Coexistence of a new type of bound state in the continuum and a lasing threshold mode induced by PT symmetry

Dimerized chain and phase diagram for ordinary BICs

The system of an infinite PT-symmetric dimerized periodic chain with period a is schematically shown in the upper inset of Fig. 1. In the absence of gain and loss, particles A and B in a unit cell are two identical spherical plasmonic nanoparticles of radius r₀ separated by a distance d. Their relative permittivity is described by the Drude model, ε(ω)=1−ωp2/ω(ω+iγ0), where ω_p is the plasma frequency and γ₀ is the collision frequency. In the study of BICs, we are only interested in the existence of a bound state that is decoupled from the far field so that the radiative Q factor diverges (6). Like previous works of BICs, we ignore the intrinsic loss γ₀ and focus on the radiation loss. In the case of d > 3r₀ and a-d > 3r₀ (32), each plasmonic nanoparticle can be treated as an electric dipole with an inverse polarizability α0−1(ω)=1r03ωp2−3ω2ωp2−2i3k03, where the imaginary part denotes the radiation loss (33) and k₀ = ω/c, with c being the speed of light in vacuum. The PT-symmetric gain and loss can be added to particles A and B by coating them with thin layers of gain and loss media, respectively. The inverse polarizability can now be written as αA−1(ω)=α0−1(ω)+iγ and αB−1(ω)=α0−1(ω)−iγ, where γ > 0 represents the amounts of gain and loss added to particles A and B, respectively.

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For a periodic chain along the z axis, a resonant state can be labeled by a Bloch momentum q inside the BZ [−π/a,π/a]. When the system is excited by an external plane wave of the form Eext=E0êyei(kxx+qz−ωt), the transverse polarizations of particles A (p_A) and B (p_B) along the y axis can be obtained from the multiple-scattering theory by solving the following equation[αA−1−SAA−SAB−SBAαB−1−SBB][pApB]=[EAEB](1)as derived in Materials and Methods, where E_A(B) = E₀e^∓iqd/2. Equation 1 can be expressed as MP = E for convenience. The expressions of the lattice sums S_ij are given in Eq. 7 (see Materials and Methods). Because of the inversion symmetry of the system, S_ij have the following symmetry properties: S_AA(q,ω) = S_BB(q,ω), S_AA(−q,ω) = S_AA(q,ω), and S_AB(BA)(−q,ω) = S_BA(AB)(q,ω) as shown in Eq. 8. Below, we denote S_AA(BB)(q,ω) by S₀₀(q,ω). The analytical study of BICs is made possible by expressing these lattice sums in terms of polylogarithm and Hurwitz-Lerch transcendent functions. Note that the material properties and geometrical parameters are contained separately in αA(B)−1 and S_ij.

To study the resonant modes of the system, we set E = 0 in Eq. 1 and look for those complex frequencies at which the matrix M has a zero eigenvalue, i.e., MP = 0. Thus, the resonant frequencies of the system can be determined from the condition det(M) = 0. It is convenient to express the polarization vector in the form P=(pA′e−iqd/2, pB′eiqd/2)T to reflect the explicit phase difference between the two particles in the unit cell for the mode q. In the case of no gain or loss, i.e., αA−1=αB−1=α0−1, the periodic part of the eigenvector, P′=(pA′, pB′)T, satisfies the following equations(α0−1−S00)pA′−SABeiqdpB′=0−SBAe−iqdpA′+(α0−1−S00)pB′=0(2)or simply Μ^′P^′ = 0. The eigenfrequencies of the resonant modes are determined by the complex frequency solutions of det(Μ′)=(α0−1−S00)2−SABSBA=0 denoted by ω(q) = ω^′(q) + iω^″(q). The function ω^′(q) gives the dispersion relation of the resonant modes, and the ratio ω^′(q)/2ω^″(q) gives the Q factor. The zero of ω^″(q) indicates the position of BICs at which the Q factor diverges. For any given q, there also exists other complex frequency solutions of det(Μ^′) = 0. Here, we are only interested in the leaky channel along which ω^″(q) is the smallest and vanishes at certain values of q when a BIC occurs.

