughness of a microdisk cavity is a wide-sense stationary stochastic process, the random variable Δr satisfies [61] ⟨Δr ()⟩ = 0 (59) where  is the standard deviation of surface roughness at the sidewall of microdisk cavities, and W( − ′) is the correlation function depending on azimuthal angle.Here the bracket y represents an ensemble average of a random variable y over all realizations of the stochastic process, where ergodicity is required.Hence, r b () has to be distinguished from the mean radius of one specific realization over one loop expressed by  [88] W where  c is the correlation angle.To ensure the 2-periodicity of the correlation function, Equation (61) has to be slightly modified. [61]Nevertheless, the modification and Equation ( 61) are nearly equivalent, when the case discussed here is restricted to the mirror-reflection symmetric microdisk cavities with 0 ≤ , ′ ≤ .
According to Equations ( 59) and ( 60), the harmonic components of the random variable Δr are random variables satisfying [61] ⟨ C q ⟩ = 0 (62) where C q as the qth dimensionless harmonic component of Δr is and W q is the roughness spectrum defined as Here R stands for the mean value of r 0 () over a loop, i.e., R = 1 2 ∫ 2 0 r 0 ()d;  q,p is the Kronecker delta.In the case of the Gaussian correlation function (61), from Equation (65) it follows that From Equations ( 62) and ( 63), the roughness spectrum (66) elucidates that few harmonic components C q with a small integer q (e.g., q ≤ 1/ c ) are statistically dominant for the surface roughness of microdisk cavities described by the Gaussian correlation function (61).For the hybrid-scattering-based EP achieved by the microdisk cavity with the deformation function (41), the harmonic orders (m 1 − p 0 ) and (p 0 − m 2 ) of the deformation function (41) might fulfill m 1 − p 0 ≤ 1/ c and p 0 − m 2 ≤ 1/ c , as  c ≪ 1 in general. [88]Otherwise, one of the relations is satisfied at least.In the present paper, 0.05 ≤  c ≤ 0.15, the two inequalities are satisfied for the calculated two degenerate mode pairs (12,1;6,3) and (9,1;6,2).Consequently, the hybrid-scatteringbased EP can be more robust against the two dominant harmonic components C m 1 −p 0 and C p 0 −m 2 of the random surface roughness by virtue of the robustness achieved in the parameter space of  m 1 −p 0 and  p 0 −m 2 , compared to the resonant-scattering-based EP.When the random surface roughness of the Gaussian correlation function ( 61) is imposed on the microdisk cavities designed for EPs, the hybrid-scattering-based EP might also exhibit more robustness than the resonant-scattering-based EP.
To validate the robustness against surface roughness, we conduct the numerical simulations of modes at hybrid-scatteringbased EPs and resonant-scattering-based EPs in microdisk cavities with identical surface roughness for comparison.To implement the simulation, we treat the boundary function of the microcavity as Equation (58).To achieve hybrid-scattering-based EPs and resonant-scattering-based EPs for the twofold quasidegenerate modes (m 1 , l 1 ; m 2 , l 2 ), the overall design boundary shape r 0 () in Equation ( 58) is respectively specified as the ones of a microdisk cavity with the deformation function (41) and its simplified form (45) Here p 0 = 11 and p 0 = 8 are also chosen to avoid the alignment of frequency for the twofold quasi-degenerate unperturbed modes (12, 1; 6, 3) and (9, 1; 6, 2) shown in Equation ( 32) and (33), respectively.For the modes, Figure 15 and 16 show that the hybrid-scattering-based EPs and resonant-scattering-based EPs are respectively attained in the microdisk cavities with the overall boundary shapes (67a) and (67b) at the configurations ( 56) and (48).Here we only focus on the HSEP+ in Figure 15 and the RSEP+ in Figure 16.On the other hand, the surface roughness Δr is theoretically constructed by the following Fourier series with the independent random variable C q When the case considered here is restricted to the microdisk cavities with mirror-reflection symmetry r b () = r b ( − ) and Δr () = Δr( − ), the roughness can be rewritten as Here C q is real with a vanishing imaginary part, which can be more straightforward given by According to the ensemble averages ( 62) and ( 63), C q can be randomly selected from a uniform distribution between − √ 3W q ∕R and √ 3W q ∕R. [61]The uniform distribution as well as the symmetric lower and upper bounds guarantee Equation (62) and the Kronecker delta in Equation ( 63), whereas the absolute values ( √ 3W q ∕R) of the lower and upper bounds make Equation (63) be satisfied for p = q.For the Gaussian correlation function (61), two randomly selected realizations of the surface roughness Δr() are shown in Figure 21.When ∕R ∼ 10 −4 , the relative surface roughness Δr()/R fluctuates by the order of 10 −4 .
