1. Introduction

The stadium resonator, which consists of two half-circles connected by two straight side walls, and the half-stadium resonator, one side of the stadium resonator, have been extensively investigated as examples of classical and quantum chaotic systems.[1–5]

A classical particle moving freely and bouncing elastically inside these structures shows two kinds of orbits. One is closed and periodic orbits that depend on the bouncing number at the straight side walls. These closed orbits are not perturbation stable. If a particle moves away from the closed orbit a little bit, the initial perturbation is rapidly enhanced as the particle repeats the bouncing at the half-circles. The other is a fully chaotic orbit which covers entire interior of the structure. In a quantum model, the wave equation is solved with appropriate boundary conditions.[2,3] In this case, “scars” patterns, which are narrow linear regions with an enhanced intensity of eigenfunction, are formed around the classical closed orbits.[1,2]

It is physically interesting to study the propagation of an optical beam inside these resonators, because the propagation can be described by a classical ray trace picture that is equivalent to the orbits of the classical particle, as long as the wavelength of the optical beam is much smaller than the cavity dimension and the beam width and divergence are negligible. To examine the beam propagation behavior, some of the authors fabricated quasi-stadium semiconductor lasers by using reactive ion etching technique[6,7] and observed the lasing characteristics, such as light output injection current characteristics, lasing spectra and output beam patterns from both end-mirrors.[8,9] Unique fringe patterns on both end-mirrors and a focused image outside the curved end-mirror were observed. Some of these characteristics are well explained by using ray trace picture based on geometrical optics.[8] However, the fringe patterns and beam divergence can not be explained by using such geometrical optics. Moreover, it is speculated that the optical beams propagating inside the laser cavity have sizable divergence and width. To better understand the beam propagation behavior, it is necessary to analyze the eigenmode of the laser cavity.

In this work, we carried out eigenmode calculations and visualized the beam propagation behavior inside the quasi-stadium laser cavity. From the view point of classification of laser cavities, the quasi-stadium resonator can be assumed to be a geometrical unstable resonator consisting of flat and curved end-mirrors with side wall reflection mirrors. We adopted the Fox and Li mode calculation method[10] with taking into account the effect of side wall reflections. In our experiment, we observed the output beam patterns just above the threshold current.[9] In this condition, the intensity of the propagating beams inside the laser cavity is relatively weak and the spatial hole burning effects are negligible. In our calculations, therefore, we do not take into account the spatial hole burning effects. We calculated the spectrum of round-trip eigenvalue, output beam patterns from both end-mirrors and beam propagation behavior inside the laser cavity for both directions (forward and backward) and visualized them. The calculated output beam patterns show excellent agreement with our experimental results.[9] This calculation method is adequate for the analysis of this kind of laser cavity, as long as the spatial hole burning effects are negligible. It is also found that the interference patterns are formed along the classical ray trajectories inside the laser cavity. These intensity patterns are similar to the “scars” patterns that are calculated in the quantum model.

2. Device structure and theoretical model

Figure 1 shows the structure of the quasi-stadium laser diode. This structure is same as the device that we measured the lasing characteristics in [9]. The flat and curved end-mirrors are connected by straight side wall mirrors. The cavity length L and width W are 660 and 60μm, respectively. The radius of curved end-mirror R is 60μm. The width of flat end-mirror W_r is 20μm. The side walls are separated from both cavity ends by distance W_s, which is 90μm. The purpose of the open unpumped corner regions is to suppress the higher order ray trajectories.

Figure 2 shows the top view of the laser cavity. The quasi-stadium resonator can be assumed to be a geometrical unstable resonator consisting of a flat end-mirror and a curved end-mirror with straight side wall mirrors. We analyzed the resonator by using the Fox and Li mode calculation method taking into account the effect of the side wall reflections. The resonator is an unstable resonator, therefore the forward propagation is different from the backward propagation.

 

Fig. 1. Structure of the quasi-stadium laser diode.

