## Abstract

Breaking the spatial symmetry in optical systems has become a key approach to the study of nonlinear dynamics, wave chaos, and non-Hermitian physics. Moreover, it enables tailoring of the spatiotemporal properties of such systems. Breaking the circular symmetry of lasers yields a more uniform light intensity profile within the optical aperture and makes uniform the spectral distribution of the optical states (modes). Those effects are known to enhance spontaneous as well as stimulated emission and consequently suppress undesired nonradiative recombination in the active region, but their importance for laser emission is not fully understood so far. In this paper, using the example of vertical-cavity surface-emitting lasers, we show that intentionally deformed optical apertures induce a more uniform light intensity distribution within the optical aperture, related to wave chaos, and a higher density of optical states, enhancing stimulated emission as predicted by quantum electrodynamics theory. These two phenomena contribute to increasing the optical output power by more than 60% and quantum efficiency by more than 10%. The results of this study are of significant importance for a variety of lasers, showing a clear link between the fundamentals of their operation and quantum electrodynamics and providing a general, robust method of enhancing emitted power for high-power broad-area lasers.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Maxwell’s equations in the paraxial approximation reduce to Helmholtz’s equation, which has the same form as Schrödinger’s equation. This similarity results in analogous phenomena in quantum mechanics and electromagnetism [1]. In quantum mechanics, it is the shape of the trapping potential that determines the symmetries of a system, whereas in electromagnetism the symmetries are determined by the distribution of the refractive index in the cavity. Reducing spatial symmetry eliminates the degeneracy of energy levels in quantum mechanics and leads to an analogous reduction in the degeneracy of resonant frequencies in electromagnetic resonant cavities. In electromagnetism, breaking the symmetry of the resonant cavity has an important impact on the emergence of wave-chaos phenomena, which can affect the spectral and spatiotemporal properties of microcavities [2–4] and semiconductor lasers [5,6]. Breaking the symmetry of the cavity can also improve resonant modes filtration in laser arrays [7] and the spectral properties of coupled resonator lasers [5]. In this paper, we show how breaking the symmetry of the electromagnetic cavity can enhance stimulated emission by modifying the spatial properties and reducing the degeneracy of the optical modes.

Since lasers were invented, there have been continual efforts to increase their emitted optical power. In general, such efforts have focused on increasing the size of the region in which stimulated emission occurs or increasing the efficiency of photon generation in that region. The first approach can be realized by increasing the size of active region of single laser [8] or by the combination of a number of lasers into an array [9]. In the case of a single laser, when the optical aperture defining the area of the active region is circular, the increase in the aperture size results in the excitation of many transverse electromagnetic modes with intensities concentrated at the perimeter of the optical aperture [10,11], known as daisy modes [12] or whispering gallery modes [13]. Those modes are usually favored in laser emission due to their lower modal losses; however, they interact more with carriers at the perimeter than at the center of the aperture, hindering linear power scaling with increase of the active region area [14].

The second approach to enhancing the emitted power is to increase the electronic density of states (eDOS) [15]. However, as postulated by Einstein in 1916, the probability of photon emission is also related to the optical density of states (oDOS). The oDOS is defined as the number of modes available per unit frequency interval. Increasing the oDOS enables stronger light–matter coupling, which in turn increases the probability of photon emission. This probability can be derived from light–matter interaction based on quantum electrodynamics theory and may be observed in experiments involving spontaneous [16,17] and stimulated [18–20] emission enhancement, known as the Purcell effect. The probability of photon emission has been theoretically investigated by Yokoyama and Brorson [21], who found that spontaneous and stimulated processes can be enhanced by an increase in the oDOS. This is a simple consequence of the fact that the intensities of both types of emission are strictly correlated via Einstein’s coefficients. The larger emission probability contributes to an increase in the radiative recombination rate and as a consequence also contributes to damping the nonradiative recombination. These processes combine to increase the quantum efficiency of the laser, as theoretically shown in [22,23].

