1. Introduction
I began studying laser diodes with a chaotic cavity while visiting Prof. Siegman’s group in the Edward L. Ginzton Laboratory at Stanford University from November 1995 to February 1996 as a visiting scholar. After that, I continued research on this topic, collaborating with researchers of Advanced Telecommunications Research Institute International, Max-Planck Institute for the Physics of Complex Systems, Purdue University, NTT Communication Science Laboratories, and so on. Here, I will introduce our major results on this research topic.

2. Lasing characteristics of laser diodes with quasi-stadium chaotic cavities
Considering the motion of a classical particle moving freely and bouncing elastically in a stadium-shaped box, one finds closed periodic orbits like those illustrated in Fig. 1(a). However, all of the periodic orbits are unstable under perturbation, which means that the trajectory of the particle changes from a regular (periodic) orbit to a chaotic trajectory, as shown in Fig. 1(b), as soon as the particle strays slightly off the periodic orbit due to a small perturbation. Accordingly, the motion of a classical particle confined in a stadium-shaped box is chaotic in general. In quantum mechanics, however, the probability for finding a particle in a box is given by the square of the wavefunction. The connection between the classical and quantum mechanics of such a particle is an interesting topic in the research fields of classical and quantum chaos [1]. If one replaces the box and the trajectories of the classical particle with a stadium-shaped optical cavity and ray trajectories confined in the cavity, the correspondence problem between the classical and quantum mechanics in the box can be replaced with the problem of ray-wave correspondence in the optical cavity.

Fig. 1 Trajectories of a classical particle confined in a stadium-shaped box.

Prof. Siegman’s group, which was interested in this problem at the time of my visit, fabricated and tested laser diodes with quasi-stadium cavities, as illustrated in Fig. 2. We measured the lasing characteristics of these laser diodes and carried out numerical calculations for obtaining resonator modes of the quasi-stadium optical cavity. Here, I introduce our experimental and theoretical results [2,3].

Fig. 2 Schematic illustration of laser diode with a quasi-stadium cavity.

Figure 3 shows the lasing characteristics of the quasi-stadium laser diode. Clear build-up of the output power was confirmed from the light output power vs. injection current characteristics, and a sharp peak appeared in the lasing spectrum (see figure insets). The images on the right-hand side of the figure show the output beam pattern observed using a lens and CCD camera. The numerical values shown in the images correspond to the distance from the output end mirror facets. Although fine interference patterns were observed, the output beams emitted from the flat end mirror split into three directions. In contrast, the output beams emitted from the curved end mirror first focus to three spots, and then diverge again and produce fine interference patterns.

Fig. 3 Lasing characteristics of laser diode with a quasi-stadium cavity.

To explain the observed output beam patterns, we calculated the lowest-loss resonator mode in the quasi-stadium cavity using an extended version of the Fox-Li mode calculation method. Figure 4 shows the output beam patterns emitted from the output mirror facets for the lowest-loss resonator mode. The calculated output beam patterns well reproduced the experimental results. We also calculated the beam propagation behavior inside the cavity and found that the optical beams are localized on unstable periodic orbits. The wavefunction localized on unstable periodic orbits in stadium-shaped boxes (called “scar modes ” in the research field of quantum chaos) was obtained from our numerical calculations. The similarities between resonator and scar modes are interesting from the viewpoint of fundamental physics [3].

Fig. 4 Calculated output beam patterns emitted from the end mirrors.

The calculation results introduced here are published in the OSA’s open access online journal [3]. In this article, one can watch short animations of beam propagation from the end mirrors. Moreover, these results were introduced in Prof. Siegman’s review article “Laser Beams and Resonators: Beyond the 1960s ” published in the millennium issue of the Journal of Selected Topics in Quantum Electronics [4].

3. Lasing characteristics of laser diodes with stadium-shaped cavity
The quasi-stadium laser diode shown in Fig. 2 is somewhat special. The lasing characteristics of the stadium-shaped cavity shown in Fig. 1 are interesting from the viewpoint of fundamental physics.
We investigated the lasing characteristics of laser diodes with the stadium-shaped cavity shown in Fig. 5, in collaboration with Advanced Telecommunications Research Institute International [5]. The laser diodes were partially fabricated by NTT Advanced Technology Corporation. Figure 6 shows the lasing characteristics of the laser diode.

Fig. 5 Laser diode with stadium-shaped cavity.

Fig. 6 Lasing characteristics of laser diode with stadium-shaped cavity. Samples A and B show the experimental results for two different samples fabricated on the same epitaxial wafer.

The threshold current of the laser diode is approximately 200 mA, above which a sharp peak appears in the lasing spectrum. The light emission in the long-axis direction is enhanced and the light emission in the short-axis direction becomes weak. Although the far-field emission pattern contains complicated interference patterns, the envelope of the far-field emission pattern agrees with the ray dynamical simulation very well.
These experimental results were reported in our invited paper published in the issue of the IEEE Journal of Selected Topics in Quantum Electronic on Nonlinear Optics [5]. To the best of our knowledge, this was the first report on the lasing of a stadium-shaped laser. Since then, the lasing characteristics of stadium-shaped lasers with different materials, cavity sizes, and excitation techniques have been reported [6,7,8].