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Fig. 2. Theoretical model for the analysis of eigenmode. (a) Virtual images of flat end-mirror, (b) virtual images of curved end-mirror.

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During the beam propagation from the flat end-mirror to the curved end-mirror (forward propagation), the side wall mirrors form some virtual images of the flat end-mirror as a result of the side wall reflections as shown in Fig. 2a. During the beam propagation from the curved end-mirror to the flat end-mirror (backward propagation), some virtual images of the curved end-mirror are formed as shown in Fig. 2b in the same manner.

To begin our calculation, we assumed a field distribution on the flat end-mirror E_f(x_f, z_f) with uniform intensity and phase as an initial condition. The field distribution on the curved end-mirror E_c(x_c,z_c) is calculated by using following one-dimensional Huygen′s integral equation from the images of the flat end-mirror including the virtual images,

Where, λ and k = 2π/ λ = 2πn_eff / λ ₀ are oscillation wavelength and wavenumber inside the laser cavity. λ ₀ is wavelength in a vacuum and n_eff is effective index of the laser diode. n indicates the number of the flat end-mirror images as shown in Fig. 2a. cosθ is the obliquity factor which depends on the angle θ between the line element (x_f,z_f)-(x_c,z_c) and the normal to the surface element dx_f.

Following that, the field distribution on the flat end-mirror is calculated by using following one-dimensional Huygen’s integral equation from the images of curved end-mirror including virtual images in the same manner,

Where n is the number of the curved end-mirror images. θ is the angle between the line element (x_f,z_f)-(x_c,z_c) and the normal to the surface element dr.

Equations (1) and (2) are obtained from the conventional two-dimensional Huygen’s integral[10] by using Fresnel approximation and separating in rectangular coordinates x and y. During these Huygen’s integrals, we took into account the effect of the open unpumped corner regions by removing the field propagating from the surface element dx_f or dr whose line element (x_f,z_f)-(x_c,z_c) crosses the open regions of the side walls. These round-trip calculations are repeated until the round-trip eigenvalue γ converges. We defined γ as,

Where m is the number of the round-trip.

Through the calculations, we obtain the field distributions on both flat and curved end-mirrors. Therefore we can calculate the beam propagation behavior inside the laser cavity in both directions (forward and backward) by using one-dimensional Huygen’s integrals. The output beam patterns from both end-mirrors are calculated by using two dimensional Huygen’s integral. Here we assumed the field distribution perpendicular to the active layer was Gaussian. In the actual device, the output beams propagating from both end-mirrors are affected by Lloyd’s mirror reflection at the surface of the substrate. To obtain good agreement with our experimental results, we also took into account the effect of the Lloyd’s mirror reflection in this calculation.

3. Calculation results

3.1 Spectrum of round-trip eigenvalue

It is speculated that the laser diode oscillates at the wavelength where the round-trip loss becomes minimum (the round-trip eigenvalue becomes maximum) near the wavelength of the gain peak. Figure 3 shows the spectrum of the round-trip eigenvalue. In this calculation, we determine the effective index n_eff as 3.3, so that the calculation results agree with our experimental results.[9] n_v is the number of the virtual images taken into account on each side of the laser cavity during the eigenmode calculation. When n_v = 0, the resonator becomes simple unstable resonator without side wall reflections. In this case, the round-trip eigenvalue is relatively small. When we take one virtual image on each side of the laser cavity (n_v = 1), the round-trip eigenvalue spectrum shows some ripples. It is speculated that these ripples are caused by the interference among the beam propagating directly and the beams reflected at the side walls. As the number of the virtual images increases, the round-trip eigenvalue increases and the spectrum becomes more complex shape, namely, some narrow and sharp peaks appear in the spectrum. However, there is no difference between the spectrums calculated for n_v = 2 and n_v = 4. The reason is that the open unpumped corner regions in the laser cavity restrict the higher order ray trajectories. It is found that n_v = 2 is enough number to analyze this laser cavity.