In the case of semiconductor lasers with active regions based on quantum wells (QWs), eDOS is not discrete (and in fact is piecewise constant), and most of the carriers occupy states that do not couple with the optical modes. Such carriers tend to recombine nonradiatively, transferring their energy into heat. Therefore, it is beneficial to have as many modes with different photon energies as possible. Competition via spectral hole burning (SHB) reduces the stimulated emission of modes with close photon energies. SHB reduces the occupancy of electron-hole pair states with energies in the vicinity of the photon energies of the modes. This results in the suppression of spectrally close modes due to their competition. If the spectral separation between the modes is sufficiently large, then photons from different modes do not compete for the same carriers. Therefore, in a laser with a QW active region, it is beneficial in terms of emitted power and efficiency to have a wide emission spectrum with evenly distributed modes. Breaking the circular symmetry of broad-area lasers by deforming the optical aperture away from a circular shape affects the transverse modes, which are characterized by more homogeneous spatial coverage of the aperture and modified spectral distributions. This can be understood based on Gutzwiller’s trace formula, which connects the spectral density of modes in cavities with nonintegrable shapes to the periodic trajectories of classical particles in billiards of the same shape [24–27].

In this paper, we demonstrate a general phenomenon whereby reducing the spatial symmetry of the optical cavity enhances stimulated emission. This is achieved via two processes, allowing for more uniform oDOS (modes) in the laser spectrum and more uniform distribution of the light intensity in the active region. We reveal that both processes contribute to increasing the optical output power and quantum efficiency of the laser. For the purpose of demonstration, we use broad-area vertical-cavity surface-emitting lasers (VCSELs) as an example structure. However, the same principles apply to a wide range of laser types, such as semiconductor edge-emitting lasers [6,28], semiconductor disc lasers [29], solid-state lasers [30], fiber lasers [31], and microcavities [3,4], in which the optical aperture is a circle with a diameter several times larger than the resonant wavelength. Broad-area VCSELs, which we consider in this paper, find applications in large power systems related to lidars, illumination systems, heating systems, and medical applications. Unlike edge-emitting lasers, their maximal emitted power is not limited by catastrophic optical damage [32]. When broad-area VCSELs are subject to selective feedback of a single transverse mode, the emission and polarization dynamics of these lasers can be considerably stabilized [33]. In what follows, we analyze the effect of intentionally deforming the optical apertures of large-aperture VCSELs away from the circular shape on enhancement of the optical output power of the lasers. We focus in particular on GaAs-based VCSELs with a resonant wavelength of 980 nm. Various deformations breaking the circular symmetry of the optical apertures are described in Section 2. In Section 3, we demonstrate that the symmetry-breaking deformations of the optical apertures result in higher quantum efficiencies and significantly higher optical output powers compared to VCSELs with circular apertures. We explain these findings by analyzing the spectral mode distributions (in Section 4) and the spatial mode distributions (in Section 5) of circular and deformed optical aperture VCSELs.

## 2. LASER CONFIGURATIONS

We consider a conventional VCSEL with a circular oxide aperture of approximately 34 µm as the reference design, and four classes of VCSELs each with oxide apertures of symmetry-broken shapes. The symmetry-broken apertures have slightly different sizes, due to their various mesa designs (see Fig. 1 and Supplement 1, Section S1). We name the symmetry-broken shapes according to the apertures obtained for the configurations with the largest mesa apertures (see the right column in Fig. 1). Their spontaneous emission near-field images resemble racket shapes used in sports: *ping-pong* (henceforth all related *ping-pong* results will be depicted as magenta symbols or lines), *padel* (all results are in blue), *squash* (green) and, though there is no clear similarity in shape, *tennis* (red). We refer to the family of all symmetry-broken structures as *rackets*. In general, all *rackets* have 34 µm-diameter oxide aperture circles with small side deformations of slightly differing sizes (Fig. 1). To remain consistent in the naming inspired by racket sports, we name the reference configuration with a simple circular aperture *ball* (black symbols or lines represent all results related to this configuration). The experimental setup used in the analysis that follows is detailed in Supplement 1, Section S2.

## 3. EMISSION CHARACTERISTICS

Figure 2(a) illustrates the maximal optical powers emitted by the studied VCSELs taken at 20ºC. The black dotted line in the figure has been drawn to emphasize the linear dependence between the optical output power and the area of the circular aperture in the case of broad-area VCSELs [14]. It is obtained from the point representing the power emitted by the ball structure extrapolated to the origin of the coordinate system. Almost all racket structures emit optical powers significantly above this line. The exceptions are two ping-pong structures (40 and 42), for which the output powers lie on and below this line. For the ping-pong structures, we also observe that the optical output powers are more strongly scattered as the aperture area increases. In the case of the other racket structures, we find much less variation in terms of the emitted power. Padel structures follow the slope of the black line. Tennis and squash structures exhibit the highest emitted powers of all the racket lasers. No abrupt variations are observed in the emitted optical powers, depending on the aperture area.

