## Abstract

We applied the Fox-Li resonator theory to analyze the mode stability of concave mirror surface-emitting lasers. The numerical modeling incorporates the oxide aperture in the simple classical cavity by adding a non-uniform phase shifting layer to the flat mirror side. The calculation shows that there is a modal loss difference between the fundamental mode and the competing modes. The amount of loss difference depends upon cavity length and the thickness of the oxide aperture. In addition to loss difference, modal gain difference plays a key role in discriminating between the fundamental mode and the higher order transverse modes. The modal gain difference heavily depends upon the size of the oxide aperture and the field intensity distribution. To summarize, the geometry of the concave cavity affects the mode profile and the unique field profile of each transverse mode makes a difference in both modal loss and gain. Finally, this leads to a side-mode suppression.

©2005 Optical Society of America

## 1. Introduction

High-power single-mode operation has been a major concern for enlarging the application area of vertical-cavity surface-emitting laser. Various methods were tried to modify the cavity structure and thus, obtain selective modal loss or selective modal gain [1–3]. Another trend to increase the power output of the fundamental transverse mode is to employ the concave cavity instead of the traditional Fabry-Perot cavity [4–6]. In a similar manner, the external concave mirror is also attached to a semiconductor-based vertical cavity to further extend the cavity length and, thus, the power limit of the micro-cavity [7,8]. The shift from the Fabry-Perot type to a concave mirror cavity seems a natural choice considering that classical solid-state laser also follows similar steps in seeking stable alignment with a low-loss cavity in the use of large mirrors. The application of a concave mirror with a long cavity length has the advantages of better transverse mode control and an increased mode size, even though it gives up the single frequency operation due to multiple longitudinal modes. The increased mode size implies less of a temperature rise for a given power consumption and it improves the performance especially, at a high-power operation since the laser performance deteriorates rapidly with an increase in temperature.

Despite the importance of the issue and the success of the device fabrications, the progress of theoretical modeling in concave mirror vertical-cavity lasers is not as rapid as the experimental one. Even though modeling attempts have existed, they seem to rely on very complicated and unconventional approaches to analyze mode competition and mode stability [9]. In this paper, we analyze the concave mirror vertical-cavity with the classical resonator theory to demonstrate that the classical resonator theory is powerful enough to provide practical information about the mode profile of the dominant transverse modes and the competition between them. We also simulate various geometrical situations including variations of cavity length and oxide aperture.

## 2. Numerical modeling

The modeling of a concave mirror vertical-cavity is not as easy as it appears since it is an open-sided cavity. Most of the textbooks dealing with lasers include resonator theory, however, they also mention the difficulty of mode calculation of the open sided cavity [10]. Mathematically, resonator eigenmodes are described by the integral equation as shown in Eq. (1), however, the round-trip propagation kernel is generally found not to be a hermitian operator. This, in turn, means that the existence of a complete and orthogonal set of eigensolutions is not guaranteed. It is just accepted that transverse eigenmodes exist empirically. As a matter of fact, real lasers have never had any difficulty in finding such modes in which to oscillate. The Fox-Li theory [11] relies on this physical process when searching for eigenmodes. The method is a computer simulation of the physical experiment of exciting a resonator externally and adjusting its length to resonate the various modes. The output at each resonance is purified by means of mode filters consisting of suitably adjusted resonators in tandem. For this reason, we adopted the Fox-Li theory to calculate the mode profile of concave mirror surface-emitting lasers.

The iterative integral equation for the field E(r, ϕ) is originally in a two-dimensional form. It is possible, however, to reduce it to a one-dimensional equation in Eq. (1) by assuming a sinusoidal azimuthal distribution of the field, as seen in Eq. (2).

$$\mathrm{where}\{\begin{array}{c}{W}_{l}^{\left(n\right)}:\mathrm{Radial}\phantom{\rule{.2em}{0ex}}\mathrm{field}\phantom{\rule{.2em}{0ex}}\mathrm{distribution}\phantom{\rule{.2em}{0ex}}\mathrm{function}\\ {K}_{l}(r,s):\mathrm{Kernal}\end{array}$$

This assumption is valid as long as the cavity has cylindrical symmetry. The reduction into a one-dimensional form saves a lot of computation time as compared with the two-dimensional form. It also means that a simulation, over a wide range of parametric space, is possible within a reasonable time.

