1. Introduction

Advances in semiconductor fabrication technology have made it possible to fabricate two-dimensional semiconductor laser cavities with various shapes [1]. This facility has motivated studies on new resonator designs, and a wide variety of two-dimensional semiconductor microlasers have been fabricated and tested, such as microdisk lasers [2], deformed disk lasers [3–5], stadium and quasi-stadium lasers [6,7], spiral lasers [8,9], and triangular-shaped lasers [10]. Understanding the morphological dependence of the lasing mode is important from both engineering and fundamental research viewpoints.

For unstrained AlGaAs/GaAs single-quantum-well lasers, lasing emission is usually TE polarized. This is understood as being due to the loss and gain difference between TE and TM modes. For example, emission involving axial or Fabry–Perot modes of quasi-stadium GaAs microlasers has been experimentally confirmed to be TE-polarized [11]. In addition, the measured far-field patterns of various GaAs microlasers have been successfully explained by assuming TE mode lasing [1].

In the present paper, we report that a GaAs microlaser can generate TM-polarized emission by the excitation of resonant modes close to the Brewster angle condition. We fabricated and tested a GaAs microlaser whose cavity shape is known as the Penrose unilluminable room (for brevity, we call it the Penrose cavity). This shape, proposed by Roger Penrose in 1958, has attracted the attention of mathematicians as a model of illumination problems [12,13], because of its peculiarity that if the shape is made with mirrored walls and a point light source is placed in the room, there always exists a dark region where the light rays cannot reach. It would be of interest in its own right to study how the unilluminable property manifests in the lasing modes, but here we focus our attention on the property that the Penrose cavity has several quasi-one-dimensional modes, like quasi-stadium cavities [6,7], corresponding to axial, diamond-shaped, and V-shaped stable closed ray orbits. By carrying out experiments with a GaAs microlaser with a Penrose cavity, we found that lasing emission is TM-polarized. We explain this finding as being the result of lasing of diamond-shaped modes whose incident angle at the cavity interface is close to the Brewster angle.

2. Cavity design and device fabrication

The geometry and parameters of the Penrose cavity are given in Fig. 1(a). The top and bottom curved mirrors consist of two half-ellipses whose semi-major axis and semi-minor axis are aand b, respectively. The points p₂ and p₇ are foci of the top half-ellipse and points p₃ and p₆ are foci of the bottom half-ellipse. The distance f between the center and the focus of a half-ellipse is expressed by

 

Fig. 1 (a) Schematic diagram of the structure of the Penrose cavity. (b) Scanning electron microscopy (SEM) image of the fabricated GaAs microlaser.

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The ray dynamics inside the cavity can be systematically studied by the method of the Poincaré surface of section (SOS) [1]. We performed ray dynamical simulations using the SOS for a Penrose cavity [14]. To obtain the SOS, we traced the ray trajectories confined inside the cavity based on the law of reflection under various initial conditions. At each reflection, we recorded the distance η measured from point O along the cavity edge and the reflection angle χ, as shown in Fig. 1(a). By plotting the combination of η and sin χ obtained at each reflection point, we obtained the SOS. In our cavity design, we used the cavity parameters a = 80 μm, b = 50 μm, d = 100 μm, g = 30 μm, and w = 60 μm, which gave a rich variety of ray dynamical trajectories [14].

Figure 2(a) shows the SOS calculated for the above parameter values. Here, the distance η is normalized by the perimeter of the cavity edge. The most distinctive feature of the Penrose cavity is its divided phase space, i.e., the SOS can be divided into three regions. The blue region in Fig. 2(a) corresponds to the ray trajectories originating from the cavity edges of regions A and A′. The corresponding ray trajectory is shown in Fig. 2(b), where the ray motion is chaotic and the ray trajectory covers almost the entire area of the cavity regions A, P, and A′. The red region corresponds to the ray trajectories originating from the cavity edges of regions B and B′. The structure of the red region is equivalent to that of the blue region due to the symmetry of the cavity shape. The corresponding ray trajectory is shown in Fig. 2(c).The trajectory is chaotic and covers almost the entire area of the cavity regions B, Q, and B′. The last region, the green region, includes several islands as well as uniformly distributed plots. The uniform plots correspond to the chaotic ray trajectory shown in Fig. 2(d), which covers almost the entire area of the cavity regions P, M, and Q. The islands in the uniform plots correspond to the ray trajectories propagating in the vicinity of the stable periodic orbits shown in Figs. 2(e)–(g), namely, an axial orbit (e), a diamond-shaped orbit (f), and two V-shaped orbits (g).

 

Fig. 2 Poincaré surface of section and ray dynamical trajectories calculated for the Penrose cavity: (a) Poincaré surface of section consisting of three colored regions; (b) chaotic ray trajectory corresponding to the blue region; (c) chaotic ray trajectory corresponding to the red region; (d) chaotic ray trajectory corresponding to the chaotic sea in the green region (i.e., excluding the islands within the green region); (e) stable axial trajectory corresponding to the islands in the green region; (f) stable diamond-shaped trajectory corresponding to the islands in the green region; (g) two stable V-shaped trajectories corresponding to the islands in the green region. The yellow lines and red arrows show the stable periodic orbits and predicted output directions, respectively.