The set of racket structures with the smallest aperture areas exhibit only between 1% (in the case of ping-pong) and 6% (in the case of tennis) larger areas than the ball structure. Nevertheless, the racket lasers with only slightly increased aperture areas enable significantly larger optical output powers than the ball laser. The racket devices are highlighted in Fig. 2(a) by a gray rectangle. The increase in optical output power of the racket lasers with respect to the ball laser ranges from 28% for ping-pong lasers to 64% for tennis lasers. These findings provide clear evidence that the slightly larger areas of the racket structures cannot alone account for their higher optical output powers. The main goal of the investigations discussed in what now follows is to identify the mechanisms responsible for the significant increase in the output powers of racket lasers.

For the purposes of detailed comparison, we select one representative VCSEL from each racket class, which differs only slightly from the ball structure in terms of aperture area but exhibits particularly high output power. These are the ping-pong, padel, squash, and tennis structures with the numbers 30, 30, 40, and 40, respectively (Fig. 1). We refer to these VCSELs as ping-pong30, padel30, squash40, and tennis40, and we refer to the four lasers collectively as *top rackets*. The aperture areas and lasing parameters of the VCSELs and ball are collected in Table 1.

Figure 2(b) illustrates the light-current characteristics of the ball and top rackets, revealing a pronounced threshold current and roll-over optical power (taken at 20ºC). The threshold current (${I_{{\rm{th}}}}$) is ${\sim}{4.2}\;{\rm{mA}}$ for ball and ranges from approximately 4.5 to 4.8 mA for the top rackets. We attribute the differences of 15% in the threshold currents to the differences in their aperture areas, of maximally 6% between ball and tennis40. The differences may also be due to reduced transverse optical confinement caused by the distortion of the circular shape of the aperture. The maximal quantum efficiency (${\eta _{{\rm{QE}}}}$) found for squash40 and tennis40 close to the lasing threshold is 0.24. Ping-pong30 and padel30 show 8% smaller values, and ball reveals the lowest quantum efficiency, which is 11% lower than that for tennis40. It is worth noting that the quantum efficiency of VCSELs typically increases with the size of the aperture from smaller apertures up to 6 µm [34]. However, broad-area VCSELs with diameters in the range of several tens of micrometers show the opposite tendency: quantum efficiency typically decreases as the aperture increases [14], primarily due to inefficient current spreading. The maximum continuous-wave (CW) optical output powers (${P_{{\max}}}$) of our devices follow the trend of the quantum efficiencies. Tennis40 and squash40 yield the highest optical output power of ${\sim}{{11}}\;{\rm{mW}}$, which is 63% larger than for ball. The maximum emitted power of ping-pong30 and padel30 is 28% higher than that for ball.

In the next sections, we discuss the underlying mechanisms that explain why lasers with deformed apertures, larger in area by 6% compared to normal circular apertures, exhibit an 11% improvement, instead of decrease, in quantum efficiency and the boost of emitted power up to 60% not comparable to the increase in aperture area.

## 4. SPATIAL DISTRIBUTION OF MODES

For circular apertures, the cavity-mode profiles can be described by products of Bessel and sine or cosine functions [10,11]. When deforming the apertures, the spatial mode profiles of VCSELs are modified. This effect will be discussed and shown to be responsible for the increases in the quantum efficiency and emitted optical powers of the top rackets. Figures 3(a)–3(e) show the emission spectra of the ball and top rackets for currents corresponding to 5 times their respective threshold currents. In Figs. 3(b)–3(e), we indicate selected modes by the integers that relate to their spectral positions. We select the five modes of the largest optical power within each of the spectra and depict their near-field emission profiles. The selected modes do not dominate throughout the entire range of pump currents, as will be discussed in the context of Fig. 5. Except the case of the ball, determination of the mode indices in top rackets is demanding due to the strong modification of the mode distributions caused by noncircular apertures. The evolution of the mode distributions while the aperture shape is modified is discussed in Supplement 1, Section S3. Therefore, in Fig. 3 we use subsequent integers to indicate the modes in the emission spectrum whose near-field patterns are illustrated in the same figure.