In this work, we choose the geometric parameters as r₀ = 20 nm, d = 70 nm, and a = 190 nm. For a particular material parameter ω_p = 0.95(2πc/a), the results of ω^′(q) and ω^″(q) are shown in Fig. 1 (A and B), respectively. Here, we only focus on the zeroth-order diffraction region, i.e., |q| < ω/c < 2π/a − |q|. From the expression of det(Μ^′) and the symmetry relations S₀₀(−q,ω) = S₀₀(q,ω) and S_ij(−q,ω) = S_ji(q,ω), it is easy to observe the symmetry of the resonant frequency, i.e., ω(q) = ω( − q), as shown in Fig. 1 (A and B). The three zeros of ω^″(q) indicate the existence of one symmetry-protected BIC at q = 0 and a pair of propagating BICs at some nonzero ±q.

The existence of BICs can also be explicitly determined via the following analytical approach. For a BIC with a real-frequency solution of Eq. 2, the nonradiative condition in the transverse plane requires a complete cancellation of far-field radiation, which in turn requires pB′=−pA′ (9, 34). Thus, the condition for BICs becomes the real-frequency solutions off(q,ω)=def(α0−1−S00)+SABeiqd=0h(q,ω)=defSBAe−iqd+(α0−1−S00)=0(3)

However, the condition h(q,ω) = 0 is not independent of the condition f (q,ω) = 0 in the determination of BICs. The proof is given in section S1. Briefly, we show that the inversion symmetry of the system gives an identity f (−q,ω) = h(q,ω), while the time-reversal symmetry gives f ^*(q,ω) = f (−q,ω). Thus, we have h(q,ω) = f ^*(q,ω), which means that the real-frequency solutions of f (q,ω) = 0 are also the solutions of h(q,ω) = 0. Therefore, at BICs, we also have S_ABe^iqd = S_BAe^-iqd. This makes the matrix Μ^′ symmetric, which in turn makes the condition det(Μ′)=[(α0−1−S00)+SABeiqd][(α0−1−S00)−SABeiqd]=0 satisfied. Thus, although f (q,ω) = 0 and det(Μ^′) = 0 have different solutions in general, they do share the same real-frequency solutions at BICs.

In Fig. 1 (C and D), we plot the real and imaginary parts of the function f(q,ω) in the q-ω plane. It can be seen that Re(f) and Im(f) are, respectively, an even and an odd function of q, consistent with the identity f ^*(q,ω) = f (−q,ω) obtained previously at BICs. This identity holds in general in the q-ω space. A rigorous mathematical proof is given in section S6. Thus, the identity h(q,ω) = f ^*(q,ω) also holds in general. In Fig. 1E, the nodal lines of Re(f) are plotted with dashed lines at different values of dimensionless plasma frequency, ω¯p=ωp(a/2πc), and the nodal line of Im(f) is plotted with red solid line. It can also be seen that, for a given value of ω¯p, the nodal lines of Re(f) and Im(f) intersect each other at three points, giving rise to three BICs: one symmetry-protected BIC at q = 0 and two propagating BICs located symmetrically on the two sides of the Γ point. To see this more explicitly, in Fig. 2A, we plot the functions Im(α0−1−S00), Im(−S_ABe^iqd), and Im(−S_BAe^−iqd) evaluated at the frequency of Re(f) = 0. This figure shows clearly that these three curves intersect at three BIC points at which Im(f) = 0, as indicated by vertical dashed lines.

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Because Im(α0−1)=−2k03/3 is independent of the material parameter ω_p, the nodal line of Im(f) depends only on the geometrical parameters, while the nodal line of Re(f) depends on the value of ω_p. In Fig. 1E, we plot the nodal lines of Re(f) at different values of ω_p, where the dimensionless plasma frequency ω¯p=ωp(a/2πc) is indicated. The nodal line of Re(f) shifts upward and moves the two propagating BICs toward the Γ point as ω¯p increases. These propagating BICs merge with the symmetry-protected BIC when some critical ω¯pc is reached and then disappear when ω¯p>ω¯pc. To obtain a phase diagram for the existence of propagating BICs, we perform the similar study but vary the geometrical parameters as well. In Fig. 1F, we show a critical surface of ω¯pc as a function of parameters a and d in the region where dipole approximation is valid. Such a surface represents a boundary separating two phases. Below this boundary, propagating BICs are guaranteed and robust against small perturbations in both material and geometrical parameters. Above the boundary, there are no propagating BICs. Thus, our system provides an analytically solvable model for studying the phase diagram of propagating BICs without resorting to a tedious numerical search for divergent Q factors through parameter tuning.