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We fabricated a microlaser with the cavity shape shown in Fig. 1 using a projection reduction i-line lithography system and a reactive-ion-etching technique on a graded-index separate-confinement-heterostructure unstrained single-quantum-well GaAs/AlGaAs structure that was grown by metal organic chemical vapor deposition. The electrode contact area was formed on the cavity regions P, M, and Q to selectively excite the cavity modes corresponding to the green regions of the SOS. The details of the layer structure and fabrication process are similar to those reported in [6]. Figure 1(b) shows a scanning electron microscopy image of the fabricated microlaser, showing that the cavity geometry and electrode pattern are defined finely.

3. Lasing characteristics

We tested the microlaser at 25°C using a current with a pulse width of 500-ns width and a repetition rate of 1 kHz. Figure 3 shows the lasing spectra at injection currents of 340 and 700 mA. A few sharp peaks appeared at wavelengths of around 840 nm at an injection current of 340 mA, as shown in Fig. 3(a). The number of peaks and the peak intensities increased as the injection current was increased, as shown in Fig. 3(b). We found that the threshold current of the microlaser is approximately 340 mA and the laser action occurred in multiple modes.

 

Fig. 3 Lasing spectra of the Penrose cavity microlaser at injection currents of (a) 340 mA and (b) 700 mA.

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Figure 4 shows the far-field emission patterns at injection currents of 340 and 700 mA for TE (red) and TM (blue) polarized components. Here, TE and TM polarization implies E_z = 0 and H_z = 0, respectively (see Fig. 1(a) for the definition of the coordinates). In contrast to previous observations for GaAs microlasers, our microlaser lased with TM polarization. For the TM component in Fig. 4(a) [i.e., just above the threshold], faint swells appeared at angles around $\pm 70°$ out of the uniform background caused by spontaneous emission, while for the TE component, slight, almost uniform emission caused by spontaneous emission is evident. As the injection current increased, the swells grew for the TM component, while the uniform emission for the TE component was largely unchanged [Fig. 4(b)]. Although fine oscillations appeared in the far-field emission patterns, the microlaser exhibited mostly directional emissions at angles around $\pm 70°$. As discussed below, these angles coincide with the output directions for the diamond-shaped modes (indicated by green lines in Fig. 4). We confirmed that the TM polarized directional emission was obtained robustly up to at least 900 mA.

 

Fig. 4 Far-field emission patterns of the Penrose cavity microlaser at injection currents of (a) 340 mA and (b) 700 mA for TE and TM polarized components. The green lines show the output direction for the diamond-shaped cavity modes.

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4. Discussion

Taking into account the fact that the chaotic ray orbits shown in Figs. 2(b) and 2(c) overlapwith the unpumped (i.e., absorbing) regions, we can expect that the modes corresponding to either the chaotic orbit [Fig. 2(d)], the axial orbit [Fig. 2(e)], the diamond-shaped orbit [Fig. 2(f)], or the V-shaped orbits [Fig. 2(g)] are likely to be excited. The output directions of these ray orbits are predicted by Snell’s law, and are indicated by red arrows in Figs. 2(d)–(g). Since the laser cavity is symmetric about the horizontal centerline, in the following, we consider the emission angles for the range $-90°\le \theta \le 90°$. First, the axial modes should have directional emission along the cavity axis corresponding to 0° in the far-field emission pattern as shown in Fig. 2(e). Second, the output directions of the diamond-shaped cavity modes are given by

The output directions corresponding to the diamond-shaped cavity modes are indicated in Fig. 4 by green lines. The swells observed in the far-field emission patterns agree well with the emission angles of the diamond-shaped modes. Therefore, we conclude that the fabricated microlaser lased using the diamond-shaped modes with TM polarization. We consider that the degraded directionality of the far-field emission patterns in Fig. 4 was mainly caused by the lasing of higher-order transverse modes. Also, we consider that the increase in the background of the far-field emission pattern was caused by the spontaneous emission generated in the outside region of the diamond-shaped mode pattern. By using a narrow electrode contact pattern along the diamond-shaped trajectory, we expect to be able to improve the output beam quality and the suppression of the spontaneous emission [7].

We next consider why the TM polarized diamond-shaped cavity modes are selectively excited. The mirror loss α for the cavity modes corresponding to the stable periodic orbits can be described by

In our estimates of mirror losses α, the chaotic ray trajectories have relatively low α−values. Thus, there may be a possibility that the corresponding chaotic modes would be excited for higher pumpings. However, to the extent we investigated, we could not observe the emission from the chaotic modes. This might be explained by nonlinear modal interactions due to the spatial overlapping of the cavity modes.

Finally we mention the possibility of the excitation of cavity modes other than the diamond-shaped modes. In previous works, we developed a partial pumping technique to selectively excite desired modes corresponding to stable periodic orbits [7]. By applying this technique, we expect to be able to selectively excite the axial modes and/or the V-shaped modes.

5. Conclusion

We fabricated and tested a GaAs microlaser having a Penrose cavity. We observed TM-polarized directional lasing emission, and attributed this to the lasing of the diamond-shaped modes based on the ray-dynamical prediction for their output directions. We also explained that the TM mode is preferentially excited because the incident angle of the diamond-shaped orbit is very close to the Brewster angle. Our result demonstrates that polarization control is possible by utilizing the two-dimensionality of a cavity shape. It would be of interest to excite both TE modes (e.g., axial modes) and TM modes (e.g., diamond-shaped modes) by the above mentioned selective excitation technique and study their interactions.