We investigate the impact of local dynamics on chaos-assisted tunneling in a highly deformed microcavity whose classical ray dynamics exhibits a small measure of trapezoidal-shaped orbit (TSO) stability islands in a main chaotic sea. These two classically completely decomposed regions in phase space can support resonance modes of their own respectively. Using numerical ray and wave analyses, we show that the emission characteristics of the TSO resonance mode are determined by local ray dynamics near TSO islands. The emission characteristics of the other high-Q resonance modes, on the other hand, are governed by usual ray-wave correspondence. We experimentally demonstrate that the TSO emission mode can be lased without selective excitations by devising a half-moon shaped highly deformed cavity. And we also show that the emission characteristics of the TSO lasing modes are well explained by numerical ray and wave analyses.
© 2013 Optical Society of America
Recent studies of microcavity lasers are mainly concentrated on finding cavity shapes in which lasing modes emit unidirectionally with high quality factors for the application to optoelectronic circuits and photonics . Besides these high expectations in practical applications, the study of microcavities is also heuristically important because it can provide us with a valuable experimental testing ground of the theoretical studies in the field of cavity quantum electrodynamics, quantum chaos, i.e., the study of quantum manifestations of classical chaos, in open systems , and purely quantum phenomena such as dynamical tunneling and chaos-assisted tunneling.
Considering the analogy of the barrier tunneling phenomena in the semiclassical solution of the one-dimensional symmetric double well, the dynamical tunneling phenomenon is originally suggested by Davis and Heller in 1981 as a subset of quantum events which cannot take place classically . Since then, many variations of the phenomena have been developed and suggested theoretically and experimentally. One of such variations of dynamical tunneling is chaos-assisted tunneling (CAT) phenomena . Since CAT is proposed by Bohigas et al. much work has been done in the field of microwave billiards [5, 6] and cold-atom systems [7, 8]. The essence of the phenomena is that the chaotic dynamics of the corresponding generic classical Hamiltonian system quantum mechanically can mediate dynamical tunneling between two regular regimes in classical phase space even though the probability of the direct dynamical tunneling is very small due to the remoteness of two regular regimes.
Most recently, the effect of CAT on the lifetime and emission patterns of high quality factor modes in deformed microcavities has been theoretically studied [9–11]. These theoretical expectations are experimentally realized by demonstrating the CAT phenomenon in a lasing mode of a microcavity laser whose cavity shape is smoothly deformed from a circle [12, 13]. They have noticed unexpected emission intensity and directionality of a special high-Q resonance mode that is supported by a classically regular submanifold embedded in the main chaotic phase space. Up to now, this has been regarded as the most decisive experimental evidence of the CAT in open systems, i.e., the microcavities . Also they have shown the mechanism of the CAT-induced directional emission by superimposing the unstable manifold onto the husimi function. In their experiment, they have performed selective excitations along the regular submanifold, that is the rectangular shape in configurational space, since various high quality modes localized along the perimeter of the cavity can be competitively excited by uniform excitations due to small and smooth deformations from a circular shape.
In this Letter, we investigate the impact of local dynamics on CAT-induced emission characteristics of resonance modes supported by stable period-four islands. The classical ray dynamics is mostly chaotic and the only relevant regular regime in the phase space is the period-four islands chain far above the critical line, which is a trapezoidal-shaped orbit (TSO) in configurational space. From wave calculations, we obtain the TSO resonance which is supported by the corresponding classical TSO islands chain and show that the emission characteristics of the TSO resonance determined by local ray dynamics near islands. In order to observe a TSO lasing mode without selective excitations, we devise a cavity shape which is highly deformed from a circle, i.e., a half-moon shaped cavity . The Q factors of the typical high-Q (WG-type and scar-type) resonances of the cavity can be considerably degraded by non-smoothness, i.e., only C1 continuity along the cavity boundary and highly deformed shape, i.e., almost a half circular shape, but the Q factor of the TSO resonance can be maintained relatively high due to total internal reflection (TIR). Therefore, we can expect to easily observe the TSO mode induced by CAT as a lasing mode in experiment without selective excitations.
2. Ray and wave simulation
The cavity shape is designed as follows: Two small circles with a radius of r = R/2 meet the boundary of a large circle with a radius R at an angle ϕ = 30° from the vertical axis. A linear section tangentially connects two small circles. Then the cavity boundary is composed of three circular arcs and one linear section as shown by the gray region in Fig. 1(b).
