1. Introduction

Microcavity lasers have many potential applications because of their low lasing threshold and small mode volume [1]. A certain microcavity laser having a circular shape generates extremely high quality factor lasing modes [2] with low threshold [3]. The laser, however, has the drawbacks that the emission direction is isotropic and the intensity is low. Recently, deformed cavities from a circular shape are an object of intensive study for the expectation that they would improve directionality and emission intensity of a microcavity laser [4, 5]. So, various deformed shapes are explored and unidirectional emissions are observed in cavities of a spiral [6], a rounded isosceles triangular [7], an annular [8], a limaçon [9], a gibbous [10], and an ellipse with a notch shape [11], and two half-ellipses jointed [12].

In addition to these characteristics, a directional single mode emission without hopping is another important property to consider for applications such as information processing using optical devices as an optical buffer [13], memory [14], and an add-drop filter [15], which are based on low loss ring resonators. In this sense, optical information processing in an optoelectronic circuit is similar to that in fiber optic communication, where a distributed feedback (DFB) [16] and a distributed Bragg reflector (DBR) semiconductor laser [17] are commonly used for a single mode in communication.

Up to now no serious attempts have been made to realize the idea of a microcavity laser, which generates a directional single mode even while the realization is indispensible to application of the laser to optoelectronic circuits. There is a certain microcavity laser, which generates a single mode directionally in a certain region, but it begins to generate another mode as the excitation increases. For example, a Limaçon cavity generates unidirectional emission but multiple modes [18]. Again, a spiral-shaped microcavity laser generates a single mode but the emission is not directional [19].

In this letter, we report a microcavity laser, which generates a directional single mode without hopping over the whole region of continuous injection currents. In the experiment, it is confirmed that a scar mode becomes a single lasing mode by considering the path length of the equidistant mode spacing near the lasing threshold. In a numerical analysis, a resonance is obtained by the boundary element method (BEM) and the far-field pattern (FFP) of the resonance agrees well with that of the experiment.

2. Results and discussions

Our laser is composed of a circle and an isosceles trapezoid as shown in Fig. 1(a). The rear boundary is circular with a radius of R = 20 μm. The two sides of the legs of the isosceles trapezoid are tangentially connected to the circular boundary at an angle (ϕ) of 8.82°. The lengths of the bottom (l₂) and the top base (l₁) are about 39.5 and 14.5 μm, respectively. The height (l₃) of the isosceles trapezoid is 80.7 μm. The length from the rear end of the circle to the top base of the isosceles trapezoid is 103.7 μm. Because of the linear sections and the circle, the ray dynamics of the corresponding billiard is completely chaotic. The scanning electron microscope (SEM) image of the fabricated InGaAsP semiconductor laser is shown in Fig. 1(b). The fabrication process and parameters are the same as in Ref. [18].

 

Fig. 1 (a) The design of our laser that is composed of a circle and an isosceles trapezoid. (b) SEM image of the fabricated InGaAsP semiconductor laser.

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In our experiment, Fig. 2 is the output power depending on the injection current. The output power is measured when the fiber facet is 50 μm apart from the center of the front boundary at θ = 45° as shown in the inset A. As the injection current increases, the output power slowly increases below 47 mA. Above 47 mA, the output power increases very rapidly and, from 130 to 160 mA, is maximized. With further increase of the injection current, the output power begins to decrease slowly. Above 215 mA, the laser terminates. To observe the lasing threshold, we obtain the output power in the range from 10 to 45 mA as shown by the inset B. The slope above 35 mA is steeper than the slope below 35 mA. This means that the lasing threshold is about 35 mA.

 

Fig. 2 The output power according to the current at 22°C. The output power is measured at θ = 45° as shown in the inset A. The inset B is the output power in the range of 10 to 45 mA, which shows the lasing threshold of about 35 mA.

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Figure 3 is optical spectra showing lasing modes above the threshold. Figure 3(a) shows several lasing mode groups when the injection current is 40 mA. In the optical spectrum, the intensity of the mode indicated by the thick red arrow, which is around 1558.52 nm, increases alone as the current increases. Above 47 mA, the mode becomes dominant while the intensity of the other modes increases slowly. When the injection current increases, the single mode begins to be red shifted due to thermal effect. To show the tuning range of the single mode, the emission spectrum for nine different injection currents is superimposed in the range from 50 to 210 mA with 20 mA step as shown in Fig. 3(b). The figure shows a tuning range of about 4.98 nm without hopping, which is similar to that of a DFB diode laser, 5 nm. When a single mode emits, the side-mode suppression ratio (SMSR) is larger than 28 dB. According to our further experiments of cooling temperature at 16°C, the tuning range is about 6 nm as shown in Fig. 3(c). The continuous red-shift without hopping is caused by the reduction of spatial hole burning [19].

