Recently, ultrahigh-Q dielectric microcavities such as a spherical [1], a toroidal [2], and a circular wedged cavity [3], whose experimental Q factors are larger than ∼ 10⁸, have attracted much attention because of their applicability to sensors for detecting bio-molecules [4, 5, 6, 7], nano particles [8, 9, 10, 11], and heavy water concentration [12].

Owing to the isotropic emission direction of these ultrahigh Q microcavities, various deformed cavities have been much studied following the reports of directional emissions in chaotic microcavities [13, 14, 15]. Recently, unidirectional emissions of high Q resonances have been found in several cavity shapes [16, 17, 18, 19, 20, 21, 22, 23, 24] and the ways of achieving high Q and unidirectional emission reported [20, 21]. However, the passive Q factors of these shapes obtained from the complex eigenvalues are less than 10⁶, which is much less than the Q factors of the circular shapes because of Q spoiling. It has been shown that as deformation increases in a half-quadrupole-half-circle microcavity, when directionality is enhanced, Q factors are spoiled [21]. According to the studies of Q spoiling [13, 25, 26, 27, 28, 29], when a cavity is fully chaotic, chaos spoils Q factors [25, 26, 27]. That is, when chaos governs the cavity, a resonance supported by an unstable periodic orbit is easily emitted from the cavity through unstable manifolds embedded in the chaotic sea. When a cavity is not fully chaotic, that is, when chaos and stable regular regimes are mixed in the classical phase space, chaotic dynamics can quantum-mechanically mediate tunneling of a resonance between two regular regimes even though they are remoted by the chaotic sea. Because of this phenomenon of chaos-assisted tunneling, even when a resonance is supported by a stable periodic orbit above the chaotic sea in the Poincaré surface of section (PSOS), the Q factor is spoiled [13].

It is known that when a microcavity is just slightly deformed not to exhibit a broad chaotic sea, high Q modes are also spoiled. In this case, Q spoiling cannot be explained by the chaos governing the phase space or by chaos-assisted tunneling. To understand the mechanism of Q spoiling in this regime, in this paper, we investigate Q spoiling of whispering gallery modes (WGMs) in a quadrupole dielectric microcavity. As the deformation increases from a circle, WGMs interact with their pair quasi-normal modes and transit to scarred resonances through an avoided resonance crossing (ARC) [30, 31, 32]. During the transition, WGMs are being deformed to scarred resonances depending on cavity deformation, which we call “resonance deformation.” In this transition, a slight resonance deformation causes a dramatic Q spoiling. To show the dependence of the Q factor on resonance deformation, we obtain Q factors of a WGM depending on resonance deformation by using the Husimi functions and show that the result agrees well with what is obtained from the complex eigenvalues.

The quadrupole cavity is given as follows:

We obtain Q factors, Q = Re(kR)/2|Im(kR)|, as a function of ε as shown in Fig. 1. The figure shows Q spoiling of WGMs as indicated by curves A, B, and C, which are the (l,m) = (1,32), the (1,30), and the (1,28) WGM, respectively. Here l and m are the radial and the angular quantum number, respectively. Q factors of these WGMs at ε = 0.0, which are obtained from the complex eigenvalues, are about 85,800, 38,700, and 17,600, respectively. The figure shows that when the cavity is just slightly deformed, Q factors are spoiled as ε increases. At ε = 0.04, the Q factors of curves A, B, and C are reduced to 29,000, 10,500, and 4,900, respectively. At this point, the narrow chaotic sea is far below the critical line.

 

Fig. 1 Q factors of the resonances depending on ε in the region from ε = 0.0 to 0.1. Curves A, B, and C are the (1,32), the (1,30), and the (1,28) WGM, respectively.

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In order to show that Q spoiling is caused by resonance deformation originated from ARCs, the real eigenvalues of the resonances are obtained in the region 0.0 < ε < 0.2 and 23.6 < Re(kR) < 24.6, which exhibit complicated interactions among the resonances through ARCs as shown in Fig. 2. Of these, we focus on the (1,32) WGM and the (2,28) quasi-normal mode as indicated by curves B and A, respectively. Although the curves do not seem to interact with each other, the difference of the two eigenvalues is minimized around ε = 0.06 as shown in the inset. This minimal difference implies an interaction of the two resonances. This type of interaction is known as the Demkov-type coupling, which can be found in some nonadiabatic processes due to collision [34, 35] and also in molecular systems [36]. Recently, it was reported that the (1,m) WGM and the (2,m − 4) quasi-normal mode in a stadium-shaped microcavity interact with each other through an ARC so that a pair of scarred resonances, which are localized on a diamond-shaped and a rectangular periodic orbit, are generated. The phenomenon is explained by the Fermi resonance [32]. For the interaction, the off-diagonal terms in the Hamiltonian are non-zero, representing the coupling for an ARC [30]. The off-diagonal terms in the Hamiltonian for the Demkov-type coupling induce an ARC for a pair of scarred resonances.