Pairing of pt-BIC and lasing threshold mode

When gain and loss are added to the system, Eq. 2 becomes(α0−1−S00+iγ)pA′−SABeiqdpB′=0−SBAe−iqdpA′+(α0−1−S00−iγ)pB′=0(4)or (Μ′ + iγσ_z)P′ = 0, where σ_z is the z component of the Pauli matrix. The conditions for the existence of real-frequency solutions of Eq. 4 in conjunction with the condition pB′=−pA′ for the cancellation of radiation fields become f (q,ω) = −iγ and h(q,ω) = iγ. Because h(q,ω) = f ^*(q,ω), we only need to consider the real-frequency solutions of f (q,ω) = −iγ. For the case of ω¯p=0.95, in Fig. 3A, we plot the nodal line of Re(f) with a dashed line and the nodal lines of Im(f) + γ with colored solid lines for different values of γ. For each value of γ, the intersection of the solid curves with the dashed curve produces three pt-BICs: one in the negative q space and two in the positive q space when 0 < γ < γ_c. They emerge from the three original BICs when the system has no gain or loss, but they are shifted in the q space. The shift of each pt-BIC can be explicitly seen from Fig. 2B, in which we choose γ=2×10−3/r03 and plot the functions Im(α0−1−S00)+γ and Im(α0−1−S00)−γ with black dashed and yellow dashed lines, respectively, in addition to the two curves Im(−S_ABe^iqd) and Im(−S_BAe^−iqd) shown in Fig. 2A for the case of no gain or loss. The three intersection points of the functions Im(α0−1−S00)+γ and Im( − S_ABe^iqd), i.e., the nodal points of Im(f) + γ, show the positions of pt-BICs, which are indicated by the vertical dashed lines. The same vertical dashed lines are obtained from the intersection of Im(α0−1−S00)−γ and Im(−S_BAe^−iqd), or the nodal points of Im(h) − γ.

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Figure 3A also shows the evolution of pt-BICs as a function of γ. As the value of γ is increased, the two pt-BICs in the positive q space move toward each other along the nodal line of Re(f), and finally, they are annihilated when γ = γ_c. The system is left with only one pt-BIC in the negative q space when γ > γ_c. Thus, it is interesting to see that the trajectory of pt-BICs follows the nodal line of Re(f) when γ is varied. On the contrary, the trajectory follows the nodal line of Im(f) + γ when the material parameter ω¯p is varied.

Accompanying each pt-BIC, a mode at the lasing threshold emerges at the same frequency but in the opposite q space. The existence of such a lasing threshold mode can also be derived analytically from Eq. 4. It is easy to show that a real-frequency solution of Eq. 4 at a point (q₀,ω₀) with pB′=−pA′ implies another solution of Eq. 4 at the point (−q₀,ω₀) with p′¯B=−SBA(−q0,ω0)eiq0dSAB(−q0,ω0)e−iq0dp′¯A. Different from the pt-BIC, we have ∣p′¯B∣<∣p′¯A∣ for this mode as observed from Fig. 2, which implies a net gain of power, i.e., P_gain > 0. Because this mode has a real frequency, the net gain must be exactly balanced by the radiation loss, implying a lasing threshold. The numerical verification of the balance between net gain and radiation loss at the lasing threshold will be given in the next subsection. It is interesting to note that at a lasing threshold, the vector P¯′≜(p′¯A,p′¯B) actually corresponds to the left eigenvector of the equation P¯′(Μ′+iγσz)=0 as can be proved directly using the identity S_ij(−q,ω) = S_ji(q,ω) for i ≠ j.

In addition to the above analytical study, we have also performed numerical studies using COMSOL Multiphysics to obtain the quasi-eigenmodes and complex eigenfrequencies ω′(q) + iω″(q) under a scattering boundary condition. For the case of ω¯p=0.95 and γ=1.7×10−3/r03, the results of ω′(q) and ω″(q) for the leaky channel calculated by COMSOL are shown by the yellow solid curve in Fig. 3 (B and C), respectively. The two red dotted curves illustrate the analytical results obtained from the condition det(Μ′ + iγσ_z) = 0 of Eq. 4. Excellent agreements are found between the theory and simulations. It is easy to show that the presence of the term iγσ_z in the determinant does not affect the symmetry of ω(q) = ω(−q). In Fig. 3B, we also plot the analytical dispersion of a higher-frequency branch ω>′(q) (red dotted curve). The corresponding imaginary part ∣ω>″(q)∣ is much larger in value and does not vanish. We are only interested in the dispersion of the leaky channel, ω′(q), along which pt-BICs and lasing threshold modes can occur at the zeros of ω″(q).

