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Chirality of a resonance localized on an islands chain is studied in a deformed Reuleaux triangular-shaped microcavity, where clockwise and counter clockwise traveling rays are classically separated. A resonance localized on a period-5 islands chain exhibits chiral emission due to the asymmetric cavity shape. Chirality is experimentally proved in a InGaAsP multi-quantum-well semiconductor laser by showing that the experimental emission characteristics well coincide with the wave dynamical ones.
© 2017 Optical Society of America
Chirality of a resonance was firstly observed in a spiral-shaped microcavity laser by Chern et al [1]. When a resonance emitting through a notch is separated into a clockwise (CW) and a counter clockwise (CCW) propagating wave, the emission intensity of one component is stronger than that of the other [1, 2]. Then soon, it was found that a triangular or star-shape resonance has a similar behavior of spatial chirality [3]. The resonance, named a quasi-scarred resonance, is a linear combination of normal modes [4]. It was also shown that the superposed state of a pair of highly nonorthogonal resonances exhibits chirality [5]. This phenomenon has been attributed to the asymmetric backscattering between the CW and the CCW component.
Recently, chirality was observed in such microcavities as a deformed Limaçon-shaped and a Gutkin’s cavity [6,7] and a circular cavity with a spiral-shaped hole [8], which are without mirror reflection symmetry. This phenomenon has a variety of advantages not only in applications but in theoretical studies. In an ultrasmall deformed Limaçon-shaped microcavity, local chirality results in unidirectional emission for directional evanescent coupling [9]. When the laser rotates, one resonance in a pair switches its chirality while the other remains [10]. Because chirality also occurs when a microcavity is perturbed by external particles due to scattering [11, 12], single particle detection is achieved by using an exceptional point [13]. Chirality was also observed at an exceptional point [14] and in a parity-time symmetric quantum ring [15].
In an asymmetric cavity, when CCW and CW rays are classically separated, there is no back-scattering leading to chirality in the survival probability distribution as in the Gutkin’s cavity [6]. In the present paper, we study chirality of a resonance, which is localized on an islands chain, in an asymmetric Reuleaux triangular-shaped microcavity, where CW and CCW traveling rays are separated. Because of localization on an islands chain, the resonance emits due to chaos-assisted tunneling [16–19]. After tunneling, due to asymmetric deformation, the CCW and the CW wave follow asymmetric manifolds in the chaotic sea and result in chirality. We experimentally prove this chiral emission by using an InGaAsP multi-quantum-wells microcavity laser.
The Reuleaux triangle is a shape formed from the intersection of three circles, each of which has its center on the boundary of the other two. It is a curve of constant width. A deformed Reuleaux triangle has six different circular arcs with the following relations:
where r0 is the radius controlling the cavity size, ∊ is the deformation parameter, θ1 is the angle of a vertex, and θ2 is the angle of another vertex. Hence, each vertex of the triangle has two circular arcs with a different radius and the six arcs are tangentially connected with each other as shown in Fig. 1(a). Here we take ∊ = 9.2, θ1 = 0.11π, and θ2 = π/2. Because the opposite arcs have the same center, the CCW and the CW rays are classically separated at p = sin χ = 0.0. In this parameter, although chirality is not maximized, we can analyze chirality of lasing modes localized on a period-5 islands chain in the absence of backscatterings.
Fig. 1 The shape of a microcavity and the positons of resonances. (a) is the shape of a microcavity according to Eq. (1). (b) is a fabricated laser. (c) is the positions of resonances. The red big dots are resonances localized on a period-5 islands chain. The inset is the resonance positions around A, which shows a pair of resonances. In (a), l is the reflection position on the boundary from x-axis, and χ is the incident angle.