 

Fig. 2 Real eigenvalues depending on deformation ε around the (1,32) and the (2,28) resonance in the region from ε = 0.0 to 0.2, respectively. Curve A is the eigenvalue of the (1,32) resonance and curve B is that of the (2,28). The inset is the difference of the real eigenvalues of the (1,32) and the (2,28) resonance. showing a progressive interaction due to Demkov-type coupling. At ε = 0.06 where the difference is minimized, an ARC takes place.

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The intensity plots of the (1,32) and the (2,28) resonance clearly exhibit resonance deformation as shown in Fig. 3. Figure 3(a) and (b) are the (1,32) WGM and the (2,28) quasi-normal mode at ε = 0.0, respectively. As ε increases, the resonances develop to a pair of scarred resonances. Figure 3(c) and (d) are the (1,32) WGM and the (2,28) quasi-normal mode at ε = 0.04, respectively, which are slightly deformed. At ε = 0.1, we can see a pair of scarred resonances localized on a rectangular and a diamond-shaped orbit as shown in Fig. 3(e) and (f), which are developed from the (1,32) WGM and the (2,28) quasi-normal mode, respectively. This transition is the phenomenon of Fermi resonance because the quantum number difference of the two resonances (|Δl|, |Δm|) = (1,4) equals the rectangular and the diamond-shaped periodic orbit (1,4) [32].

 

Fig. 3 (1,32) and (2,28) resonance for the three cases of a deformation parameter. (a) and (b) are the (1,32) and the (2,28) quasi-normal mode at ε = 0.0, respectively, (c) and (d) are the resonances at ε = 0.04 developed from the (1,32) and the (2,28) quasi-normal mode, respectively, and (e) and (f) are those at ε = 0.1 developed from the (1,32) and the (2,28) quasi-normal mode, respectively. (g) and (h) are the Husimi functions of the (1,32) and the (2,28) resonance at ε = 0.04, respectively. The blue lines are the critical line. In the SOS, the narrow chaotic sea is much below than the critical line.

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Incident Husimi functions inside the cavity H_i(θ, p) clearly exhibit resonance deformation, where θ is the angle from the x-axis. Figure 3(g) is H_i(θ, p) of the (1,32) resonance superimposed on the PSOS at ε = 0.04. The four spots, placed far above the critical line p = sin (χ_c) = 1/n ∼ 0.65, are the result of resonance deformation, where χ_c is the critical angle for total internal reflection. Although the bright spots are far above the unstable period-4 orbit, we can affirm that the (1,32) resonance is localized on a rectangular-shaped unstable period-4 orbit since the spots are placed around the unstable period-4 orbit in the θ -axis. The position of the resonance far above the unstable periodic orbit on the p-axis is caused by the slight deformation of the cavity. Figure 3(h) is H_i(θ, p) of the (2,28) resonance superimposed on the PSOS, which shows that the wave is localized on a diamond-shaped stable period-4 orbit. The four spots overlapping with the critical line imply that this resonance is leakier than the (1,32) resonance.

The PSOS shows that the chaotic sea around the period-2 islands chain is far below the critical line and that the Husimi function of the (1,32) resonance is above the critical line. Hence the probability of the direct dynamical tunneling of the resonance localized on the period-4 islands chain to the chaotic sea [23] is very low because the two regimes are divided by the broad Kolmogorov-Arnold-Moser barrier. In addition, at ε < 0.03, even though there are small chaotic regions around the unstable periodic orbits in the PSOS, the Q factor of the (1,32) resonance is spoiled as shown in Fig. 1. This result implies that there is another dominant Q spoiling mechanism in this deformation parameter region.