The function ω″(q) has zeros at three pairs of discrete q points, i.e., q₀ = ±0.090, ±0.400, and ± 0.442, indicated by colored polygons in Fig. 3B. By using COMSOL, we numerically simulate the cross-sectional field profiles of |E| for these three pairs of singularities and show the results in Fig. 3D. Taking the pair split from the original BIC at the Γ point as an example, the two states at q₀ = ±0.090 have completely different eigenstate profiles. The E field is spatially localized in the transverse plane for the state at q₀ = 0.090 without far-field radiation, indicating a pt-BIC (black pentagon). However, an extended state is observed at q₀ = −0.090, indicating a lasing threshold mode (red pentagon) having far-field radiation, in agreement with the analytical prediction. Similar conclusions can be drawn for the other pairs of states marked by the triangles and squares in Fig. 3 (B and D). A quantitative analysis of these field patterns in terms of the powers of net gain/loss and radiation loss will be given in the next subsection.

It is important to note that the slope of ω″(q) at the zeros of ω″(q) is not zero as shown in Fig. 3C, unlike the slope at the zeros of ω″(q), which is also zero shown in Fig. 1B. The difference suggests two different behaviors of ω″(q) in the vicinity of a zero ω″, which in turn give two different Q factor divergence rates, i.e., Q⁻¹ ∝ ω″(q) ∝ q − q₀ for the former case and Q⁻¹ ∝ ω″(q) ∝ (q − q₀)² for the latter. The quadratic behavior of Q⁻¹ is a generic result of all known BICs (35–37), and as such, the linear behavior of Q⁻¹ found here suggests that pt-BICs belong to a new class of BICs. An analytical discussion of the Q factor divergence rate as well its numerical verification is given in section S2. A direct consequence of the linear behavior of ω″(q) at pt-BICs is the creation of an unstable region on one side of q₀, in which ω″(q) is positive and lasing can occur. In this region, the net gain obtained from the particles exceeds the radiation loss, and therefore, the energy of a mode grows exponentially in time. For the case of q₀ = −0.090, an unstable region exists between this lasing threshold and the accompanying pt-BIC at q₀ = 0.090 as shown in Fig. 3C. Together with the existence of two other pairs of pt-BICs and lasing threshold modes, the q space is divided into four stable passive regions and three unstable lasing regions. Besides the difference in the Q factor divergence rate, there exists another important difference between an ordinary BIC and a pt-BIC. It is well known that an ordinary BIC cannot be excited by an external field, nor can it radiate. However, this is no longer true for pt-BIC. As we will show in a later subsection that a pt-BIC can be excited by an external field even though it does not radiate itself.

Net gain/loss and radiation loss of the resonant modes

Here, via COMSOL simulation, we study numerically the decay or growth rate of the resonant modes and compare it with the analytical result of ω″(q) shown in Fig. 3C. In particular, we wish to numerically verify that ω″ vanishes at both the pt-BICs and the lasing threshold modes. To find this rate at any given q, we first evaluate separately the radiation power P_rad and absorption power P_abs per unit cell from the eigenfields generated by COMSOL under the scattering boundary condition (see Materials and Methods). Here, a negative P_abs represents the power due to net gain in the system.

To find the decay/growth rate of a resonant mode, we further calculate the energy stored in the resonant mode, U_eff. The fields E and H used to calculate U_eff are the numerical results obtained from COMSOL simulations minus the analytical far-field radiation fields (see section S3). The decay rates due to both radiative and nonradiative processes can be obtained and denoted, respectively, by γ_rad = P_rad/2U_eff and γ_abs = P_abs/2U_eff. A negative γ_abs denotes the growth rate. The results of −γ_abs and γ_rad are shown in Fig. 4B. In Fig. 4C, we plot their sum γ_tot = γ_rad + γ_abs (red dotted line) together with the analytically determined total decay rate (yellow solid line), i.e., −ω^″(q), which is obtained directly from Fig. 3C. Excellent agreement between the theory and simulation is clearly seen. Although γ_tot is symmetric in the q space, its physics is very different. For example, in the positive q space, both γ_rad and γ_abs vanish at the two pt-BICs, i.e., q = 0.090 and 0.400. This confirms that, like ordinary BICs, pt-BICs have neither radiative loss nor nonradiative loss. In the negative q space, however, there exist two lasing threshold modes at q = −0.090 and −0.400, at which both γ_rad and γ_abs are nonzero although γ_rad + γ_abs = 0. This confirms that at lasing thresholds, the power obtained through the net gain from the particles is exactly balanced by the power loss due to radiation. Thus, the condition of ω″ = 0 for real eigenfrequency solutions cannot tell the difference between the two possible solutions: pt-BICs and lasing threshold modes. Their difference can only be seen by studying γ_rad and γ_abs separately.

