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

Wave-chaos-induced single-frequency lasing in microcavities
Satoshi Sunada1 , Susumu Shinohara2 , Takehiro Fukushima3 , and Takahisa Harayama4
1

Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University
Kakuma-machi Kanazawa, Ishikawa 920-1192, Japan
2
NTT Communication Science Laboratories, NTT Corporation
2-4 Hikaridai Seika-cho Soraku-gun, Kyoto 619-0237, Japan3
Department of Information and Communication Engineering, Okayama Prefectural University
111 Kuboki Soja, Okayama 719-1197, Japan
4
Department of Applied Physics, School of Advanced Science and Engineering, Waseda University
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email: sunada@se.kanazawa-u.ac.jp
Abstract—We experimentally and numerically demonstrate that stable single-frequency (i.e., single-mode) lasing
can be achieved in microcavities that have wave-chaoticmodes. The single-frequency lasing is explained as a result of strong competitive interactions among wave-chaoticmodes.
1. Introduction
Various active devices ranging from musical instruments
to lasers generate oscillating states with well-defined frequencies from the interplay between resonator geometry
and an active nonlinear element [1, 2]. Understanding andcontrolling the formation of such self-organized oscillatingstates is important in device physics and related applications. As a specific example, two-dimensional (2D) microcavity lasers have attracted considerable attention overthe past decades [3, 4, 5]. Depending on the cavity shapes,
they can exhibit a variety of lasing states through the interaction between the light field and active gain material [2].
The studies of 2D microcavity lasers have led to a wide
range of applications, including low-threshold microlasers
with unidirectional emission [6], low-coherence microlasers [7], and fast random signal generation [8]. Moreover,
2D microcavity lasers have served as a platform for experimentally addressing fundamental issues such as quantum/wave chaos in open systems [5] and non-Hermitianphysics [9, 10].
The lasing emission patterns and spectra are fundamental and important characteristics of 2D microcavity lasers.
Previous experimental and theoretical studies have shownthat the lasing emission patterns can be well characterized
by resonant modes, which are determined only by the cavity shape and refractive index [3, 4]. Meanwhile, the spectral characteristics are still poorly understood because nonlinear interaction among resonant modes plays an crucialrole in determining lasing frequencies.
In this presentation, we report our recent finding on a relationship between shapes of 2D microcavities and the lasing frequencies [11]. We show that stable single-frequency(i.e., single-mode) lasing can be achieved in fully chaoticcavities, where all of the internal ray orbits are chaotic in a
ray optics picture and the spatial patterns of the modes arewave-chaotic, whereas multi-frequency (i.e., multimode)
lasing is exhibited in non-chaotic cavities. The achievement of single-mode lasing in fully chaotic cavities is explained as a result of strong mode competition among wave
chaotic modes.

2. Fully chaotic and non-chaotic microcavities
A specific shape of the fully chaotic cavity that we study
here is a stadium [12], which is widely used for classicaland quantum chaos studies. As shown in Fig. 1(a), the stadium cavity consists of two straight lines of length l andtwo half circles of radius R. We define the aspect ratio parameter p = W/L, where W = 2R is the length of the minor
axis and L = 2R + l is the length of the major axis. For thenon-chaotic cavity, we focus on the elliptic cavity definedin Fig. 1(b). It is known that a closed elliptic cavity is an
integrable system [13], thus exhibiting no chaotic behavior.
In the same manner as for the stadium cavity, we define theaspect ratio parameter for the elliptic cavity as p = B/A,
where A and B are the lengths of the major and minor axes,respectively.
We fabricated semiconductor microcavities with thestadium and elliptic cavities by applying a reactive-ionetching technique to a graded index separate-confinementheterostructure(GRIN-SCH)
single-quantum-well
GaAs/Al x Ga1−x As structure grown by MOCVD (SeeRef. [14] for details on the layer structures and fabrication
process). The fabricated lasers are shown in Figs. 1(c) and1(d). In our experiments, the lasers were soldered onto
aluminum nitride submounts at 20 ± 0.1 ◦ C and electricallydriven with cw current injection. The optical outputs werecollected with anti-reflection-coated lenses and coupled toa multimode optical fiber via a 30-dB optical isolator.

- 557 -Figure 1: (a) Stadium cavity and (b) elliptic cavity. (c)(d)Optical microscope images of the fabricated lasers, where
both cavities have an area S = 8748 µm2 and an aspect ratio
p = 0.5.
3. Experimental results

4. Discussion and Analysis

Figure 2: Spectra of the stadium and elliptic cavity lasers
with a cavity area S = 8748 µm2 and an aspect ratio p =0.5 for the injection current I = 140 mA. The thresholdcurrents of the stadium and elliptic cavity lasers were 74
and 34 mA, respectively.

25
Number of peaks

Figure 2 shows typical lasing wavelength spectra for thestadium cavity laser and elliptic cavity lasers. Because thelasing wavelength λ is related to a frequency f by c = f λ,where c is the light velocity, we dicuss the spectral characteristics in the wavelength regime. As shown in Fig. 2,
there is a remarkable diﬀerence in the number of peaks between the two lasers. The spectrum of the stadium cavity laser exhibits only a single sharp peak, suggesting single frequency (single-mode) lasing, despite the fact that thenumber of modes within the gain band is more than a few
thousand because of the large cavity area. Meanwhile, thespectrum of the elliptic cavity laser always exhibits multiple peaks, i.e., multimode lasing.
We systematically investigated the spectral characteristics of the two lasers with various cavity areas S and aspectratios p. Figure 3 shows the number of peaks whose intensities were larger than −20 dB of the maximum peakintensity in each spectrum as a function of the current I
normalized by the threshold current Ith for each laser. The
spectra of the elliptic cavity lasers always exhibit multiple
peaks, i.e., multimode lasing, and the number of peaks increases as I increases. On the other hand, the spectra of
the stadium cavity lasers always exhibit a single peak, i.e.,single-mode lasing, regardless of the cavity areas and aspect ratios, and the single-mode lasing is maintained evenfor high injection current values I. However, as a slight exception, we observed two peaks for the stadium cavity laser
for I/Ith = 2.1 and 2.7 in Fig. 3. We attribute this to mode
hopping caused by a thermal eﬀect of the current injection,such as a gain shift and a change in the refractive index.