Now to explain why high Q resonances are spoiled even when a small chaotic sea exists, we study the effect of resonance deformation on Q spoiling. Figure 4(a) is the reflectivity of a ray on a planar and a curved boundary for the effective refractive index 1.54. As is shown by curve A, the critical angle is about p = 0.65 for the planar boundary. However, in the circular boundary, there is no critical angle for the (1,32) WGM as shown by curve B. The reflectivity obtained according to Ref. [20, 37] reaches about 0.98 at p = sin χ = 0.8. This implies that the (1,32) WGM can be emitted due to the curved boundary. Figure 4(b) is the emission Husimi function H_e(θ, p) of the (1,32) WGM at ε = 0.0, which is given by that H_e(θ, p) ≡ H_i(θ, p) − H_r(θ, p), where H_r(θ, p) is the reflected Husimi function inside the cavity [38]. The figure shows that the emission of the (1,32) WGM at ε = 0.0 is above p = 0.8, which means that the main leakage of the WGM is caused by the reflectivity on the curved boundary. This is why the leakage is so low.

 

Fig. 4 Reflectivity and emission Husimi functions. (a) Curves A and B are the reflectivity of the (1,32) resonance on the plane and the circular boundary, respectively. (b) and (c) are the emission Husimi function of H_i(s, p) − H_r(s, p) of the (1,32) resonance at ε = 0.0 and 0.4, respectively. The straight lines are p = 0.65 and p = 0.8, respectively.

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Figure 4(c) is H_e(θ, p) of the (1,32) resonance at ε = 0.04. In contrast to Fig. 4(b), the figure shows that the four bright spots are around p = 0.8. When we compare the emission Husimi function with the incident one, we see that while the latter is above p = 0.8, the former is around p = 0.8. This means that the leakage of the resonance is caused by the position of the emission Husimi function on the PSOS, which is moved downward, in comparison with the incident Husimi function, due to resonance deformation. The figures clearly show that the main leakage is caused by resonance deformation. Hence we can say that when the (1,32) WGM interacts with the (2,28) quasi-normal mode, its morphology is deformed and its Q factor is spoiled.

We can obtain the Q factor of a resonance depending on resonance deformation as follows [37]:

where T and χ are the transmittance and the incident angle of a resonance, respectively, and n_ekR is the real eigenvalue. In this equation, T is crucial in obtaining the Q factor depending on resonance deformation. Some attempts have been made to obtain Q factors [39, 40, 41, 42] but none to obtain Q spoiling in relation to resonance deformation. To include information of resonance deformation, the transmittance T is obtained from the Husimi functions as follows:

(3)$$T\equiv \frac{{\displaystyle \int H_e(\theta ,p)d\theta dp}}{{\displaystyle \int H_i(\theta ,p)d\theta dp}}.$$

The incident angle χ of a resonance can be obtained from the semi-classical relation m = n_ekR sin χ, where m is the angular quantum number. Because a wavefunction of a resonance can be expanded by

(4)$$\psi (r,\varphi )={\displaystyle \sum _{m=-\infty }^{\infty }{\alpha }_m{J}_m(n_{e}kR)}\exp (im\varphi ),$$

α_m can be determined through the Fourier transformation of a resonance. Because m can be determined from the relation ⟨m⟩ = ∑(m|α_m|/∑|α_m|), cos χ can be determined.

Figure 5 is α_m of the (1,32) resonance at ε = 0.0 and 0.04 and the average m depending on ε. As is shown in Fig. 5(a), when, ε = 0.0, α_m has two peaks at m = ±32, which coincide with the angular quantum number. When ε = 0.04, there are several peaks around m = ±32 as shown in Fig. 5(b). These peaks are caused by resonance deformation. Hence, ⟨m⟩ of the (1,32) and the (2,28) resonance are obtained depending on ε as shown in Fig 5(c).

 

Fig. 5 α_m and ⟨m⟩ depending on ε. (a) and (b) are α_m of the (1,32) resonance at ε = 0.0 and 0.04, respectively. (c) is the averge m of the (1,32) and the (2,28) resonance depending on ε.

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Figure 6 is the Q factor of the (1,32) resonance depending on ε, where curves A and B are obtained from the complex eigenvalues and from the Husimi functions, respectively. The Q factors are rapidly spoiled at the same rate as in Fig. 5(a). This result implies that the Q factor obtained from the Husimi functions coincides well with what is obtained from the complex eigenvalues although there is a deviation of about 10%. We attribute the deviation to artificial effects such as a Gaussian profile in obtaining the overlap of a coherent state [38]. We note here that we can diminish the deviation by adjusting the standard deviation of the Gaussian profile σ. Although there are some reports on determining σ, for the sake of integrity, we use the σ used in Ref.[38, 43]. The Q factors, which are obtained from the Husimi functions, clearly show the effect of resonance deformation. When the two curves are normalized to be 1.0 at ε = 0.0, they are almost the same as shown in Fig. 5(b). The half-width at half maximum of curves A and B are 0.027 and 0.0273, respectively. Also the two curves fit the Gaussian profile well. This means that Q spoiling is caused by the deformation of the (1,32) resonance during the transition to a scarred resonance. This spoiling mechanism works all the same with the other high Q WGMs.