To coincide with the condition of our fabricated laser as shown in Fig. 1(b), we take transverse electric polarized waves and the effective refractive index of 3.3. The Helmholtz equation is solved by using the boundary element method [20] in the region of 391 < nL/λ < 395 in consideration of the fabricated laser size, where n is the effective refractive index, λ is the vacuum wavelength, and L is the total boundary length given by that L = θ1(r0 + r3) + θ2(r1 + r4) + (π − θ1 − θ2)(r2 + r5). To study chirality, four pairs of resonances localized on the period-5 islands chain are obtained as shown in Fig. 1(c).
We take a resonance, whose eigenvalue is Re(nL/λ) = 393.4524 and Im(nL/λ) = −0.0039 (a red circle in the inset). To separate the CCW and the CW propagating wave component, the coefficient of the angular momentum αm is obtained by using the following Fourier transformation:
where Jm is the m-th order Bessel function and is the Hankel function of the first kind. Equations (2) and (3) are the resonance inside and outside of the cavity, respectively.Figure 2(a) is the angular momentum distribution, |αm|2 versus m. The inner (two red dashed curves) and the outer (two blue solid curves) distributions are the spatial modal distribution outside and inside of the cavity, respectively. The separation is caused by the effective refractive index 3.3. At a glance, we see the CW and the CCW spatial modal distribution are not symmetric. The CCW propagating wave is the sum over all positive m’s and the CW one the sum over all negative m’s. The total spatial distribution of the resonance shown in Fig. 2(b) is separated into a CCW and a CW propagating wave as shown in Figs. 2(c) and 2(d), respectively. The resonance inside of the cavity shows a pentagonal orbit, which is a period-5 islands chain. The far-field patterns (FFPs) of the total resonance and the CCW and the CW propagating wave are obtained as shown in Figs. 2(e)–2(g), respectively. The main emission intensity of the CCW propagating wave is stronger than that of the CW one. The chirality of this resonance is about 0.102, which is obtained by using the following relation:
Fig. 2 Emission of a resonance localized on a period-5 islands chain. (a) is the angular momentum distribution for |αm|2 versus m. The inner (two red dashed curves) and the outer (two blue solid curves) distributions are the spatial modal distribution outside and inside of the cavity, respectively. (b) is the resonance localized on a period-5 islands chain. (c) and (d) are the CCW and the CW propagating wave, respectively. (e) is the FFP of the resonance. (f) and (g) are the FFP of the CCW and the CW propagating wave, respectively.
The Husimi function of the resonance superimposed on classical trajectories is obtained as shown in Fig. 3(a). Because our laser is selectively excited along the path of the period-5 periodic orbit as shown in Fig. 1(b), 10, 000 initial rays are uniformly distributed around (S0, p0) = (0.236, ±0.775), regions A and B, respectively. When each trajectory gets under the critical line, evolution is stopped for the next trajectory. The classical trajectories less than the critical line is the emission of rays. As each of the CCW and the CW traveling rays has three emission routes similar to the FFPs in Fig. 2, the propagating waves well follow the routes as shown by the Husimi function. Here the emission routes denoted by D and E are the strongest emission of the CCW and the CW traveling rays, respectively. The Husimi function indicates that even though the resonance is localized on an islands chain, a portion of wave tunnels to a chaotic region, follows manifolds in the chaotic region, and then emits out. This phenomenon is chaos-assisted tunneling [16–19]. The other emission is relatively weak.
Between the two main emission directions, the intensity at D is stronger than the one at E. Because the emission direction is related to the manifold structure, manifold structures around the period-5 islands chain are obtained. The structures in Figs. 3(b) and 3(c) are the CCW (Region A) and the CW traveling rays (Region C), respectively. The three lines at each figure are caused by the unstable manifolds. The manifold structure in the two figures are not symmetric. We note here that when a microcavity is symmetric, the manifold structures of the two regions should be symmetric. The insets show three stable and three unstable manifolds. The stable manifolds in the CCW trajectories are unstable manifolds in the CW trajectories and vice versa. Each inset, which is similar to the Henon-Heiles map, clearly shows an island.
