2016 International Symposium on Nonlinear Theory and Its Applications,
NOLTA2016, Yugawara, Japan, November 27th-30th, 2016

Asymmetric emission caused by chaos-assisted tunneling andsynchronization in two-dimensional microcavity lasers
Yuta Kawashima1 , Susumu Shinohara2 , Satoshi Sunada3 , Takehiro Fukushima4 and Takahisa Harayama1
1

Department of Applied Physics, School of Advanced Science and Engineering, Waseda University,
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan2NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai,Seika-cho, Soraku-gun, Kyoto 619-0237, Japan3
Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University,Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
4
Department of Information and Communication Engineering, Okayama Prefectural University,
111 Kuboki, Soja, Okayama 719-1197, Japan
Email: louzoulbouce@asagi.waseda.jp

It has been experimentally reported that low threshold lasing and directional emission can be simultaneously achieved in a two-dimensional microlaser with thequadrupolar cavity [1]. The quadrupolar cavity is definedin the polar coordinates (r, θ) asr(θ) = r0 (1 + ϵ cos 2θ),

(1)

where r0 is the size parameter and ϵ is the deformation parameter fixed as ϵ = 0.09. In this cavity, a resonant modethat localizes along a pair of stable triangular orbits is confined by total internal reflection. This strong light confinement leads to a low lasing threshold, while weak light emission occurs by chaos-assisted tunneling [2]. The tunneling
induces light intensity leakage from the stable triangular orbits to chaotic orbits that eventually escape from the cavity
violating the critical angle condition for total internal reflection, where the chaotic dynamics governed by unstablemanifolds results in directional emission [3].
The emission patterns of two-dimensional microcavity
lasers can be theoretically studied by analyzing the resonant modes of the two-dimensional Helmholtz equation [4]with the parities a, b ∈ {−, +}. Therefore the emission patterns of individual resonant modes have the same symmetry
as the cavity shape (i.e., symmetric with respect to both x
and y axis). However, when we consider nonlinear modal
interaction through a lasing medium, the locking, or synchronization of two diﬀerent parity modes can occur, and it
yields an asymmetric emission pattern [6].
In this presentation, we show that the experimentally observed asymmetric emission pattern can be explained by
the locking of a nearly degenerate pair, ψ++ and ψ+− (i.e.,the modes with slightly diﬀerent wave numbers). First, wedemonstrate that although ψ++ and ψ+− are localized along
the pair of stable triangular orbits, their superposition turns
out to be localized only along one of the pair. Secondly,
by using the Maxwell-Bloch model that takes account thenonlinear eﬀect of a lasing medium [7], we actually simulate the locking of the nearly degenerate pair as well as theappearance of the asymmetric emission patterns.
References(2)[1] N.L. Aung et al., Appl. Phys. Lett. 107, 151106 (2015).

where n(x, y) is the refractive index and k is the vacuum
wave number. The real part of k represents the resonant
wave number, while the imaginary part represents the decay rate of the resonant mode. We consider transversemagnetic polarization, namely, ψ(x, y) represents the zcomponent of the electric field Ez . The refractive index
inside the cavity is nin = 3.3, while it is nout = 1 outside
the cavity. Equation (2) can be numerically solved by, forexample, the extended boundary element method [5].
In the experiment in Ref. [1], asymmetric directionalemission patterns were observed despite the symmetry ofthe quadrupolar cavity. The wave functions of the resonantmodes are divided into the four symmetry classes[2] V.A. Podolskiy and E.E. Narimanov, Opt. Lett. 30, 474(2005).

[∇2 + n2 (x, y) k2 ] ψ(x, y) = 0,

ψab (−x, y) = a ψab (x, y)ψab (x, −y) = b ψab (x, y),[3] H.G.L. Schwefel et al., J. Opt. Soc. Am. B 21, 923(2004).
[4] T. Harayama and S. Shinohara, Laser Photonics Rev.
5, 247 (2011).
[5] J. Wiersig, J. Opt. A : Pure Appl. Opt. 5, 53 (2003).
[6] T. Harayama et al., Phys. Rev. Lett. 91, 073903 (2003).
[7] T. Harayama et al., Phys. Rev. A. 72, 013803 (2005).

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