Stadium
Ellipse

20
15
105
0
1

1.5

2
2.5
3
3.5
Normalized current I/Ith4Figure 3: Injection current dependence of the numberof peaks in the spectra of the stadium cavity lasers (redcrosses) and the elliptic cavity lasers (blue circles) withcavity area S = 4463, 6427, and 8748 µ m2 and aspectratios in a range of 0.3 ≤ p ≤ 1.

The above results indicate that strong suppression ofmultimode lasing is a common feature of stadium cavity- 558 -

0.7
Stadium
Ellipse0.6
Frequency

0.5
0.4
0.3
0.2
0.1
00

Figure 4: (a) Spatial intensity pattern of a low-loss mode ina quarter of the stadium cavity. (b) Spatial intensity patternof a low-loss mode in a quarter of the elliptic cavity.
lasers. As discussed in Refs. [15, 16], a competitive interaction occurs among modes that are overlapped not only
spectrally but also spatially. In a fully chaotic cavity, modes
typically have complex spatial patterns that spread throughout the entire cavity due to the ray dynamical property [fora typical example, see Fig. 4(a)], and therefore result inlarge spatial overlaps with other modes. Actually, previous numerical simulations of stadium cavity lasers demonstrated a strong selection of lasing modes owing to a competitive interaction [17, 18, 19]. On the other hand, nonchaotic cavities typically support spatially localized modes[e.g., Fig. 4 (b)]. Interestingly, diﬀerent modes are localized in diﬀerent areas. Thus, when the spatial overlap
among modes is small, the competition among them canbe avoided. Indeed, the simultaneous lasing of multiple
modes for a non-chaotic cavity laser with a circular shape
was numerically demonstrated in Ref. [20].
To quantify the spatial overlaps between two modes in acavity, we introduce the following cross-correlation for theamplitude distributions of resonance modes:∫|ϕ(r)| |ψ(r)| w(r)drC = √(∫(1)) (∫),
|ϕ(r)|2 w(r)dr|ψ(r)|2 w(r)drwhere ϕ(r) and ψ(r) are the modal wave functions andw(r) represents a pumping region. For uniform pumping,w(r) = 1 inside the cavity, whereas w(r) = 0 outside. The
correlation is essentially similar to a spatial contribution to
the cross-gain saturation (i.e., intensity cross-correlation)
between two modes [15, 16, 21]. Using the boundary element method [22], we calculated the resonances of the stadium and elliptic cavities with p = 0.5, imposing a refractive index of 3.3 inside the cavities and transverse electric(TE) polarization. Because of computational power limitations, we set a size parameter 2πR/λ ≈ 100, where R and λ

0.2

0.4
0.6
Overlap C

0.8

1

Figure 5: Histogram of overlap C defined by Eq. (1) between two low-loss modes for the stadium cavity (solid)
and elliptic cavity (dotted).
are the characteristic radius and wavelength, respectively.
This size parameter value is smaller than that of a real laser
cavity used in the experiments but is suﬃciently large todiscuss the properties of the wave functions in the shortwavelength regimes [23]. We obtained approximately 100
low-loss modes with a quality factor Q ≥ 3000 for the stadium cavity, whereas Q ≥ 3 × 105 for the elliptic cavity.
Typical examples of the wave functions for stadium and elliptic cavities are shown in Fig. 4.
Figure 5 shows the histogram of the spatial overlap C
between two low-loss modes in stadium and elliptic cavities. The C-values for the stadium cavity are distributed
around 0.77. In contrast, the C-values for the elliptic cavity
are widely distributed with a mean value of 0.45. The relatively low C-values come from the small spatial overlaps
between the localized modes. In particular, the overlaps aresmall between two modes with diﬀerent radial mode numbers nr , which characterize the number of field maxima in
the radial direction. For instance, the C-value between themodes with nr = 5 and nr = 1 was only 0.14. As seenin Fig. 2 (b), the lasing peaks in the spectra of the elliptic cavity laser are not always equally spaced. This resultmeans that modes with diﬀerent nr -values were involved in
the lasing, which supports our interpretation of the relation
between the spatial overlaps and the spectral characteristics.
5. Conclusion
We experimentally investigated the diﬀerence in thespectral characteristics between fully chaotic cavity lasers
with a stadium shape and non-chaotic cavity lasers with anelliptic shape. In the stadium cavity lasers, only a single
mode was excited at high pumping regimes regardless ofthe size and aspect ratio, whereas many modes were excited in the elliptic cavity lasers. The strong suppression ofmultimode lasing observed in the stadium cavity lasers can- 559 -be explained by the large spatial overlaps among the lowloss modes. Because a common feature of modes in a fully
chaotic cavity is the spatial pattern that spreads throughout the entire cavity, we expect that the modal suppression
leading to single-mode lasing is a universal feature of fully
chaotic cavity lasers.
Acknowledgments
This work was supported by JSPS KAKENHI Grant
Numbers 26790056 and 16K04974.
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