## Abstract

Optical modes in deformed dielectric microdisk cavities often show an unexpected localization along unstable periodic ray orbits. We reveal a new mechanism for this kind of localization in weakly deformed cavities. In such systems the ray dynamics is nearly integrable and its phase space contains small island chains. When increasing the deformation the enlarging islands incorporate more and more modes. Each time a mode comes close to the border of an island chain (separatrix) the mode exhibits a strong localization near the corresponding unstable periodic orbit. Using an EBK quantization scheme taking into account the Fresnel coefficients we derive a frequency condition for the localization. Observing far field intensity patterns and tunneling distances, reveals small differences in the emission properties.

© 2017 Optical Society of America

## 1. Introduction

The ray-wave (classical-quantum) correspondence has been the objective of extensive research. Semiclassical methods such as the Einstein-Brillouin-Keller (EBK) quantization scheme [1, 2] for integrable systems and trace formulas for the periodic orbits (POs) of classically chaotic systems [3–6] have proven to be remarkably useful. One particular interesting aspect is the phenomenon of “scarring”, first observed by Heller in a quantized stadium billiard [7]. It refers to the existence of energy eigenstates with strong concentration along unstable POs of the underlying classical system. In optical microcavities, the localization of optical modes along short unstable POs has been observed experimentally in various kinds of strongly deformed microlasers [8–11]. These observations are of interest because such modes can have relatively high quality factors (long photon lifetimes) even in the presence of chaos since the corresponding PO is located entirely outside the leaky region, which is the region where the condition for total internal reflection is not fulfilled. Numerical simulations indicate that scarring in open systems with chaotic ray dynamics, such as deformed microdisk cavities, is rather the rule than the exception; see, e.g., [12–18].

A related phenomenon observed in deformed microdisk cavities is the appearance of “quasi-scarred modes” showing a strong localization on simple geometric structures with no underlying PO [19, 20]. Lasing on quasiscarred modes has been successfully realized for spiral-shaped InGaAsP microcavity lasers [21]. Quasiscars find a natural explanation in terms of an extended ray dynamics [22] which incorporates Fresnel filtering [23].

Yet another phenomenon is the appearance of “scarlike modes” in systems with marginal stable POs which have been slightly perturbed [24–29]. These states form near avoided resonance crossings as a perturbation parameter is varied. This happens in particular when an integrable system is slightly perturbed [25]. According to the Poincaré-Birkhoff theorem [30,31] the family of marginal stable POs is replaced by pairs of stable and unstable POs. The stable POs are embedded into small islands in phase space. Figure 1 illustrates this situation in the Poincare surface of section (SOS) [31], a section through the full phase space obtained by plotting the intersection points of a set of trajectories with the cavity’s boundary. The unstable POs are located on the enclosure of the island which is called separatrix. For very weak deformation the two modes involved in the avoided resonance crossing hybridize such that one mode is located near the stable POs and the other mode near the unstable POs. This scenario can be considered as a quantum version of the Poincaré-Birkhoff theorem [25], see also [32].

In this article, we reveal another type of localization along unstable POs which appears naturally in weakly deformed microdisk cavities. Consider a weakly deformed cavity with a small island chain in phase space, possibly with a scarlike mode localized near the unstable PO. Increasing the deformation further leads to an increasing of the size of the island chain. The phenomenon occurs as a transition process of modes getting into an increasing island chain from the outside. Each time a mode comes close to the separatrix the mode exhibits a strong localization near the corresponding unstable PO. Such a scenario has been predicted already in the context of atomic physics, namely for hydrogen in a circularly polarized electromagnetic field [33]. Interestingly, the localization at the unstable PO is of classical nature: the classical motion slows down near the unstable point [34]. Similar localization has also been observed in quantum maps [35].