Figure 22 illustrates the intracavity intensity distributions of modes at a hybrid-scattering-based EP and a resonant-scatteringbased EP in microdisk cavities with surface roughness.In contrast to the case without surface roughness shown in Figure 19, subtle variations can be observed for the similarity between the degenerate mode pairs in the microcavities with roughness, implying that the surface roughness leads to the deviation of the microcavity system from the EP.However, a rigorous conclusion is not drawn from a few randomly chosen realizations of the surface roughness, as optical properties of modes might fluctuate significantly for the microdisk cavities with random surface roughness. [61]To quantitatively evaluate the impact of random surface roughness on the modes at EPs, the normalized pairwise overlap S in Equation ( 57) can be utilized.67) with the configurations ( 56) and (48), which in (a-d), respectively correspond to the ones in Figure 19a-d, in order to form the hybrid-scattering-based EPs (the HSEP+ in Figure 15) and resonant-scattering-based EPs (the RSEP+ in Figure 16); the surface roughness Δr() in (a-d) corresponds to the one shown in Figure 21a.56) and (48), in order to achieve the hybrid-scattering-based EP (the HSEP+ in Figure 15) and the resonant-scattering-based EP (the RSEP+ in Figure 16); the surface roughness consisting of 30 randomly picked realizations for a given standard deviation  and a given correlation angle  c is the same for both the two classes of EPs.Filled circle and square points respectively denote an individual result for hybrid-scattering-based EPs and resonant-scatteringbased EPs, while dashed and dotted lines respectively depict the results of averaging over the filled circle and square points.Note that empty points △▽◇☆ mark the results under the realizations of surface roughness shown in Figure 21a-d, respectively.from random surface roughness are so significant that the general rules of modes can be overwhelmed by the fluctuation. [61]espite the significant fluctuation, it is noteworthy that S of the hybrid-scattering-based EP surpasses that of the resonantscattering-based EP for almost every realization of surface roughness with the given standard deviation  and correlation angle  c .The advantage of hybrid-scattering-based EPs over resonantscattering-based EPs is not limited to the given regime of  c and  shown in Figure 23, e.g., such feature can be observed for the EPs in Figure 23a when /R increases to /R = 3 × 10 −4 (not shown).Furthermore, the average value of S for hybrid-scattering-based EPs is higher than that of resonant-scattering-based EPs.As EPs are associated with high nonorthogonality reflected by large S, the comparison implies that the hybrid-scattering-based EP is more robust against random surface roughness at the sidewall of microdisk cavities compared to the resonant-scattering-based EP.
To statistically demonstrate the impact of random surface roughness on the two classes of EPs, the average value S and standard deviation ΔS of the normalized pairwise overlap S as the functions of  are shown in Figure 24.The comparison of the average value S illustrates that the hybrid-scattering-based EP prevails over the resonant-scattering-based EP for the regime of 0 ≤ /R ≤ 0.0004.Combining S with ΔS, the lower and upper bounds of S can be approximately evaluated by ( S − ΔS) and  56) and (48), in order to achieve the hybrid-scattering-based EP (the HSEP+ in Figure 15) and the resonant-scattering-based EP (the RSEP+ in Figure 16); the surface roughness consisting of 30 randomly picked realizations for each  is the same for the two classes of EPs.Filled circle and square points respectively represent the average value of S over the 30 realizations on each  for the hybridscattering-based EPs and resonant-scattering-based EPs, while solid and dotted error bars correspondingly depict the standard deviations of S over the 30 realizations.
( S + ΔS), respectively.The hybrid-scattering-based EP maintains higher nonorthogonality reflected by larger S with respect to the resonant-scattering-based EP, while comparing the lower and upper bounds of S between the two classes of EPs.Therefore, the comparisons confirm that the hybrid-scattering-based EP is more robust against the subwavelength scaled random surface roughness as a primary fabrication error of microdisk cavities compared with the resonant-scattering-based EP.
Note that the average value S and standard deviation ΔS of the hybrid-scattering-based EP degrade in spite of its robustness over the corresponding resonant-scattering-based EP, when the random surface roughness is imposed.The reason for the degradation is that the effect of the other dominant harmonic components C q (q ≤ 1/ c ) of the random surface roughness cannot be eliminated except C m 1 −p 0 and C p 0 −m 2 .Thus, the random surface roughness can still drive the microcavity system away from the hybrid-scattering-based EP.This can also be intuitively explained by using the theory of the response strength of EPs, when the random surface roughness is regarded as an external perturbation for the EP. [82]rthermore, the robustness of the hybrid-scattering-based EP can be further improved and optimized by using the 2PT, particularly when only the modest superiority is achieved for the hybridscattering-based EP with respect to the resonant-scattering-based EP like the cases in Figure 24c,d.On one hand, more optimized hybrid-scattering-based EPs with higher nonorthogonality could be searched and determined in the parameter space of higher dimensions.Specifically, the configuration (56) that is only achieved in the 2D parameter space of  m 1 −p 0 and  p 0 −m 2 in Figure 15 can be substituted with the one obtained in the 4D space of  m 1 −m 2 ,  2m 1 ,  m 1 −p 0 , and  p 0 −m 2 for the microdisk cavities with the deformation function family (41).On the other hand, the robustness of hybrid-scattering-based EPs could be improved by taking advantage of more multiple non-resonant scattering paths, which provide more degrees of freedom.The scheme is applicable for the twofold quasi-degenerate unperturbed modes (m 1 ,l 1 ; m 2 ,l 2 ) with large |m 1 − m 2 |.Specifically, the deformation function (41)