The round-trip eigenvalue becomes maximum at 857nm. We believe that the laser diode oscillates at this wavelength when the gain peak is located near the wavelength. We carried out following calculations at the wavelength of 857nm.

 

Fig. 3. Spectrum of round-trip eigenvalue.

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3.2 Output beam patterns propagating from both end-mirrors

Figures 4a and 4b that are short animations show the variation of the output beam patterns against the distance from the flat and curved end-mirrors, respectively. A unique fringe pattern is formed on the flat end-mirror (Fig. 4a). Upon leaving the flat end-mirror, the fringe pattern diverges and eventually split into three main lobes. On the curved end-mirror (Fig. 4b), a fringe pattern is also formed. Upon leaving the curved end-mirror, however, the fringe pattern focuses to three spots at the distance of 24μm. Beyond this distance, the spots again diverge and reform a fringe pattern. The variation of the output beam patterns well agrees with our experimental results.[9]

3.3 Beam propagation behavior inside the laser cavity

Finally we calculated the beam propagation behavior inside the laser cavity. Figure 5 shows the intensity patterns of the propagation beams inside the laser cavity for both directions. In this calculation, the field distribution on the line perpendicular to the laser cavity was calculated by using one dimensional Huygen’s integral from the end-mirrors.

 

Fig. 4. Click in the space to start the animations that show the variations of output beam patterns versus the distance (a) from the flat end-mirror and (b) from the curved end-mirror, respectively. [Media 1] [Media 2]

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Then the intensity distribution is calculated and normalized on the line so that the maximum intensity becomes unity. These calculations are then repeated, changing the location of the line from one end-mirror to the other end-mirror. After leaving the end-mirror, the intensity of the propagating beam decreases because the round-trip eigenvalue is less than unity and the beam divergence is relatively large. The purpose of the normalization of the intensity is to show the intensity patterns clearly through the entire laser cavity.

It is found that the interference patterns are formed along the closed and repetitive trajectories that is expected from the classical ray trace picture.[8] These intensity patterns look like the “scars” patterns that are obtained from the quantum model.[1,2] It is speculated that these intensity patterns are formed by the interference between the beams propagating along the classical ray trajectories.

 

Fig. 5. Beam propagation behavior inside the laser cavity, (a) forward propagation and (b) backward propagation.

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The round-trip beam propagation can be explained as follows. The fringe pattern on the flat end-mirror diverges and eventually splits into three directions, each beam propagating toward the curved end-mirror. After bouncing on the side wall mirrors a fringe pattern is formed on the curved end-mirror. The reflected beams at the curved end-mirror then focus to three spots at the point R/2 away from the top of the curved end-mirror. These spots again diverge and propagate toward the flat end-mirror with large beam divergence. Eventually the same fringe pattern is formed on the flat end-mirror after one round-trip.

In the classical ray trace picture, the closed repetitive trajectories are independent of each other. In the actual quasi-stadium laser cavity, however, it is speculated that the beams propagating along the classical ray trajectories have sizable divergence and they are coupled to each other.

4. Conclusions

We analyzed the beam propagation behavior in a quasi-stadium laser diode by using the Fox and Li mode calculation method taking into account the effect of the side wall reflections. Unique fringe patterns from both end-mirrors, and a focused image outside the curved end-mirror are obtained. These results show the excellent agreement with our experimental results. We also visualized the beam propagation behavior inside the laser cavity. It is found that the interference patterns are formed along the classical closed ray trajectories. These patterns are formed by the interference between the beams propagating along different ray trajectories. This calculation method is useful for the analysis of this kind of laser cavity.

The laser structure analyzed in this work is specialized compared to a conventional stadium resonator. For example, the laser cavity has open unpumped corner regions to suppress the higher order ray trajectories, and the radius of the curved end-mirror is larger than the half of the cavity width. Further research will be conducted to analyze dependency of the beam propagation behavior on the structural parameters of the quasi-stadium laser cavity.