All the modes of the top rackets exhibit noncylindrical spatial distributions, in contrast to the modes of ball, where the ${{\rm{LP}}_{n,1}}$ modes dominate. The ball modes mainly overlap with the periphery of the aperture, whereas the top racket modes penetrate the centers of the apertures to a greater extent. The near-field images of the total light intensity, shown in the rightmost column of Fig. 3, confirm this observation, revealing noticeable differences in the spatial intensity distribution between the ball and top rackets. This is further emphasized in Fig. 4(a), which shows the total light intensity of each VCSEL averaged over the angle and the quantum efficiency in Fig. 4(b) and maximal output power in Fig. 4(c) of the VCSELs versus the integral of the near-field light intensity. As depicted in Fig. 4(a), the intensity is more evenly distributed spatially in the apertures of all top rackets than in the case of ball. However, there is no monotonic dependence between the light intensity integral and the quantum efficiency, nor with the maximal emitted power. The only clear difference is that the top rackets show greater quantum efficiency and maximal emitted power compared to the ball, and also reveal light-intensity integrals from 5% to 13% larger than for ball.

In the experimental spectrum of ball, one can observe mode ${{\rm{LP}}_{12,1}}$ and another lower wavelength mode ${{\rm{LP}}_{13,1}}$, the wavelengths of which are ${\sim}{{2}}\;{\rm{nm}}$ shorter than the longest wavelength mode ${{\rm{LP}}_{0,1}}$. In the top racket spectra, the spectral distance between the two outermost modes is noticeably less than 2 nm. This suggests that the highest-order modes observed in the spectra of the top rackets are of lower order than in the case of ball.

We compared the experimental findings to our numerical model, which was calibrated based on the experimental modal characteristics of ball (see Supplement 1, Section S3). The numerical model shows that, due to the deformation of the aperture shape, the largest-order ${{\rm{LP}}_{n,1}}$ modes suffer increased transverse optical leakage compared to modes obtained with circular apertures [35]. These additional modal losses result in spectral narrowing by eliminating the shortest wavelength modes. Moreover, lower-order modes show modified spatial distributions while sustaining optical losses (see Supplement 1, Fig. S4 and Supplement 1, Fig. S5), thereby facilitating increased overlap between the light intensity and the central part of the active region.

Although current crowding contributes to higher carrier concentrations at the periphery of the optical apertures, the differences in the carrier concentrations in the center and periphery are minor, as indicated by the near-field images of spontaneous emission in Fig. 1. Therefore, the more uniform distribution of light intensity in the top rackets is expected to contribute to enhance stimulated emission and quantum efficiency compared to the ball VCSEL.

## 5. SPECTRAL DISTRIBUTION OF MODES

The exact oDOS cannot be determined experimentally for our devices, due to nonlinear competition between the lasing modes. Hence, not all optical states are sufficiently filled with photons to be above the noise level. However, the number of modes registered in the laser spectra reflects the number of available, nondegenerate optical states that can be calculated.

Figure 5(a) shows the measured normalized intensity of all registered modes in the domain of the injected current ($I$) and resonant wavelength ($\lambda$) for the ball VCSEL.