To clarify this localization in our setting, we quantize the POs in terms of a semi-classical approach taking into account the degree of the deformation. Through the analysis, we show that a mode is maximally localized in the vicinity of the unstable PO when its quantized frequency matches the natural frequency of the mode. In addition, by examining intensities outside the cavity, surprising similarity of far field patterns are revealed even though the mode structures, inside the cavity, are completely opposite as in the island and on the separatrix. On the other hand, further studies on the emission properties of these modes near the cavity boundary reveal different tunneling distances of them.

This paper is organized as follows. Section 2 specifies the system that serves as an example. In Sec. 3 we show that this system possesses for very weak deformation the already known scarlike modes. In Sec. 4.1 we semiclassically quantize the unstable PO. Numerical results are presented in Sect. 4.2. A further analysis in terms of a pendulum approximation is given in Sect. 4.3. Section 5 addresses properties of far field intensity patterns. In Sec. 6 tunneling characteristics of emissions are analyzed. The work is summarized in Sect. 7.

## 2. The system

For our analysis we introduce an oval-shaped cavity

where*ε*is the deformation parameter. For

*ε*= 0 one obtains a circle of radius

*R*. We choose the effective refractive index to be

*n*= 3.3.

_{e}Maxwell’s equations are solved in two dimensions within the effective index approximation with Sommerfeld outgoing wave conditions at infinity. The optical modes are defined as the solutions with time dependence *e*^{−iωt}. We express the complex-valued frequency *ω* by the dimensionless frequency *ωR/c* = *kR* where *c* is the speed of light in vacuum and *k* = *ω/c* is the wave number. The real part of *kR* is the conventional frequency whereas the imaginary part determines the line width (decay rate) *γ* = −2 Im *kR* and the quality factor *Q* = −Re *kR*/[2 Im *kR*] of the given mode.

Only in the limiting case *ε* = 0, the modes can be computed analytically and can be characterized by two mode numbers (*l*, *m*), see e.g. [36]. Here, *l* is the radial mode number and *m* is the azimuthal mode number. In the general case, the modes are determined numerically. Here, we employ the boundary element method (BEM) [37] and focus on transverse magnetic (TM) polarization, on even modes (positive parity with respect to the symmetry line *y* = 0 of the cavity) and on the regions 95.0 ≤ Re(*n _{e}kR*) ≤ 96.2 and 0 ≤

*ε*≤ 0.05.

## 3. Scarlike modes

In this section we demonstrate that for very small *ε* the oval-shaped cavity (1) possess scarlike modes in the sense of [24]. An example is shown in Figs. 2(a) and 2(b) for *ε* = 0.0025. Both modes are developed from the (*l*, *m*) = (11, 48) and the (10, 51) modes in the circular cavity by continuously increasing *ε* from 0 to 0.0025.

The connection to the ray dynamics is most conveniently illustrated by the Husimi function [38] of a given mode. It can be considered as quasi-probability distributions of a mode at the boundary of the dielectric cavity [39]. We use here the incidence Husimi function hu(s,p) in the interior region of the cavity [39]. Note that for the Husimi functions we set the squeezing factor *ξ* of the coherent state as
${\sigma}^{\prime}=\xi \sigma =(2\pi /2)\left(\sqrt{2}/{n}_{e}kR\right)$ [39, 40] in order to reflect our rectangular phase space region [*s _{min}*,

*s*] × [

_{max}*p*,

_{min}*p*] = [0, 2

_{max}*πR*] × [−1, 1]. Figures 2(c) and 2(d) are Husimi functions of the two modes. By comparing the Husimi functions with the superimposed ray orbits it can be seen that one mode is localized in the island with the period-3 stable PO in its center, whereas the other mode is localized near the period-3 unstable PO. The good agreement between mode and ray calculations may be even improved by incorporating the Goos-Hänchen shift into the ray dynamics [41].