Fig. 3 Husimi function and the classical trajectories on Birkhoff coordinate. (a) is the Husimi function superimposed on classical trajectories. (b) and (c) are the enlarged structures of two corresponding islands, which are marked by A and C in (a), respectively. The insets show islands of a period-5 periodic orbit. D and E are the main emission directions of the CCW and the CW propagating wave, respectively. Here, S = l/L.
Chirality is experimentally confirmed by using an InGaAsP multi-quantum well semiconductor laser, whose cavity size is R0 = L/2π = 30μm. The process of fabrication is the same as in [21], Yi et al. The laser emission is launched into a fiber, whose facet is placed about 50 μm apart from the cavity boundary for measurement of the spectra. For FFP, the fiber facet, which is 300 μm apart from the boundary, is rotated. The input facet of the fiber is of a cone type whose angle is 70 degrees and that of the fiber core is spherical with the radius of 15 μm for coupling. The launched power is measured with a power meter (Newport 818IR) connected to a multi-function optical meter (Newport 1835c) and the spectrum with an optical spectrum analyzer (Agilent 86142B).
Figure 4(a) is the FFP measured at the injection current of 35 mA, which is near above the threshold. We can see two strong emissions as marked by A and B, which correspond to the CCW and the CW propagating wave, respectively. In order to show the coincidence between the experimental and the numerical FFP, the FFP of the resonance is superimposed on the experimental FFP. The FFP of the resonance plotted by a black line well coincides with the experimentally obtained FFP. Because of the selective excitation, the laser emits resonances localized on the period-5 periodic orbit.
Fig. 4 Experimental results in our microcavity laser. (a) is the FFP at the injection current of 35 mA. The black and the red curve are the resonance and the experimental FFP, respectively. (b) and (c) are the optical spectrum of directions A and B, respectively. A and B are the CCW and the CW propagating wave.
Lasing spectra are also obtained as shown in Figs. 4(b) and 4(c) for directions A and B at 35 mA, respectively. Each spectrum shows four lasing modes around 1575 nm. The spectra of directions A and B are the same, which implies that two emission directions are originated from the same periodic orbit. From the mode spacing in the spectra, the path length of the lasing modes are obtained by using the following equation:
ng is the group refractive index. In our measurement, because λ1 = 1570.9, λ2 = 1574.8, λ3 = 1578.7, and λ4 = 1582.7 nm, the average path length is Lavg ∼ 171.78 μm for ng = 3.68 [22]. The path length well coincides with the length of the period-5 periodic orbit ∼ 175.35 μm for R0 = 30 μm. The deviation is about 2 percent. From the spectrum analysis, we confirm that the lasing modes are localized on the period-5 periodic orbit.In a further wave analysis, among the four resonance pairs in Fig. 1(c), when we compare chirality of two resonances in each pair, the chiralities are similar. But chiralities of four resonance pairs are different while the emission directions are the same. Each resonance pair have their own chirality. That is, although the emission directions of the resonances are the same, the emission intensities of the CCW and the CW propagating wave are different according to pairs. In the experiment, CCW direction of the third lasing mode is weaker than the CW one as shown in Figs. 4(b) and 4(c). However, the CCW propagating waves have the tendency of stronger emission intensities than the CW ones. When we obtain emission intensities and directions in an experiment with other lasers, chirality, intensity ratio between the CCW and the CW direction, is also independent while the emission directions are the same.
In conclusion, we have studied chirality in a Reuleaux triangular shaped microcavity laser, where the CCW and the CW rays are separated. A resonance localized on the period-5 islands chain emits with chirality due to the asymmetric deformation of the microcavity. These numerical characteristics well agree with the experimental results in chirality. Although each resonance has its own chirality, the trend of chirality in experiment is similar to the results in a numerical analysis. Therefore, we can say that even if there are classically no backscatterings, an asymmetric microcavity can have chiral emission.
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.
References and links
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