 

Fig. 6 Q-factors of the (1,32) resonance as a function of ε. Curves A and B in (a) are obtained from the complex eigenvalues and from Eqs. (2) and (3), respectively. Curves in (b) are normalized to be 1.0 at ε = 0.0. The squares and the circles are obtained from the complex eigenvalues and from the Husimi functions, respectively. The thick curve with “+” is the Gaussian profile. The two curves fit the Gaussian profile well as shown by the thick curve.

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We also study Q spoiling in a stadium-shaped dielectric microcavity. In this cavity, even though chaos governs the system, the spoiling mechanism works the same as in the quadrupole cavity. When high Q WGMs interact with their pair quasi-normal modes through ARCs, pairs of scarred resonances are generated as the cavity deformation increases. For example, the (1,32) WGM and the (2,28) quasi-normal mode interact with each other and a pair of scarred resonances localized on a diamond-shaped and a rectangular orbit each are generated due to Fermi resonance [32]. During the transition to a scarred resonance, the high Q WGM is spoiled due to deformation of the resonance depending on the cavity morphology. Even in this cavity, when we obtain Q spoiling, the Q factors obtained from Eqs. (2) and (3) agree well with those obtained from the complex eigenvalues. Also we note that Q factors of several low Q resonances are enhanced as deformation increases like the Q enhancement in Ref.[27] due to interaction of a pair of resonances.

It is known that as the cavity size increases, the Q factor exponentially increases in a circular and an elliptic microcavity. The effect of cavity size on the Q factor is not the motivation of our study. We are motivated by Q spoiling in a chaotic microcavity depending on cavity deformation in a slightly deformed region. As mentioned in the above, up to now, Q spoiling was attributed to chaos or chaos-assisted tunneling. As far as a broad chaotic sea appears in the phase space, the effect of chaos is dominant in Q spoiling [13, 26]. However, when a narrow chaotic sea is far below the critical line or small chaotic regions appear around unstable periodic orbits, it is negligible on Q spoiling in comparison with resonance deformation as shown in Fig. 5. In order to prove the dominancy of resonance deformation in effecting Q spoiling, we obtain Q factors by using the Husimi functions and show a good agreement of our Q factors with those obtained from the complex eigenvalues.

In our study, even while confined to Q spoiling of the (1,32) WGM because of the limit of computational power in obtaining imaginary eigenvalues, we can clearly explain the mechanism of Q spoiling. In a large cavity size, although interactions among resonances are very complicated, the Fermi resonance relation is alway satisfied in formation of scarred resonances [32]. For examples, when a (1,m) WGM interacts with its pair (2,m − 6) quasi-normal mode, a pair of hexagonal scarred resonances are generated to satisfy the Fermi resonance relation such that (|Δl|, |Δm|) = (1,6), where |Δl| and |Δm| are the radial and the angular quantum number difference of the two quasi-normal modes, respectively, and (1,6) is the hexagonal periodic orbits. Also when a (l,m) quasi-normal mode interacts with its pair (l + 1, m − 4), where l > 1, a rectangular and a diamond-shaped scarred resonance are generated to satisfy the Fermi resonance relation. During the transition of quasi-normal modes to scarred resonances, Q spoiling takes place as deformation increases. The transition mechanism hereof will soon appear elsewhere.

In conclusion, our findings show that even when a circular microcavity is slightly deformed to a chaotic one, ultrahigh Q WGMs in a circular cavity are dramatically spoiled due to the transition to scarred resonances. During the transition, while directionality is improved, Q factors are spoiled due to resonance deformation. Our results will provide theoretical backgrounds for the development of ultrahigh Q microcavity lasers, which emit directionally.

Acknowledgments

This research was supported by Basic Science Research Program ( NRF-2013R1A1A2060846) and High-Tech Convergence Technology Development Program ( NRF-2014M3C1A3051331) through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning.

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