The localization happens when the mode pair satisfies a necessary requirement [25,27] which is, essentially, a rational winding number

*ω*are classical frequencies of the rays and Δ

^{i}*n*are mode number differences in the

^{i}*i*-th degrees of freedom and

*r*and

*t*are integers, respectively. This phenomenon can be understood by the well known Fermi resonance interpretation. In our case, a relation (Δ

*l*, Δ

*m*) = (

*t*,

*r*) is used where Δ

*l*≡ Δ

*n*and Δ

^{l}*m*≡ Δ

*n*are the radial and the angular mode number differences. According to this definition, (

^{m}*t*,

*r*) represents the classical POs performing

*r*-reflections during

*t*-cyclic motions. In this article, the considered mode pairs have the mode number difference (Δ

*l*, Δ

*m*) = (1, 3) and hence the PO should be also (

*t*,

*r*) = (1, 3), which corresponds to the orbits having 3-reflections during 1-rotating motion along the angular direction for a period (i.e., triangular orbit). In fact, we have already observed this relation in the previous example; the (

*l*,

*m*) = (11, 48) mode and the (10, 51) mode at

*ε*= 0.0025 together form the modes that are localized on the period-3 stable and unstable PO, respectively.

## 4. Separatrix modes

In this section we show that by increasing the deformation further the scarlike mode disappears and that at certain values of the deformation parameter a new type of localized mode appears.

#### 4.1. Periodic orbit quantization

As a first step the period-3 POs are quantized according to an one-dimensional EBK quantization scheme. In analogy to quantum mechanics, we consider the momentum *n _{e}k* (

*ħ*is set to unity) along the entire path of the PO, leading to [42,43]

*N*= 0, 1, 2, . . . and The Maslov index

*μ*counts the number of classical caustics crossed plus twice the number of hard-wall reflections [1, 31, 43–45]. For both period-3 orbits

*μ*= 9 because each orbit crosses the inner caustics three times and gets reflected at the cavity’s boundary also three times. The phase

*ϕ*belongs to the monodromy matrix which measures the stability of the PO. In Birkhoff coordinates it is expressed by [46]:

*s*

_{0(1)}and Δ

*p*

_{0(1)}are the initial (final) deviations in

*s*and

*p*axes, respectively. The eigenvalues of the matrix

*M*are

*λ*

_{±}=exp(±

*iϕ*). For the unstable PO the phase

*ϕ*is zero, because in that case the eigenvalues of the monodromy matrix are purely real. Finally, the Fresnel phase

*α*arises due to the reflection of the ray at the dielectric boundary. From the well-known Fresnel coefficients we get for TM polarization

*χ*is the incident angle of the PO at the

_{j}*j*-th bouncing point on the cavity’s boundary. The quantized frequency is given by

*l*is the length of the given PO. The quantities

_{po}*l*,

_{po}*ϕ*, and

*α*are numerically determined depending on the value of the deformation parameter.

#### 4.2. Numerical results

Figure 3(a) shows the quantized frequencies (7) of stable (squares) and unstable (triangles) POs as functions of the deformation parameter *ε*. Also shown are the modes in the regime 95.0 ≤ Re(*n _{e}kR*) ≤ 96.2. The solid line and the dashed one mark the two modes which are developed from the (11, 48) and the (10, 51) mode of the circular cavity. For very small

*ε*≈

*ε*

_{1}= 0.002 their frequencies agree well with the quantized frequencies. Moreover, one of the modes show a strong localization near the unstable PO, (

*s′*,

*p′*) in phase space, in terms of the averaged Husimi functions $\text{hu}({s}^{\prime},{p}^{\prime})={\sum}_{n=1}^{3}\text{hu}\left({s}_{n}^{\prime},{p}_{n}^{\prime}\right)/3$ at that location [see Fig. 3(b)]. This mode is the scarlike mode discussed in Sect. 3.

Upon further increasing the deformation the frequency of the former scarlike modes starts around *ε* = 0.006 to deviate from that of the quantized unstable PO, Fig. 3(a). This is fully consistent with the decrease of localization at the unstable point in Fig. 3(b) at around the same value of *ε*. The interpretation is that the increasing island swallows up the scarlike mode. It becomes a mode in the island close to the stable PO as it is confirmed in Figs. 4(b) and 4(e). However, in contrast to the mode localized around the stable PO [see Figs. 4(a) and 4(d)], it is localized on a torus surrounding the stable PO. It can be considered as a first excited island mode or as a first transversal excitation of the mode along the stable PO.