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In section S5, we also show two examples of excitation by external waves. The scattering and absorption for the excitation of a plane wave and point sources are given in figs. S4 and S5, respectively. It can be observed that both the extinction cross section C_ext due to the plane wave excitation shown in fig. S4B and the extinction power P_ext due to the excitation of point sources shown in fig. S5B exhibit the same behavior for q and −q modes, although the scattering and absorption behaviors are rather different for ±q. This is consistent with the result shown in Fig. 4 that γ_rad and γ_abs for the resonant modes are asymmetric in the q space, but the total decay rate γ_tot is symmetric.

Coupling of BICs and pt-BICs to external fields

When the system is excited by a plane wave ∣E〉 at a frequency ω with a wave vector q in the z direction, the response of the system can be obtained from the equation∣P〉=∑n1λn∣Pn〉〈P¯n∣E〉=∑nWn(q,ω)λn∣Pn〉(5)where λ_n = λ₋ and λ₊ are two eigenvalues of the matrix M in Eq. 1 with parameters q and ω, and ∣P_n〉 and 〈P¯n∣ are, respectively, the right and left eigenvectors of M, i.e., M∣P_n〉 = λ_n∣P_n〉 and 〈P¯n∣M=λn〈P¯n∣, and Wn(q,ω)=〈P¯n∣E〉 (33). The dispersion ω(q) obtained from the condition Re(λ₋) = 0 corresponds to the leaky channel we are interested in, i.e., the lower-frequency branch shown in Fig. 3B. The dispersion obtained from Re(λ₊) = 0 corresponds to the higher-frequency branch in Fig. 3B. Close to a BIC at the point (q₀,ω₀), i.e., q ≅ q₀, ω ≅ ω₀, λ₋ is vanishingly small and the response of the system ∣P〉 is mainly determined by the right eigenvector ∣P₋〉 and the quasi-mode coupling strength W−(q,ω)=〈P¯−∣E〉 (16).

For the case of no gain and loss, because a BIC cannot be coupled to the external plane wave, the coupling strength 〈P¯−∣E〉 is zero at BICs. If we rewrite the left eigenvector 〈P¯−∣ as (p′¯Aeiqd/2,p′¯Be−iqd/2) and the external plane wave ∣E〉 as (EA′e−iqd/2,EB′eiqd/2)T, we have 〈P¯−′∣E′〉=0 or p′¯B=−p′¯A as ∣E′〉 ∝ (1,1)^T. Because the matrix M′ in Eq. 2 is symmetric at the BICs as shown in Fig. 2A, its right eigenvector has the same entries as the left eigenvector, i.e., ∣P−′〉=(〈P¯−′∣)T. This also implies a vanishing radiation loss at BICs, i.e., 〈E′∣P−′〉=0. The equivalence between zero coupling strength and zero radiation loss is expected for all ordinary BICs.

However, this is no longer true for pt-BICs. We will see that a pt-BIC can be excited by an external field, but it does not radiate. This is because when gain and loss are introduced into the system, the matrix Μ′ + iγσ_z in Eq. 4 is no longer symmetric at the pt-BICs as shown in Fig. 2B. The left and right eigenvectors of a pt-BIC are hence different. Because the right eigenvector of a pt-BIC, ∣P−′〉, satisfying Eq. 4 has an antisymmetric form, i.e., pB′=−pA′, the corresponding left eigenvector, 〈P¯−′∣, will not be antisymmetric, i.e., p′¯B≠−p′¯A, implying 〈P¯−∣E〉≠0. This shows that a pt-BIC can be excited by external fields, although it does not radiate.