Conclusion
In this paper, we have derived a generalized perturbation theory unifying the originally developed nondegenerate and degenerate treatments for weakly deformed microdisk cavities.We have demonstrated that the two originally developed treatments are only the simplified forms of the present perturbation theory special cases.In contrast to the two originally developed treatments, the present theory is applicable for describing a more generalized scenario, in which both non-resonant and resonant surface scattering coexist.The validity of the derived perturbation theory is verified by the agreement with the numerical FEM at two generic classes of microdisk cavities: locally notched microdisks and globally deformed microdisks.Through the simulation on the two generic classes of microdisk cavities, the generalized perturbation theory unveils the distinct physical pictures from the ones predicted by the originally developed nondegenerate and degenerate treatments.For locally notched microdisk cavities, quasi-degenerate modes are strikingly different from the unperturbed ones or the modes in the nondegenerate regime described by the originally developed nondegenerate perturbation theory, even though the notch size is much smaller than wavelength.The difference lies in the efficient coupling of quasi-degenerate modes induced by resonant surface scattering processes, which can be well analyzed by the present perturbation theory.For smooth globally deformed microdisk cavities, we found that the coupling of quasi-degenerate modes dominated by resonant scattering processes can be modified by non-resonant scattering processes.The modification is not limited to the coupling of modes.The complex frequencies of quasi-degenerate modes can be tuned by controlling non-resonant scattering processes.The dependence on non-resonant scattering processes cannot be described by the originally developed degenerate perturbation theory but the generalized one derived here.Utilizing non-resonant scattering processes as an additional degree of freedom, the hybrid-scattering-based EP is introduced by using the generalized perturbation theory.Compared to the EP solely via resonant scattering, the hybrid-scattering-based EP exhibits higher nonorthogonality by virtue of the additional degree of freedom.Furthermore, the hybrid-scattering-based EP is more robust than the resonant-scattering-based one, when random surface roughness as a fundamental fabrication error of microdisk cavities exists at the sidewall of cavity boundary.
Although the present perturbation theory is only applicable for the microdisk cavities with mirror-reflection symmetry, the perturbation theory provides a paradigm for treating quasi-degenerate modes in microdisk cavities.According to the paradigm, the counterpropagating degenerate unperturbed modes in asymmetric microdisk cavities can be treated, provided that the standing-wave basis here for mirror-reflection symmetric microdisk cavities is transformed into the traveling-wave basis for asymmetric microdisk cavities.Thus, the further extended perturbation theory for asymmetric microdisk cavities is expected.Another interesting direction is to extend the perturbation theory for TE polarization.The two extended theories and the aforementioned optimization of the hybrid-scattering-based EP will be part of our future work.
the mirror-reflection symmetry r b () = r b ( − ), as shown in Figure 2g,h.