As explained, the increasing size of the island chain is responsible for the destruction of the scarlike mode. Interestingly, the very same mechanism gives birth to a new mode being localized around the unstable PO: By increasing the size of the island chain further a new mode comes closer to the separatrix of the island chain until it becomes a “separatrix mode”. Since the classical motion becomes very slow near the unstable PO, the separatrix mode mainly has intensity near this unstable PO [see Figs. 4(c) and 4(f)]. This scenario can be observed in Fig. 3. Upon increasing *ε* the mode originating from the (9, 54) mode of the circular cavity (dot-dashed curve) starts to come closer in frequency to the quantized unstable PO and accumulate intensity at the unstable point. The maximum localization appears at *ε* = 0.0175.

This mechanism goes now on and on. Upon further increasing the deformation the (9, 54) mode will be incorporated into the island. It is no longer a separatrix mode. On further increasing the deformation the (8, 57) mode (dotted curve in Fig. 3) comes close to the separatrix and shows strong localization near the unstable PO. The maximum localization of this separatrix mode is at *ε* = 0.049.

The whole progression of creation and annihilation of localization is summarized in Fig. 5. For *ε* ≈ 0 there is no localization along the boundary of the cavity. For *ε* ≈ 0.002 one mode is localized at the stable and one at the unstable PO (scarlike mode). Around *ε* ≈ 0.0175 two modes are located inside the island and one mode is located on the separatrix (separatrix mode) showing large intensity at the unstable PO. For *ε* ≈ 0.049 three modes stay inside the island and one on the separatrix.

#### 4.3. Pendulum approximation

For a further analysis we consider the pendulum approximation for an isolated island chain in phase space [47–51]

*H*

_{0}and the strength of potential are

*ω*(

*p*) = 2 cos

^{−1}(

*p*) and

*p*

_{t:r}= cos(

*πt/r*). It should be pointed out that

*V*

_{t:r}(

*ε*) is governed by the trace of the monodromy matrix

*M*

_{t:r}(

*ε*) calculated on the stable (

*t*,

*r*)-PO depending on the deformation parameter

*ε*. From the given Hamiltonian, we estimate a measure of the size of the island chain. In order to figure out this measure we, again, use an approximation in [48, 49, 51] which is optimized to determine the separatrix

*p*(

_{sep}*s*) of the island chain. The size of this separatrix is expressed in terms of momentum distances Δ

*p*(

*s*) from

*p*

_{t:r}as

*p*(

*s*

^{*}) which is reached whenever

*s*

^{*}≡

*s*(2

_{max}*z*+ 1)/2

*r*with integer

*z*.

Based on the pendulum approximation, we compute the inside-islands intensities ratio of the Husimi functions as shown in Fig. 6(a). As the deformation parameter increases, the modes are gradually occupying more area in the island chain. The figure confirms the discussed scenario: starting from weak deformation each mode first becomes a separatrix mode with high intensity around the unstable PO [cf. Fig. 3] and low intensity in the island chain [see Fig. 6(a)]. With increasing deformation the localization at the unstable PO reduces and at the same time the intensity in the island chain increases.

Another illustration of the capturing mechanism is shown in Fig. 6(b). The figure shows the expectation value 〈*δp*〉 = ∫ d*s*d*p* hu(*s*, *p*)|*p* − *p*_{t:r}|/ ∫ d*s*d*p* hu(*s*, *p*) of the distance from the *p*_{t:r} using Husimi functions as probability density in phase space and a resonance width using the pendulum approximation. As the deformation parameter increases, the distance of each mode to the island center, 〈*δp*〉, at some deformation becomes smaller then the size of the island chain, Δ*p*(*s*^{*}) consistent with Figs. 3 and 6(a).