In the modeling, the oxide aperture is incorporated into a simple concave mirror cavity by adding one layer next to flat mirror. This is a typical situation observed in real electrically-pumped devices [12]. The flat side usually consists of a distributed Bragg reflector(DBR) and active medium. In order to supply the current to active medium, it needs some current confinement structures. For AlGaAs material, the oxide aperture is one of the most commonly used structures for current confinement. Even though implantation can be used for current confinement, it still suffers from thermal lensing and it experiences a similar optical effect as the oxide aperture. The thermal effect induces a higher refractive index at the center than at the border. Optically pumped lasers are not an exception since they are also influenced by the thermal effect. Therefore, the index profile of the oxide aperture, as depicted in Fig. 1, can be a situation, which is commonly faced by real devices. The numerical implementation of an oxide aperture is done by adding different phase shifts to a flat mirror, depending on the location in the mirror.

The DBR structure is not included in the modeled cavity since it causes a uniform phase shift in a lateral direction, unlike the oxide aperture. It is assumed that the constant phase shift is absorbed into the flat mirror side. The tuning process in the Fox-Li theory adjusts the cavity length within the wavelength order so as to locate the resonance frequencies of the longitudinal modes. Therefore, ignoring the DBR structure in modeling may not incur serious problems, at least, in terms of phase shift. On the other hand, the omission of the DBR structure greatly simplifies numerical calculations.

For examples of calculations, we present the field intensity profile and power loss of a mode in Fig. 2 and Fig. 3, respectively. These calculations were carried out for the cavity geometry described by the parameter values in Table 1. There are many eigensolutions under the same geometry. In order to specify the eigensolutions, we adopt TEM_{nl} notation used in the reference[11]. The mode number n and l are related to the field intensity modulation in the radial direction and in the azimuthal direction, respectively. For instance, the mode in Fig. 2 is a fundamental mode labeled as TEM_{00}. Adjusting the cavity length in the order of the wavelength corresponds to the tuning process in a real laser. The tuning process results in mode selection of the radial mode in the Fox-Li theory. The low-loss radial modes appear as resonant peaks as the cavity length varies. In the concave mirror cavity, the ${\mathrm{TEM}}_{{0}_{l}}$ mode becomes the dominant mode in each azimuthal symmetry family due to its relatively low loss. For the mode in Fig. 2, a phase shift of 22 degrees should be added to the original cavity length for resonance, where 360 degrees is equal to one wavelength change in cavity length.

The convergence of the solution can be verified by checking the power loss per iteration. When the mode profile is stabilized with increasing iteration steps, the power loss per round trip comes close to a certain number. This always happened in our calculations, as seen in Fig. 3, except for the high loss modes. Good convergence is attributed to the use of the mode filtering technique, as suggested by reference [11]. The filtered mode profile is fed back to the initial field profile of the next batch of iterations. In general, a mode filtering of five repetitions is sufficient in obtaining good convergence.

## 3. Results

As mentioned earlier, we concentrated on modal loss and modal gain of each transverse mode in order to analyze the mode competition behavior. The output power ratio between the two modes above threshold is proportional to the modal loss/gain difference [13, 14]. The modal loss of the individual mode is calculated from the mode profile. Once the mode profile is stabilized, a fixed amount of optical power is lost whenever the mode is reflected from each mirror. Loss stems from the finite size of the mirror and the infinite tail of the optical mode. Although field intensity decays rapidly with an increase in the radius, a small portion of light still exists outside the circular mirror. Therefore, cavity loss depends on the size of the mirror and optical mode. In general, the higher order transverse mode contains more energy at the border than does the fundamental mode. As such, we can expect that a fundamental mode suffers from a lower loss than the high-order modes. As a matter of fact, the real problem concerns the parameter range the concave cavity maintains the fundamental mode and the mode suppression ratio over the parameter range.

In order to investigate mode stability, we scan several parameters over a range of values while the values of other parameters are held constant, as indicated in Table 1. One of the variable parameters is the cavity length and the results are presented in Fig. 4, Fig. 5 and Fig. 6. The three plots correspond to three different oxide apertures illustrating the effect of the oxide aperture on mode stability. The first one indicates the case where there is no oxide aperture, while the other two results refer to oxide apertures of different thicknesses. The thicknesses of the oxide in Fig. 5 and Fig. 6 are 0.03 µm and 0.06 µm, respectively.

First, the modal loss in Fig. 4 maintains a lower value than that of the oxide aperture cavity, especially for the fundamental mode. It can be ascribed to the absence of scattering loss caused by the oxide aperture edge. An oxide aperture generates a spatially high frequency component in an optical mode. The high frequency component is likely to propagate at a large angle from the cavity axis, thus resulting in an escape from the cavity. For a single-mode operation, the modal loss difference between TEM_{00} and TEM_{01} should be significant. Considering that the modal loss difference of 0.5 % is enough to discriminate between the two modes in the case of VCSEL, the stable single-mode operation region can be where the cavity length is larger than 150 µm. The upper limit is determined by the optical loss of the fundamental mode. If the optical loss is too large for the optical gain to overcome, the laser cannot satisfy lasing conditions. Assuming that the active medium can make up for 2 % of modal loss, the upper limit of cavity length is about 250 µm. As the cavity length increases, the mode suppression ratio and the threshold current increase according to the calculations.