 

Fig. 3 (a) The spectrum at 40 mA. The red arrows show a lasing mode group having a mode spacing of 2.64 nm. The lasing mode marked by the thick red arrow develops into a single mode. (b) The single mode emission at 22°C in the range of 50 to 210 mA with 20 mA step. (c) The single mode emission at 16°C in the range of 60 to 240 mA with 20 mA step.

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To analyze the lasing mode, we obtain a path length of the lasing mode groups in Fig. 3(a). Of the several mode groups in the spectrum, we take the group where the single mode belongs. The lasing modes marked by the red arrows are equidistantly spaced by 2.64 nm. From the four modes, we obtain the path length by using the equation $L=\frac{{\lambda }_{\mathit{avg}}^2}{n_g\Delta \lambda }$, where L is the path length, n_g is the group refractive index, λ_avg is the average wavelength between two neighboring modes, and Δλ is the mode spacing of the two neighboring modes. The equidistance mode spacing of scarred modes was explained in a quadrupole microcavity [20]. By considering the group refractive index of n_g = 3.68, we obtain the path length of about 250 μm. Here, n_g = 3.68 is obtained from an elliptic-shaped microcavity laser [21].

To show directional emission of the single mode, the FFP is measured at 80 mA, when single mode emits. The dashed curve in Fig. 4(a) shows an angularly distributed FFP obtained experimentally. In the measurement, the fiber is rotated at 600 μm apart from the center of the cavity as shown in the inset. Two highly directional emissions are observed around ±50° with a narrow divergence angle of 15°. The trend of a FFP is independent of injection currents, except weak backward emission caused by measurement.

 

Fig. 4 (a) FFPs of the experiment (dashed line) at 80 mA and the resonance (solid line). The angle (θ) is defined in the inset. (b) Inside intensity patterns of the typical scar type resonance. (c) Outside emission intensity patterns of the typical resonance from the cavity.

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To analyze the emission direction, resonances are obtained by solving Helmholtz equation using the BEM. The size of the cavity is about n_eL/λ ∼ 521 for the wavelength of λ = 1560 nm, where n_e is the effective refractive index of 3.3 for TE polarization and L is the boundary length. Of the resonances in the range of 71.3 ≤ Re[L/λ] ≤ 71.9, we take a resonance, which coincides with the experimentally obtained path length. The resonance is shown in Fig. 4(b), whose Re[L/λ] = 71.416 and the quality factor is about 1,800. When a path is drawn on the obtained resonance, the path length is about 249 μm. The error rate is about 0.4 percent. In the resonance, the intensity pattern is mainly localized near unstable periodic orbit (UPO) of the corresponding ray dynamics, and so it can be called a scar mode. This scar mode can have a high quality factor in comparison with the other modes since the supporting UPO is located above the critical line in Birkhoff phase space and the direction of the emission is supposed to be determined by the unstable manifold near the critical line. Hence, the FFP of the scar mode agrees well with the experimentally obtained result as shown in Fig. 4(a). The emission direction from the cavity is shown in Fig. 4(c). The difference between experimental and numerical results in emission direction and divergence angle is caused by surface roughness along the sidewall [22].

It is well known that the generation of the single mode lasing is responsible for the reduction of spatial hole burning by the impact of a traveling wave component in the resonance. We, therefore, have estimated the traveling wave components of the resonance by introducing a standing wave ratio (SWR), which is a commonly used concept in telecommunications, and then the result is compared with that of a resonance of the Limaçon cavity, i.e., a typical whispering gallery type resonance from a near circular shape cavity (see the inset in Fig. 5). The SWR is as follows: SWR = |A_a|/|A_n|, where a local amplitude of antinodes and nodes inside the cavity are represented by A_a and A_n. By defining a discrete grid inside of the cavity we obtain the amplitude of wavefunction based on a grid point. We take a node as the average value of two consecutive antinodes. Figure 5 shows the normalized probability distribution function (PDF) of SWR for resonances of the isoceles trapezoid (black circle) and the Limaçon (red rectangle) cavities. It clearly shows that the PDF of the resonance of the isoceles trapezoid cavity is largely distributed for smaller SWR compared to that of the Limaçon. This indicates that the scar mode in the isoceles trapezoid cavity can have more travelling wave components than resonances from near circular shape cavities. Therefore, the scar mode can be lased as a single mode above the lasing threshold.

 

Fig. 5 Normalized Probability distribution function (PDF) of the standing wave ratio (SWR) in the resonance of the trapezoid (black) and Limaçon (red) cavity. The PDF of the resonance of the trapezoid cavity is largely distributed for smaller SWR compared to that of the Limaçon. This indicates resonance of the trapezoid cavity has more travelling wave components than that of the Limaçon. The inset shows the typical whispering gallery type resonance of the Limaçon cavity.

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3. Summary

We report an InGaAsP microcavity laser, which generates a directional single mode without hopping. When a whole cavity is excited, the laser generates the single mode in two directions around ±50°. The scar mode becomes the single lasing mode by considering the path length of the lasing mode groups in the emission spectrum. We also obtain the scar mode by the BEM and the FFP of the mode agrees well with that of the experiment.