Figure 10 .
Figure10.Simulated amplitude ratio and phase difference between two dominant angular momentum components of modes versus harmonic strengths   of cavity boundary for twofold quasi-degenerate modes in a globally deformed microdisk cavity.The dominant mode number of modes is (9, 1; 6, 2) as shown in Equation(33).The boundary of the deformed cavity is defined by the deformation function(37) with m 1 = 9, m 2 = 6, and p 0 = 8, i.e., f()/R =  1 cos () +  2 cos (2) +  3 cos (3).The effective index of the cavity is n = 3.2.Filled (square or circle) points are the numerical results obtained by using the FEM, whereas solid and dashed curves respectively depict the results calculated by the 2PT and the 1PT.

Figure 14 .
Figure14.Amplitude and phase analysis of dominant angular momentum components a m i (i = 1, 2) for twofold quasi-degenerate modes in deformed microdisk cavities in the case of avoided resonance crossing.In (a,b) and (c,d), the modes correspond to the ones shown in Figure13a,b,c,d, respectively; the deformed microcavities in (a,b) and (c,d) correspond to the ones shown in Figure13a,b,c,d, respectively.Filled square and circle points are the numerical results obtained by using the FEM, whereas dashed and solid curves respectively depict the results from the 1PT and the 2PT.

Figure 15 .
Figure 15.The contour maps of a,b) the auxiliary function g and c,d) its ratio value g/g 0 versus  m 1 −p 0 and  p 0 −m 2 for twofold quasi-degenerate modes in a deformed microdisk cavity with the deformation function(41) and the configuration(51).In (a) and (b), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation (32) and(33), respectively.The deformation functions of the microcavity defined by Equation (41) are (a) f()/R =  1 cos (1) +  5 cos (5) +  6 cos (6) +  24 cos (24) with the configuration (51a) as well as (b) f()/R =  1 cos (1) +  2 cos (2) +  3 cos (3) +  18 cos (18) with the configuration (51b).Here the auxiliary function g is calculated by using the normalized vectors (54b) within the 2PT, to take the impacts of non-resonant scattering processes on modes into consideration.The minima of the auxiliary function g that indicate hybridscattering-based EPs are marked as "HSEP±".(c) and (d) illustrate the ratio g/g 0 of the auxiliary function respectively corresponding to (a) and (b).In (c) and (d), the regions with g ≤ g 0 and g > g 0 are depicted by green and transparent color, respectively.Here g 0 is the auxiliary function value of the corresponding resonant-scattering-based EP calculated by Equation (44) and (54a), e.g., g 0 is respectively g 0 ≈ 10 −5.23 and g 0 ≈ 10 −5.34 for the modes in (c) and (d).

Figure 18 .
Figure 18.Evolution of the dimensionless frequency x of a deformed microdisk cavity along the trajectory enclosing the hybrid-scattering-based EP predicted by the 2PT.The hybrid-scattering-based EPs corresponding to the configuration (56) are the "HSEP+" marked in Figure 15.In (a,b) and (c,d), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation (32) and (33), respectively.The deformation functions of the microcavity are (a,b) f()/R =  1 cos (1) +  5 cos (5) +  6 cos (6) +  24 cos (24) as well as (c,d) f()/R =  1 cos (1) +  2 cos (2) +  3 cos (3) +  18 cos (18).Here, the trajectories enclosing the EPs twice are the circles with a,b) the radius  = 0.001 in the parameter space of  6 and  24 as well as c,d) the radius  = 0.0005 in the parameter space of  3 and  18 .For comparison, the results are computed by using both the 2PT and the numerical FEM, as respectively depicted by sphere and cube dots.

Figure 19 .
Figure 19.Intracavity intensity distribution | i | 2 of modes at EPs in deformed microdisk cavities simulated by the numerical FEM.In (a,c) and (b,d), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation (32) and (33), respectively.In (a-d), the boundary functions of the deformed microcavities correspond to the ones shown in Figure17a,b,c,d, respectively.In (a,b) and (c,d), the EPs belong to the hybrid-scattering-based EP designed by using the 2PT and the resonant-scattering-based EP designed by using the 1PT, marked by HSEP+ and RSEP+ in Figure15and 16, respectively.

Figure 20 .
Figure 20.The Poynting vectors ⃗ J ∝ Im( * ∇) of the modes at the hybrid-scattering-based EP calculated by the numerical FEM.In (a) and (b), the modes are the ones in Figure 19a,b, respectively.The EP are the ones marked as "HSEP+" shown in Figure 15.Long black arrows mark dominant directions of Poynting vectors; short blue arrows mark directions of Poynting vectors at local positions, and the length of the blue arrow is proportional to the modulus | ⃗ J| of Poynting vectors; black red shading depicts | ⃗ J| normalized to be unity at maximum.