For each current, the intensity of all modes in the spectrum is normalized relative to the intensity of the dominant mode at this current. The red points in all Figs. 5(a)–5(f) represent the dominant modes at the given CW bias currents, regardless of their absolute power. The black dashed line in Fig. 5(a) represents the polynomial fit (${\lambda _f}$) of the function that approximates the dependence of the wavelength versus the bias current for the 10 modes of longest wavelengths at each current. Figure 5(b) illustrates the normalized intensity of modes for ball in the domain of current and the relative wavelength ($\Delta \lambda$), which is specified as the difference between the wavelengths ($\lambda$) of the modes and ${\lambda _f}$. Figures 5(c)–5(f) are constructed similarly to Fig. 5(b) and present the emission spectra of all top rackets. The value of ${\lambda _f}$ is determined separately for each top racket VCSEL. Such a representation enables more convenient observation of the spectral width and the number of modes that evolve with increasing bias current. Figure 5(b) shows that the spectrum of ball is almost twice the width of the spectrum for tennis40 [Fig. 5(f)] and significantly broader than those of the other rackets [Figs. 5(c)–5(e)] for injection currents significantly above the threshold. The main reason for this is the absence of modes of very high order (short wavelengths), which are present in the spectrum of ball. It is also worth noting that the wavelength shift of the emission spectrum from threshold to light-current (LI) rollover is almost above 5 nm and is equal for all devices. This suggests that all devices reach their LI rollover at similar active region temperatures, although there are different currents at rollover. The designs of all the VCSELs differ only in the shape of the mesa, which does not influence the process of heat spreading nor of heat sinking. Hence, the devices must differ in terms of the heat generated by nonradiative recombination in the active region.Figure 6(a) shows the density of observed modes (${\rho _m}$) for all the VCSELs as a function of the current determined based on Fig. 5 (see formula (17) in Supplement 1, Section S3: Numerical model and its calibration). We define ${\rho _m}$ as the ratio between the total number of observed modes and the emission spectrum width. We associate the width of the spectrum with the difference between the two modes with the outermost wavelengths at a given current. Figure 6(a) reveals that the largest density of modes is observed for tennis40 and is about twice as large as for ball. The densities of modes for other top rackets are lower than for tennis40, but still significantly larger than for ball. All the VCSELs exhibit a decrease in the density of modes close to the lasing rollover. This effect is expected to be responsible for the reduced probability of stimulated emission in favor of nonradiative recombination. The current at which the density of the modes starts to reduce is the lowest in the case of ball and gradually increases for the top rackets exhibiting higher emitted powers, suggesting again less severe heating. We find a clear dependence between the emission properties of the VCSELs and the density of their modes, as illustrated in Fig. 6. Figure 6(a) depicts the density of modes ${\rho _m}$ versus current. Figures 6(b) and 6(c) provide evidence for a nearly linear dependence of both quantum efficiency and maximal emitted optical power on the maximal density of modes.

Summarizing the experimental characterization, the top rackets compared to ball reveal enhanced quantum efficiency that is an effect of an increased stimulated emission rate due to the higher density of modes present in the emission spectrum. This effect corresponds to the theoretical investigations of Yokoyama and Brorson [21], indicating an enhancement of the stimulated emission by an increase in the density of optical states in the cavity. A higher rate of stimulated emission reduces the number of carriers contributing to nonradiative recombination, which reduces the generation of heat. This results in an enhancement of the maximal optical output power.

To reveal the mechanisms responsible for elimination of higher-order modes and higher density of modes in the VCSELs with deformed apertures (top rackets) compared to the circular aperture VCSEL (ball), we use our numerical model that is detailed in Supplement 1, Section S3. Each ${{\rm{LP}}_{n,m}}$ mode in an anisotropic, circularly symmetric VCSEL is represented by two spectrally closely spaced modes of orthogonal polarizations and two rotations, respectively. The symmetry-broken optical apertures affect the mode distributions and their spectral separation but do not affect the total number of the modes if the sizes of the apertures are comparable. In the numerical simulations, we consider two VCSEL configurations: one with a circular aperture and the other with a broken circular symmetry determined by parameter $b = {1.5}\;{\rm{\unicode{x00B5}{\rm m}}}$ (defined in Supplement 1, Fig. S4). In the circular aperture VCSEL, ${{\rm{LP}}_{n,1}}$ modes with $n\; \ge \;{{9}}$ reveal the lowest optical losses and hence the highest quality ($Q$) factor (see Supplement 1, Fig. S3 for definition of $Q$-factor) among all the modes confined by the oxide aperture. These modes are also among the modes that dominate in the experimental emission spectrum of ball [see Fig. 3(a)], which is in qualitative agreement with [36], where the dependence between the light intensity of modes and their ${Q^2}$ was demonstrated in the occurrence of stimulated emission. For the VCSEL with the deformed circular aperture, the calculated $Q$-factors of ${{\rm{LP}}_{n,1}}$ modes with $n\; \ge \;{{9}}$ become lower than $Q$-factors of all the other modes, whose $Q$-factors remain at approximately the same level as in ball. This behavior, observed in the numerical analysis, according to [36], justifies the absence of the modes with $n\; \ge \;{{9}}$ in the case of top rackets and the domination of the lower-order modes in their emission spectra.