As for the concave cavity with an oxide aperture, the cavity loss of the fundamental mode is larger than that of the cavity without it, as shown in Fig. 5. The competing higher order transverse modes, however, also experience a larger loss, which means a larger mode suppression ratio. By using the same criteria for mode stability and lasing conditions, the laser calculated in Fig. 5 can operate at a cavity length between 100 µm and 270 µm with mode stability. The loss difference between TEM_{00} and TEM_{01} changes depending upon the cavity length. The range between 170 µm and 230 µm is a window whereby the loss difference is small and thus, mode competition is intense. In this region, considerable optical power of the side mode can be observed along with that of the most dominant mode.

If the thickness of the oxide layer increases further, the cavity loss possibly exceeds the maximum modal gain which an active medium can provide. An increased power loss of an individual mode, in the case of a thick oxide aperture, is illustrated in Fig. 6. Moreover, the mode stability in the middle window deteriorates due to the reduced loss difference between the two dominant transverse modes. The loss of the fundamental mode grows rapidly, along with the thickness of the oxide layer, so that it may affect lasing operations, according to the simulation. Considering that thermal lens can easily add to the index change in real devices, by the same order of magnitude as oxide does, the exaggerated cavity loss may be largely attributed to the abrupt index step at the aperture boundary. A similar phenomenon is observed in VCSEL and there have been many successful theoretical works explaining the scattering effect of the oxide aperture in the conventional VCSEL structures [15, 16]. In practice, a tapered oxide aperture is used to reduce scattering loss caused by abrupt boundary. Therefore, introduction of the tapered oxide may suppress the modal loss in a concave cavity through the same mechanism, especially when using a thick oxide layer. The deterioration of the mode suppression ratio in the middle window, however, may not be avoided with such a thick oxide layer, following the trends found in Fig. 5 and Fig. 6. Further research regarding the study of the effect of a tapered oxide aperture and thermal lens is required.

Along with loss difference, gain difference is also frequently cited as a source of mode selection mechanism. It is worthwhile to calculate the gain difference between the transverse modes for a complete analysis. Modal gain is a product of material gain and the gain confinement factor. If we assume that the current flows into aperture uniformly, the material gain profile is uniform over the oxide aperture and is independent of the transverse mode profile. Thus, the mode dependence of the modal gain comes only from the confinement factor, which is a ratio of photon energy inside an active medium to the total photon energy. The confinement factor is contingent upon the mode profile and the radius of the oxide aperture since it is in proportion to the overlap integral between the field distribution and the carrier profile. The calculated confinement factors of various transverse modes are plotted as a function of cavity length in Fig. 7. The distinctive gain difference between TEM_{00} and TEM_{01} is expected. The confinement factor of TEM_{00} is almost twice that of TEM_{01}. It means that the modal gain of TEM_{00} is twice that of TEM_{01}. For a comparison with the loss difference, it amounts to a 1 % loss difference when the cavity loss of the fundamental mode is 2 %. In any event, the second mode should increase the current up to twice that of the fundamental mode in order to reach the same lasing conditions.

A large modal gain difference in Fig. 7 can be easily understood by noticing the distinctive mode profiles, as shown in Fig. 8. TEM_{00} maintains a higher photon intensity at the center than in the outer region, as compared with other modes. An appropriate choice of the oxide aperture radius boosts the difference in the confinement factor and modal gain. An aperture radius of 6 µm induces an appreciable gain difference in the simulation.

In this paper, we only demonstrate the simplicity and the feasibility of Fox-Li resonator theory in the analysis of a concave mirror vertical-cavity surface-emitting laser. More accurate modeling requires further research on thermal effect and cavity geometries conforming to real devices [17]. The sophistication of this theoretical work will be the topic of the future research.

## 4. Summary

We applied the Fox-Li resonator theory to analyze the mode stability of concave mirror surface-emitting lasers. The numerical modeling incorporates the oxide aperture into the simple classical cavity by adding a non-uniform phase shifting layer to the flat mirror side. The calculation shows that there is a modal loss difference between the fundamental mode and the competing modes. The amount of loss difference depends upon cavity length and the thickness of the oxide aperture. In addition to the loss difference, modal gain difference plays a key role in discriminating the fundamental mode and the higher order transverse modes. The modal gain difference heavily depends upon the size of the oxide aperture and the field intensity distribution. To summarize, cavity geometry affects mode competition through modal loss and gain difference. Through the analysis, the simulation based on Fox-Li resonator theory turns out to be an efficient tool for optimizing the oxide aperture concave cavity.

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