In Fig. 1(a), ray dynamics shows the mostly chaotic behavior in Birkhoff surface of section (BSOS), where s is the arclength measured along the cavity boundary and its conjugate momentum p = sinχ, i.e., the tangential momentum of an incident ray for an incident angle χ. Here, the ray trajectories propagates clockwise (p < 0) or counterclockwise (p > 0) direction due to the chaotic behavior in BSOS. There are two island chains embedded in the main chaotic sea. The centers of islands for each chain correspond to the period-four TSO and V-shaped orbit, respectively, as shown on the right in Fig. 1(a). The impact of the V-shaped orbit on high-Q resonances is negligible because of the normal incidency to the cavity boundary. However, the trajectories of the TSO islands chain are preserved by TIR. This island chain, i.e., the classical submanifold embedded in the main chaotic sea, can support the resonance in Fig. 1(c) that shows a typical TSO bare cavity resonance mode which is numerically obtained from the Helmholtz equation by using the boundary element method (BEM) . We assume the effective refractive index n = 3.3 and transverse electric (TE) polarization in order to maintain an integrity of our analyses with a fabricated InGaAsP semiconductor microcavity laser. The resonance position is Re(nkR) = 653.9354 and Im(nkR) = −0.0077, where k and R are the vacuum wavenumber and the radius of the large circle, respectively. Then, the Q factor, defined by Re(nkR)/|2Im(nkR)|, is about 42, 500, which is about 4 times larger than that of the typical high-Q resonances due to TIR. We can immediately see that the inside intensity pattern of this mode is supported by the chain of TSO islands. However, notice that the outside intensity pattern is not tangential to the cavity boundary which is expected from the evanescent emission. This is the refractive emission along the unstable manifold near the critical line of the cavity as Shinohara et al. reported in Ref. . Here, we will show more specifically that the emission characteristics of the TSO resonance mode induced by CAT are determined by the local dynamics of the beach area of the TSO islands, i.e., the initial values on the chaotic sea near the TSO islands determines the emission directions of the TSO resonance mode. The beach area implies the neighboring chaotic sea of the TSO islands. To give phenomenological explanation by classical ray dynamics, we will assume that trajectories in the beach area is continuously supplied by dynamical tunneling from the TSO islands.
A survival probability distribution (SPD) gives us intuitions on the overall ray dynamics of an open cavity and uncovers the foliated structures of the unstable manifolds of the cavity which is usually covered up by topological transitivity of chaotic trajectories in BSOS of a closed billiard [17, 18]. We prepare an ensemble of rays of 109 initial points with equal probability which uniformly covers the phase space (s, |p| ≥ 0.9). Then the SPD is obtained by summing up the survival probability of the ray ensemble upon each reflection to the cavity boundary after a transient time as shown in Fig. 2(a). The SPD reveals the unstable manifold that determines the route of refractive escape. Rays from the main escape regions of A and B below the critical line practically gives the main peaks (A,B) of the far-field pattern (FFP) in the ray calculation, as shown by the red dashed line in Fig. 2(c). Moreover, the SPD provides more information about the high-Q resonances since they are supported by the long term surviving ray trajectories of an open cavity . So, we obtain the high-Q resonances by the BEM in the range of 653.4 ≤ Re(nkR) ≤ 656.6, which are compatible with our cavity size. By averaging the emission intensities of 42 high-Q resonances with Q ≥ 5, 000, we show that the FFP of the wave calculation agrees well with that of the ray calculation as shown in Fig. 2(c). In our nkR regime, it is expected from the ray-wave correspondence in microcavities . The husimi functions for 42 high-Q resonances are obtained in order to be compared with the SPD, which is the wave analogy of the SPD. We classify them into two groups based on their inside intensity pattern as shown in Fig. 2(b) and 2(d). The groups of Fig. 2(b) and 2(d) are supported by the main chaotic sea and the classical TSO islands, respectively.
From the ray and wave calculations, we note that (i) the SPD is robust against variations of the initial ensemble of points because their overall ray dynamics is governed by the chaotic sea. The high-Q resonances such as WG-type, scar-type, and generic irregular-type resonances are supported by main chaotic sea. That is manifested by the close correspondence between the SPD of Fig. 2(a) and the husimi function of Fig. 2(b). (ii) As shown in the husimi function of Fig. 2(d), the high probability regions, composed of the TSO islands and neighboring chaotic sea, i.e., beach area, support the TSO resonance modes. The boundary of the high intensity regions in Fig. 2(d) is outlined by white solid curves in Fig. 2(b) for clarify. We note here that the ray trajectories in the chaotic sea can not enter the TSO islands in the SPD because two regions are completely decomposed in classical dynamics. So it is not surprising that the overall ray dynamics using SPD does not give good explanation to the emission characteristics of the TSO resonance.