## 5. Far fields patterns

Until now, our discussions have been showing no distinct openness feature of the dielectric microcavity. Now, far field emission patterns of the modes are investigated. By these studies, an almost identical directionality is discovered which seemingly not related to the mode localization regions. Figure 7 shows comparisons of far-field intensity patterns between the modes (*l*, *m*) = (11, 48) localized on the stable points in the islands and successively formed separatrix modes (10,51), (9,54), (8,57) at *ε* = 0.0020, 0.0175, 0.049, respectively. Hereafter we call the mode (*l*, *m*) = (11, 48) the island ground mode for convenience.

As we can see in the first row of Fig. 7, emission directions of the island ground and separatrix modes are difficult to discern. This surprising similarity of the far fields can be conceived via the phase space distributions of the modes. It is instructive to point out that there are some achievements to hybridize a directional low-Q mode with a high-Q weak directional mode [52]. The back bone of this idea is based on the superposition property of the modes at the avoided resonance crossing point and in that sense the avoided resonance crossing may serve the similar far field patterns of the interacting modes. However, the modes we are discussing in the present paper do not belong to that case since the similarity of the far field patterns is not restricted to the avoided resonance crossing point at all. It is more clear when we see, e.g., Figs. 7(c) and 7(f) which are the far fields of the modes having a large frequency difference at *ε*_{3} in Fig. 3.

It is obvious that a tunneling path from the localization points to the emission regions would take the shortest distance to the critical line of the leaky region. Thus, an emission tail takes the straight extension toward the critical line for the island ground mode while the tail of the separatrix mode occurs at the lowest point of the separatrix in phase space, which are at the same phase space coordinate *s* as the island mode.

Figure 8 shows Husimi functions of the island ground (above row) and separatrix (below row) modes. In the above row, the three straight stretched Husimi functions from the main localization regions to the critical line are evident while the below row exhibits emission tongues at the same *s* to the above row which are different from the main localization of the separatrix mode. Note that this mismatch between localization inside the cavity and the emission characteristics outside has already been treated in a different system exhibiting a similar phase space structure [53]. However we emphasize that their works are based on the perturbative treatment of complex ray orbits to describe the emission process while the main focus of the present paper is on the properties of the separatrix mode formation process depending on the deformation parameter.

## 6. Tunneling distances

In this section, we will address differences of tunneling emission distances of the modes which are measured from the cavity boundary. The distant emission from the cavity boundary is caused by the evanescent leakage fulfilling an angular momentum continuation across the cavity boundary [54–56]. The tunneling distance is given as

*p*= 1/

_{c}*n*is the critical line of the leaky region in phase space. The quantity Γ

_{e}*=*

_{p}*p*−

*p*is a tunneling distance from

_{c}*p*to

*p*in phase space (cf. Fig. 8) while

_{c}*δ*is a real space distance of the emission point measured from the cavity boundary.

Now, we recall the previous discussions that the emissions of the island ground mode and the separatrix mode share the same emission regions because the separatrix modes tunnel from the lowest points of the separatrix contours in phase space. Consequently, Γ* _{S}* of the separatrix mode is smaller than Γ

*of the island ground mode (see rightmost panels in Fig. 8) so that, from Eq. (12), we can expect the smaller*

_{I}*δ*for the separatrix mode than that of the island ground mode. In order to verify this claim we use, first, Lee’s “emission Husimi function” [55] which manipulates a sum of Gaussian overlap of the wave function and its derivative with respect to a certain virtual plane in the exterior region of the cavity. For the emission Husimi function we set the plane at

*y*

_{0}= 1.5

*R*to detect the field emitting vertically from the left end of the cavity boundary

*x*. In Fig. 9(a), this emission direction is guided by a vertical line with arrows and we can see the emission is parallel to the cavity boundary. By comparing all the emissions in Figs. 9(a) and 9(b) we can confirm the very similar emission behaviors of the island ground and separatrix modes again. In the followings we concentrate on the tunneling distances of the vertical emissions radiating from the left end of the cavity boundary.