4. Observation of the chaos-assisted tunneling using a chaotic cavity[9]
Although this work was organized by Dr. Shinohara of the Max-Planck Institute (Currently at Waseda University), I will briefly introduce our experimental results here, because this work has attracted considerable attention in the research field of quantum chaos.
The egg-shaped cavity shown in Fig. 7 supports two types of ray trajectories: the regular ray trajectories (blue line) in the vicinity of the rectangular stable periodic orbit (red line) shown in Fig. 7(a), and the chaotic ray trajectories (blue line) shown in Fig. 7(b). The optical rays that slightly deviate from the rectangular stable periodic orbit propagate in the vicinity of the stable periodic orbit, as illustrated by the blue lines in Fig. 7(a), and never convert to chaotic ray trajectories. In the same manner, the chaotic ray trajectories never convert to regular ray trajectories. This means that, in the case of ray dynamics, the regular ray trajectories in the vicinity of the stable periodic orbit and the chaotic ray trajectories are independent from one another. However, when considering wave propagation, a small amount of energy of the light propagating along the stable periodic orbit can be transferred to that of the chaotic ray trajectories. This phenomenon is called chaos-assisted tunneling.

Fig. 7 Ray trajectories confined in the egg-shaped cavity.

We confirmed the existence of chaos-assisted tunneling in the egg-shaped cavity. In particular, we prepared laser diodes with the egg-shaped cavity shown in Fig. 7 and excited the resonator mode corresponding to the rectangular stable periodic orbit selectively using an electrode contact formed along the stable periodic orbit. Then, the far-field emission pattern from the laser diode was observed. The optical beam propagating along the stable periodic orbit is totally reflected at the cavity edges and confined in the cavity, because the incident angle of the optical beam is much larger than the critical angle. In this case, if chaos-assisted tunneling does not occur, it is expected that only the evanescent wave formed outside the cavity edge near the reflection points is emitted slightly in the tangential directions (±45^o and ±135^ofrom the horizontal axis) of the cavity edge, as indicated by the blue arrows in Fig. 7(a). In contrast, if chaos-assisted tunneling occurs, it is expected that a portion of the energy propagating along the rectangular stable periodic orbit will transfer to the chaotic trajectories and spread in the cavity. The light following the chaotic trajectories eventually breaks the condition of total internal reflection and is emitted from the cavity edges. Our calculations show that the light is mainly emitted in the directions ±90^o from the horizontal axis, as indicated by the blue arrows in Fig. 7(b). Accordingly, one can confirm the presence or absence of chaos-assisted tunneling by observing the far-field emission pattern of the laser diode.
Figure 8 shows a scanning electron microscope image of the fabricated laser diode and the observed far-field emission pattern. The laser diode has an electrode contact along the rectangular stable periodic orbit to excite the resonator modes corresponding to the periodic orbit. We confirmed that the lasing spectrum exhibits equidistant peaks whose spacing agrees very well with that estimated from the path length of the rectangular stable periodic orbit. The observed far-field emission pattern exhibits two broad peaks around ±90^o. Therefore, we concluded that chaos-assisted tunneling occurs in the egg-shaped cavity.

Fig. 8 Scanning electron microscope image of laser diode with an egg-shaped cavity and its far-field emission pattern.

This work was introduced in Prof. Stone’s review article “Chaotic Billiard Lasers ” published in Nature in the News & Views Section in 2010 [10].

Acknowledgements
I would like to thank the late Prof. A. E. Siegman of Stanford University as well as Dr. C. G. Fanning, Dr. Y. Cheng, Dr. Y. Sun, and Dr. S. A. Biellak of the Siegman laboratory, for giving me the opportunity to begin researching chaotic cavity lasers in the early stage of my career. I also want to thank my long-term collaborators, Dr. S. Shinohara of NTT Communication Science Laboratories, Prof. T. Harayama of Waseda University, and Prof. S. Sunada of Kanazawa University, and my collaborators in select research topics, Prof. E. E. Narimanov of Purdue University and Prof. J. Wiersig of Otto von Guericke University at Magdeburg. Finally, I would like to thank all contributors to this research at Advanced Telecommunications Research Institute International, NTT Communication Science Laboratories, and NTT Advanced Technology Corporation.

References
[1] M. C. Gutzwiller, Chaos in classical and quantum mechanics, Springer Verlag, New York, 1990.
[2] T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, Lasing characteristics of a quasi-stadium laser diode, Conference on Laser and Electro-Optics (CLEO’97) OSA Technical digest (Optical Society of America), paper CWF6, 1997.
[3] T. Fukushima, S. A. Biellak, Y. Sun, and A. E. Siegman, Beam propagation behavior in a quasi-stadium laser diode, Optics Express, vol. 2, no. 2, pp. 21-28, 1998.
[4] A. E. Siegman, Laser beam and resonators: Beyond the 1960s, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, pp. 1389-1399, 2000.
[5] T. Fukushima and T. Harayama, Stadium and quasi-stadium laser diodes, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 5, pp. 1039-1051, 2004.
[6] M. Lebental, J. S. Lauret, R. Hierle and J. Zyss, Highly directional stadium-shaped polymer microlasers, Applied Physics Letters, vol. 88, no. 3, pp. 031108, 2006.
[7] W. Fang, H. Cao, and G. S. Solomon, Control of lasing in fully chaotic open microcavities by tailoring the shape factor, Applied Physics Letters, vol. 90, no. 8, 081108, 2007.
[8] H.-G. Park, F. Qian, C. J. Barrelet, and Y. Li, Microstadium single-nanowire laser, Applied Physics Letters, vol. 91, no. 25, 251115, 2007.
[9] S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, Chaos-assisted directional light emission from microcavity lasers, Physical Review Letters, vol. 104, no. 16, 163902, 2010.
[10] A. D. Stone, Chaotic billiard lasers, Nature, vol. 465, no. 7299, pp. 696-697, 2010.