Let us take the state at q = 0.090 in Fig. 3B as an example. For the pt-BIC marked by the black pentagon, the right eigenvector has the form of zero radiation condition, i.e., ∣P−′〉0.090=12{1,−1}T. From the symmetry relations S₀₀(−q,ω) = S₀₀(q,ω) and S_ij(−q,ω) = S_ji(q,ω) shown in Eq. 8, it is easy to show that the left eigenvector of the pt-BIC has the same entries as the right eigenvector of the accompanying lasing threshold at q = −0.090, namely, 〈P¯−′∣q=∣P−′〉−qT. Because it has been shown that the amplitude of the dipole moment of particle A is larger than that of particle B at q = −0.090, we have ∣p′¯B∣<∣p′¯A∣ for q = 0.090, implying a nonzero coupling strength 〈P¯−∣E〉 at the pt-BIC. A nonzero coupling strength W₋ makes the response vector ∣P〉 of Eq. 5 divergent when a pt-BIC is approached as λ₋→0. This is a typical result of the excitation precisely at the frequency of a resonant mode, which has zero spectrum width. Thus, the PT-symmetric perturbation changes the singular behavior of a pt-BIC, which not only reduces the Q factor divergence rate but also violates the equivalence between zero coupling strength and zero radiation loss.

To demonstrate the nonzero coupling strength at pt-BICs, we have simulated the dipole moments in a unit cell close to a pt-BIC excited by both point sources and plane waves as shown in fig. S6. For both types of sources, the magnitude of the simulated dipole moments (p_A, p_B) diverges at the same rate as 1/λ₋, confirming the analytical prediction that the coupling strength is nonzero at pt-BICs.

A generic phenomenon

The splitting of an ordinary BIC into a pair of pt-BIC and lasing threshold mode under a PT-symmetric perturbation and the associated phenomena discussed above are consequences of the identity h(q,ω) = f *(q,ω), which makes the real-frequency solutions of Eq. 4 possible. Because this identity does not depend on the dielectric constant of nanoparticles, the phenomena also apply to dielectric nanoparticles as long as the dipole approximation is valid. Furthermore, because the identity holds independent of the choice of the system geometric parameters, as mathematically proved in the section S6, the phenomena are guaranteed and robust in our system. For the experimental realization, we can add gain and loss to particles A and B separately by coating them with a thin shell of dielectric material with relative permittivity ε1′+iε1A″ for particle A and ε1′+iε1B″ for particle B. The relations between the amount of gain or loss, i.e., ε1A″<0 or ε1B″>0, and the value of γ or −γ in Eq. 4 are given in section S4.

The novel phenomena found here are generic for other 1D PT-symmetric system having BICs. Here, we propose another system for the possible experimental realization of the found phenomena. The system considered is a photonic crystal fiber with periodic index modulations (38) shown in the inset of Fig. 5A, where the two components of the dielectric core are of equal thickness, i.e., d = 0.5a, and have relative permittivities ε₁ = 4.9 and ε₂ = 1. The radius of the core is chosen to be the same as the lattice constant, i.e., r₀ = a. The real and imaginary parts of the eigenfrequencies of the transverse electric (TE) modes simulated by COMSOL Multiphysics are shown, respectively, in Fig. 5 (A and B). A symmetry-protected BIC can be found at the Γ point. The corresponding E_y field is spatially localized in the lateral direction as shown in the inset of Fig. 5B. To make the system PT symmetric, we divide each unit cell into four layers of equal thickness and introduce gain or loss into each sublayer in the order ε₁ − iη, ε₂ + iη, ε₂ − iη, and ε₁ + iη as shown in the inset of Fig. 5C so that the PT symmetric perturbation ε(z) = ε^∗(−z) is satisfied. For the case of η = 0.2, the real and imaginary parts of the eigenfrequencies are shown, respectively, in Fig. 5 (C and D). It can be clearly seen that the PT-symmetric perturbation splits the original BIC into a pair of pt-BIC and lasing threshold mode located symmetrically on the two sides of the q space at which ω″ = 0. Although the pt-BIC and the lasing threshold mode have the same eigenfrequency, they have completely different eigenstates that are, respectively, localized and extended in the lateral direction as shown in the inset of Fig. 5D. To experimentally achieve the PT-symmetric modulation of the dielectric constant, one can place a periodic loss structure on the modulated dielectric core with gain, similar to a previous work (39).

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