Figure 21 .
Figure 21.The surface roughness Δr() in the case of the Gaussian correlation function with different roughness standard deviation .a,b) Two randomly selected realizations for the surface roughness with /R = 0.0001,  c = 0.05; c,d) two randomly selected realizations for the surface roughness with /R = 0.0002,  c = 0.05.

Figure 23
illustrates the normalized pairwise overlap S of twofold degenerate modes at hybridscattering-based EPs and resonant-scattering-based EPs under 30 randomly selected realizations of surface roughness, along with the corresponding average values.When the 30 random realizations of surface roughness as the ones in Figure 21 are imposed on the microdisk cavities with the deformation functions (41) and (45) respectively designed for the hybrid-scattering-based EPs and resonant-scattering-based EPs, S fluctuates dramatically for both the two classes of EPs.In general, the fluctuations resulting

Figure 22 .
Figure 22.Intracavity intensity distribution | i | 2 of modes at EPs in a rough deformed microdisk cavity calculated by the numerical FEM.In (a,c) and (b,d), the dominant mode numbers of modes are (12,1;6,3) and (9,1;6,2) as shown in Equation (32) and(33), respectively.The boundary functions r b () of the microcavity are defined by Equation (58); the overall boundary shapes r 0 () are defined by Equation (67) with the configurations (56) and(48), which in (a-d), respectively correspond to the ones in Figure19a-d, in order to form the hybrid-scattering-based EPs (the HSEP+ in Figure15) and resonant-scattering-based EPs (the RSEP+ in Figure16); the surface roughness Δr() in (a-d) corresponds to the one shown in Figure21a.

Figure 23 .
Figure 23.The normalized pairwise overlap S of degenerate mode pairs at EPs in microdisk cavities with random surface roughness calculated by the numerical FEM.In (a,b) and(c,d), the modes are twofold degenerate modes labelled by the mode numbers (12,1;6,3) and (9,1;6,2), respectively.The boundary functions of the microcavity are defined by Equation (58); the overall boundary shapes r 0 () are defined by Equation (67) with the configurations (56) and(48), in order to achieve the hybrid-scattering-based EP (the HSEP+ in Figure15) and the resonant-scattering-based EP (the RSEP+ in Figure16); the surface roughness consisting of 30 randomly picked realizations for a given standard deviation  and a given correlation angle  c is the same for both the two classes of EPs.Filled circle and square points respectively denote an individual result for hybrid-scattering-based EPs and resonant-scatteringbased EPs, while dashed and dotted lines respectively depict the results of averaging over the filled circle and square points.Note that empty points △▽◇☆ mark the results under the realizations of surface roughness shown in Figure21a-d, respectively.

Figure 24 .
Figure 24.The average value S and standard deviation ΔS of the normalized pairwise overlap S of degenerate mode pairs at EPs in rough microdisk cavities versus roughness standard deviation  calculated by the FEM.In (a,b) and (c,d), the dominant mode numbers of the degenerate mode pairs are (12,1;6,3) and (9,1;6,2), respectively.The boundary functions of the microcavity are defined by Equation (58); the overall boundary shapes r 0 () are defined by Equation (67) with the configurations (56) and(48), in order to achieve the hybrid-scattering-based EP (the HSEP+ in Figure15) and the resonant-scattering-based EP (the RSEP+ in Figure16); the surface roughness consisting of 30 randomly picked realizations for each  is the same for the two classes of EPs.Filled circle and square points respectively represent the average value of S over the 30 realizations on each  for the hybridscattering-based EPs and resonant-scattering-based EPs, while solid and dotted error bars correspondingly depict the standard deviations of S over the 30 realizations.
are modified by adding two new harmonic modulations  m 1 −p 1 cos[(m 1 − p 1 )] and  p 1 −m 2 cos[(p 1 − m 2 )] (m 2 < p 1 < m 1 ) in order to extend the 2-dimensional region of robustness along the space of  m 1 −p 0 and  p 0 −m 2 to the counterpart in the space of higher dimensions composed of  m 1 −p 1 ,  p 1 −m 2 ,  m 1 −p 0 , and  p 0 −m 2 .The hybrid-scattering-based EP in the higher dimensions might be more robust against the fabrication error dominated by the harmonics of cos[(m 1 − p 1 )] and cos[(p 1 − m 2 )] besides cos[(m 1 − p 0 )] and cos[(p 0 − m 2 )].The optimization and improvement of the hybrid-scattering-based EP is the topic of our future work.
EP are the deformation parameters  2m 1 and  m 1 −m 2 of the cavity boundary with respect to  2m 1 , EP and  m 1 −m 2 ,EP , respectively.