The density of the modes in the spectra of top rackets is higher than in ball. We hypothesize that a lower competition between the modes is responsible for this effect. Figure 7 illustrates numerically simulated cumulative nearest-neighbor eigenvalue spacing expressed in energy for circular and noncircular aperture VCSELs. In both cases, the determined numbers of the modes confined within the apertures were close to 80. A significant number of modes in the circular aperture VCSEL are closely spaced or nearly degenerated due to the almost circular symmetry of the aperture (in a real device perfect circular symmetry is impossible), and hence similar interaction with oxide aperture of two ${{\rm{LP}}_{n,m}}$ modes of the same $n$ and $m$, but of different rotation. The spectral distance between these modes increases when circular symmetry is broken. This effect is illustrated by black and red curves in Fig. 7, demonstrating a significantly greater number of neighbors with larger eigenvalue spacing in the case of the VCSEL with broken circular symmetry. Closely spaced modes compete for the same electronic states, which suppresses some of them in the laser spectrum. The strength of the modes’ interaction can be expressed by the Lorentzian function,

where $\sigma$ is an eigenvalue spacing and $\Delta E$ is the range of energies of the electronic states contributing to the mode, resulting from the time–energy uncertainty principle. $\Delta E$ can be approximated to be in the range from ${2.8} \cdot {{1}}{{{0}}^{- 6}}$ to ${1.6} \cdot {{1}}{{{0}}^{- 5}}\;{\rm{eV}}$, as discussed in Supplement 1, Section S7. Blue curves in Fig. 7 represent two Lorentzian functions, with $\Delta E$ corresponding to both limits, expressing the strength of the competition between the modes. In the case of the circular aperture VCSEL, a significant number of neighbors is in the range of the highest values of the Lorentzian function. This suggests stronger competition between a large number of modes, which inevitably leads to a lower number of the modes observed in the spectrum with respect to a VCSEL with noncircular aperture. Therefore, symmetry breaking of the aperture increases the distances between the modes and reduces their interaction, which in turn enables a larger number of modes to be present in the laser spectrum.The effect of boosting the emitted power by breaking the cylindrical symmetry is not limited to broad-area VCSELs with large aperture sizes only. Shifting the position of the oxide layers results in a modification of the mode competition and, moreover, changes also the lateral confinement induced by the selective thermal wet oxidization. In Supplement 1, Fig. S7, we present the results of the numerical analysis of VCSEL structures with an oxide aperture diameter of 4 µm. The spectral nearest-neighbor statistics show that the nearest-neighbor median increases from ${4.1} \cdot {{1}}{{{0}}^{- 5}}\;{\rm{eV}}$ in a VCSEL with circular aperture to ${1.75} \cdot {{1}}{{{0}}^{- 4}}\;{\rm{eV}}$ in VCSELs with broken cylindrical symmetry. This example shows that the effect, which is found to be crucial for boosting the emitted power in symmetry-broken broad-aperture VCSELs, is also present in the case of small-aperture VCSELs with carefully tuned position of the oxide aperture.

In the case of very-large-aperture VCSELs, larger than those considered in this study, boosting the emitted power can be hindered by the current crowding effect that causes a nonuniform gain distribution. However, the incorporation of a tunnel junction, resulting in a more uniform current distribution [37], is expected to lead to a boost of the emitted power also in configurations with even larger optical apertures.

The spatial and spectral behavior of the optical modes in the top rackets may also be explained from the viewpoint of nonintegrable cavity shapes, which result in the onset of wave chaos with its characteristically more uniform spectral spacing (spectral level repulsion) between the modes [38]. In Supplement 1, Fig. S8, we confirm that the modes of the top rackets show wave chaotic features, in contrast to ball*.*

## 6. CONCLUSIONS AND DISCUSSION

In this work, we presented a detailed spectral emission analysis of GaAs-based, electrically injected, broad-area oxide-confined VCSELs with a resonant wavelength of ${\sim}{{980}}\;{\rm{nm}}$. We demonstrated that VCSELs with specifically deformed oxide apertures (rackets)—with aperture areas not more than 6% larger than the area of the circular device (ball)—exhibit an increase in optical output power of more than 60% and an increase in quantum efficiency of more than 10%. This significant power boost is remarkable, since it is based on a deformation that only slightly increases the aperture area. More importantly, the deformation breaks the symmetry of the aperture, away from a circular shape toward a racket structure that only exhibits one-mirror symmetry. Thus, our asymmetric “ugly ducklings” become “beautiful swans,” shining much more brightly.