Figure 3 shows the logarithmic husimi function of the TSO mode. The major intensity regions cover the classical TSO islands and their beach areas because we assume that the wave can continuously tunnel to the neighboring chaotic sea by a classically forbidden process, i.e., dynamical tunneling. Thus, the ray dynamic contributions of the beach areas can be included in the analysis to determine the emission characteristics of the TSO resonance mode. We prepare an initial ensemble of rays with an highly eccentric elliptic shape that encircles the TSO island and its beach area as shown by the green region marked by 1 in Fig. 3. We choose the size of the initial ensemble that corresponds to that of the high probability region in SPD (Fig 2(a)). Then the ensemble of rays are propagated to the clockwise direction and the ray trajectories are superimposed onto the husimi function until they cross the critical line, i.e., refractively emitted. The ray trajectories marked 2, 3, and 4 move along each TSO island during 3 iterations. After that the trajectories 5, 6, and 7 follows the chaotic dynamics and finally cross the critical line along the unstable manifold at 6 iterations. When the rays emit out, the direction is counter clockwise as shown the 7-th iteration in Fig. 3 so that the emission direction coincides with the experimental result as shown the inset. That is, the initial ensembles on the beach area well follow the husimi function. We find the significant overlap between the ray trajectories and the intensity pattern of the husimi function as shown in Fig. 3. The difference between the local dynamics near islands and the overall chaotic dynamics can be noticed by the distributions of the trajectories at the emission gate. While the structure of the emission gate due to overall ray dynamics is presented as a broad band tongue as shown in Fig. 2(a), that due to local dynamics is presented as an outline of the tongue because the refractive emission occurs from the island in a short time. The FFPs obtained from the local ray dynamics and TSO resonance are shown in the inset of Fig. 3. The qualitative correspondence is very good although the FFP of TSO modes consists of two sharp main peaks while that of local ray dynamics near islands consists of two broad peaks. This difference may come from the fact that the only two slightly separated TSO modes are taken into account in order to compare with a single TSO lasing mode of the experimental results (Fig. 4(a)). Therefore, we find that the dynamics of beach area, i.e., local dynamics, can play a key role to characterize the emission characteristics of the TSO resonance mode.
3. Experimental results
We perform a lasing experiment using a half-moon shaped InGaAsP semiconductor microcavity laser to ensure that the TSO mode can be lased by uniform excitations. By defining metal electrode on the whole cavity of R = 50 μm, the whole region is uniformly excited. The details of the fabrication process are described in Ref. . In the lasing experiment, we operate the microcavity laser in pulsed mode with 0.9 μsec pulse width at 1.0 MHz repetition rate. When the current is 70 mA, that is above the lasing threshold, the emission spectrum exhibits two lasing mode groups of A and B as shown in Fig. 4(a). The mode groups of A and B have the equidistant mode spacing of 2.8 nm and 2.7 nm, respectively. From the mode spacing, we obtain the path length L by using the equation , where ng = 3.68 is the group refractive index , and λavg and Δλ are the average wavelength and mode spacing of two neighboring modes, respectively. The equidistant mode spacing of 2.8 nm and 2.7 nm implies the path length of about 236 μm and 242 μm, which closely corresponds to the orbit length of 232 μm for the TSO and 252 μm for the perimeter of the cavity, respectively.
Experimental FFPs are obtained from measurements by rotating the fiber 600 μm apart from the center of the cavity which is shown in the inset of Fig. 4(b). Figure 4(b) shows the obtained FFPs in experiment and in TSO resonance mode. We notice that two main peaks near 60° which correspond to our theoretical ray and wave emission characteristics of the TSO mode. Therefore, as we expected, we can conclude that the TSO lasing mode is excited without selective excitations and that is confirmed by considering both the emission spectrum and FFP. However, WG-type mode, whose far field emission pattern is characterized by a single main peak, can be also excited for different configurations of pulse widths and injection currents . Based on our observation, the TSO lasing mode tends to excite when the pulse width is getting longer.
We have explored the impact of local dynamics on the CAT-induced emission characteristics of the resonance modes supported by the chain of TSO islands in the half-moon shaped microcavity. There is a good correspondence between the global SPD and husimi function of high-Q modes and the local dynamics near islands and husimi function of the TSO mode, respectively. It is well known fact that the emission route in microcavities is along the unstable manifold near the critical line. However, in our case, even though the emission route is similar, there is a little difference in the structure of the emission gate between the overall ray dynamics and local dynamics near islands. This difference is manifested by the FFP of the emission patterns both experimentally and numerically. We have performed an experiment to ensure that the TSO mode can be lased by uniform excitations. Based on the longitudinal mode spacing and FFP, we conclude that the TSO mode is lased as a dominant lasing mode. By comparing the emission direction of the TSO mode with that of the rest of high-Q modes, we have shown that the difference in emission direction is indeed caused by the CAT-induced emission as described by local dynamics of near TSO islands. Therefore, this fact, in turn, can be interpreted as another strong evidence of the CAT since the dynamical tunneling is most likely to occur near the island region in phase space.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning ( NRF-2013R1A1A2060846).
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