_{b}The emission Husimi function *τ*(*x*) for our setup is :

*x*) = 0.5 [

*ψ*(

*x*,

*y*

_{0}) −

*i∂*(

_{y}ψ*x*,

*y*

_{0})/

*k*] and

*G*(

*x*,

*x′*) = 1/(

*σπ*)

^{1/4}exp [−(

*x*−

*x′*)

^{2}/2

*σ*]. Here

*ψ*(

*x*,

*y*

_{0}),

*∂*, and

_{y}*σ*are the wave function along

*x*-axis at fixed

*y*=

*y*

_{0}, partial derivative with respect to

*y*and width factor of the Gaussian set to $\sigma =\sqrt{2}/k$ for the wave number

*k*.

Figure 10 exhibits the comparisons of *τ*(*x*) between the island ground and separatrix modes in Fig. 7. From the behavior of the peak position of *τ*(*x*), we can observe two kinds of characteristics of the tunneling distances. The first one is a decreasing tendency of the tunneling distances in both modes as the deformation increases and the other one is an increasing difference of the tunneling distances between the modes depending on the increasing deformations. These are organized in Table 1 and explained below.

When the deformation is very weak Γ* _{I}* and Γ

*are either close to the resonant torus sin*

_{S}*χ*= 0.5 corresponding to

*δ*= 0.65

*R*when certain modes are on the torus. For strong deformation, the more curved KAM tori [57] result in smaller Γ

*and Γ*

_{I}*. Consequently, tunneling distances, in real space, have relatively closer values to 0.65*

_{S}*R*for small deformation and decrease toward zero following the increasing deformation as we can perceive in Table 1.

As a final remark we relate the far field intensities of Fig. 7 and the near field tunneling distances. By taking a Fourier transform of the far field intensity pattern, we can obtain an amplitude of an angular momentum number *m* as

*I*= |

*ψ*|

^{2}not amplitude

*ψ*for the purpose of mimicking the practical measurement. A normalizing factor

*𝒵*is introduced in order to manipulate a square absolute value of

*ℱ*(

*m*) as a weight function of

*m*, i.e.,

*P*(

*m*) = |

*ℱ*(

*m*)|

^{2}. Note that by comparing the Fourier transform of the wave amplitudes and of the wave intensities, we have observed significant additional side peaks in the intensity case (not shown), in particular, around

*m*= 3 corresponding to the 6 maxima in the far field intensity pattern. Thus, in order to abandon the effects from these contributions we consider only the range from

*m*= 20 to

_{i}*m*= 50.

_{f}Figure 11 shows weight functions of the far field intensities in Fig. 7 with respect to the angular momentum number *m*. According to the increasing deformation, from Figs. 11(a) to (c), a clustered distribution of *m* moves from high to low values for both modes while the distributions in the clusters are individually changing. Using this weight function, we can obtain the tunneling distances in Eq. (12) with averaged values of sin *χ* = *m/n _{e}kR* as below

## 7. Summary

We presented a new type of mode localization along unstable periodic orbits in deformed microdisk cavities. In the regime of weak deformation the near-integrable ray dynamics exhibits small island chains in phase space. When increasing the deformation the enlarging islands incorporate more and more modes. Each time a mode comes close to the separatrix of the island chain the mode exhibits a strong localization near the associated unstable periodic orbit.

We demonstrated this interesting effect by studying an oval-shaped cavity. For very weak deformation this cavity exhibits the conventional scarlike modes. We showed that by increasing further the deformation the revealed mechanism first destroys the scarlike mode and then subsequently creates and destroys the new type of localized separatrix modes. Using an EBK quantization scheme for the periodic orbits we derive a frequency condition for the localization. The mode shows the localization whenever its frequency is comparable to that of the quantized frequency of the unstable periodic orbit.

Through the investigations of the mode emission properties, very similar far field intensities and different tunneling distances are revealed.

## Funding

This research was supported by 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 and the DFG (project WI1986/7-1).

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