We analyzed in detail the underlying physical mechanisms responsible for the increase in emitted optical power. Our analysis showed that the quantum efficiency and the optical power emitted by deformed aperture VCSELs increase due to more uniform distributions of the light intensity in the apertures and more uniform density of the modes. Regarding the spatial distribution of the light intensity, we demonstrated that the intensity is more uniformly distributed in racket devices than in the ball device as noncylindrical modes emerge. Consequently, stimulated emission is more uniformly distributed over the entire surface of the aperture, and is not predominantly at the periphery as in the ball. Concerning the spectral distribution of modes, we observed that symmetry breaking eliminates modes that are spectrally closely spaced, as observed in the ball, and creates a more uniform density of optical states available for laser emission as observed in rackets. With the more uniform spectral distribution of modes, the modes experience less competition, increasing the probability of stimulated emission. Those mechanisms lead to an increase in the number of electrons per unit time recombining through stimulated emission, as well as to a reduction in the number of electrons recombining through nonradiative processes. Consequently, the heating of the VCSELs with deformed apertures is reduced, and we achieve higher quantum efficiencies and higher rollover currents, resulting in higher maximal emitted optical power. In future works, our research efforts will focus on determining the optimal oxide aperture shapes expected to enhance the emitted power by design. Figure 2(a) shows the enhancement of the maximal emitted power in a majority of the rackets with respect to the ball; however, a more detailed understanding of the dependence of the emitted power on the parameters determining the deviation of the aperture from the circular shape has not yet been obtained.

In the research presented here, we excluded dynamical phenomena, although broad-area VCSELs often exhibit spatiotemporal emission dynamics and even spatiotemporal chaos [35,38,39]. Recently, Bittner *et al*. [6] demonstrated that wave-chaotic or disordered cavities can be used to suppress such spatiotemporal instabilities in wave-chaotic semiconductor microcavity lasers. In future work, it is hoped to explore whether the spatiotemporal instabilities in racket structures can be suppressed as well, resulting in both higher output optical powers and more stable emission.

More generally, the possibility of boosting the emitted power by symmetry breaking may apply to other high-power lasers in which circular apertures are commonly used, such as broad-area edge-emitting lasers, solid-state lasers, multimode-fiber lasers, and amplifiers. Nor is this possibility limited to the optical aperture shapes discussed here; other shapes may be used with reduced spatial symmetries and suitable distributions of the refractive index that induce wave chaos. This work thus opens up a new avenue of research, whereby the efficiency of stimulated emission can be enhanced by engineering the spectral structure of the resonator. Such an approach is used already to enhance spontaneous emission; but it has for too long been left unexplored in terms of its potential for improving the stimulated emission of lasers.

## Funding

Narodowe Centrum Nauki (2015/18/E/ST7/00572); German Research Foundation (Collaborative Research Center 787).

## Acknowledgment

I. F. and T. C. are grateful for countless incredible squash and tennis matches inspiring also their scientific activities, and they are looking forward to padel and ping-pong matches in the future.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

## REFERENCES

**1. **T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. **5**, 247–271 (2011). [CrossRef]

**2. **M. Tang, Y.-D. Yang, H.-Z. Weng, J.-L. Xiao, and Y.-Z. Huang, “Ray dynamics and wave chaos in circular-side polygonal microcavities,” Phys. Rev. A **99**, 033814 (2019). [CrossRef]

**3. **Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. **108**, 243902 (2012). [CrossRef]

**4. **C. Liu, A. Di Falco, D. Molinari, Y. Khan, B. S. Ooi, T. F. Krauss, and A. Fratalocchi, “Enhanced energy storage in chaotic optical resonators,” Nat. Photonics **7**, 473–478 (2013). [CrossRef]

**5. **H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science **346**, 975–978 (2014). [CrossRef]

**6. **S. Bittner, S. Guazzotti, Y. Zeng, X. Hu, H. Yılmaz, K. Kim, S. S. Oh, Q. J. Wang, O. Hess, and H. Cao, “Suppressing spatiotemporal lasing instabilities with wave-chaotic microcavities,” Science **361**, 1225–1231 (2018). [CrossRef]

**7. **M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science **363**, 623–626 (2019). [CrossRef]

**8. **J. Mawst, A. Bhattacharya, J. Lopez, D. Botez, D. Z. Garbuzov, L. DeMarco, J. C. Connolly, M. Jansen, F. Fang, and R. F. Nabiev, “8 W continuous wave front-facet power from broad-waveguide Al-free 980 nm diode lasers,” Appl. Phys. Lett. **69**, 1532–1534 (1996). [CrossRef]

**9. **S. Uchiyama and K. Iga, “Two-dimensional array of GaInAsP/lnP surface-emitting lasers,” Electron. Lett. **21**, 162–164 (1985). [CrossRef]

**10. **D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. **33**, 1205–1215 (1997). [CrossRef]

**11. **H. Wenzel and H.-J. Wunsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **33**, 1156–1162 (1997). [CrossRef]

**12. **C. Degen, I. Fischer, and W. Elsäßer, “Transverse modes in oxide confined VCSELs: influence of pump profile, spatial hole burning, and thermal effects,” Opt. Express **5**, 38–47 (1999). [CrossRef]

**13. **C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. **124**, 1807–1809 (1961). [CrossRef]

**14. **M. Grabherr, M. Miller, R. Jager, R. Michalzik, U. Martin, H. J. Unold, and K. J. Ebeling, “High-power VCSELs: single devices and densely packed 2-D-arrays,” IEEE J. Sel. Top. Quantum Electron. **5**, 495–502 (1999). [CrossRef]

**15. **H. Kroemer, “A proposed class of hetero-junction injection lasers,” Proc. IEEE **51**, 1782–1783 (1963). [CrossRef]

**16. **E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express **17**, 7790–7799 (2009). [CrossRef]

**17. **H. Altug, D. Englund, and J. Vuckovic, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. **2**, 484–488 (2006). [CrossRef]

**18. **A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. **67**, 437–440 (1991). [CrossRef]

**19. **M. Djiango, T. Kobayashi, and W. J. Blau, “Cavity-enhanced stimulated emission cross section in polymer microlasers,” Appl. Phys. Lett. **93**, 143306 (2008). [CrossRef]

**20. **W. Wei, X. Zhang, X. Yan, and X. Ren, “Observation of enhanced spontaneous and stimulated emission of GaAs/AlGaAs nanowire via the Purcell effect,” AIP Adv. **5**, 087148 (2015). [CrossRef]

**21. **H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. **66**, 4801 (1989). [CrossRef]

**22. **M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B **87**, 205310 (2013). [CrossRef]

**23. **T. Suhr, “Modeling of coupled nano-cavity lasers,” Ph.D. diss. (Technical University of Denmark, 2012).

**24. **H.-J. Stöckmann, *Quantum Chaos* (Cambridge University, 2000).

**25. **H. Schomerus, J. Wiersig, and M. Hentschel, “Optomechanical probes of resonances in amplifying microresonators,” Phys. Rev. A **70**, 012703 (2004). [CrossRef]

**26. **S. Shinohara and T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E **75**, 036216 (2007). [CrossRef]

**27. **T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. **94**, 233901 (2005). [CrossRef]

**28. **D. J. H. C. Maas, A.-R. Bellancourt, B. Rudin, M. Golling, H. J. Unold, T. Südmeyer, and U. Keller, “Vertical integration of ultrafast semiconductor lasers,” Appl. Phys. B **88**, 493–497 (2007). [CrossRef]

**29. **L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. **60**, 289–291 (1992). [CrossRef]

**30. **M. C. Collodo, F. Sedlmeir, B. Sprenger, S. Svitlov, L. J. Wang, and H. G. L. Schwefel, “Sub-kHz lasing of a CaF_{2} whispering gallery mode resonator stabilized fiber ring laser,” Opt. Express **22**, 19277–19283 (2014). [CrossRef]

**31. **Y. X. Zhang, X. Y. Pu, K. Zhu, and L. Feng, “Threshold property of whispering-gallery- mode fiber lasers pumped by evanescent waves,” J. Opt. Soc. Am. B **28**, 2048–2056 (2011). [CrossRef]

**32. **R. Michalzik, *VCSELs: Fundamentals, Technology and Applications of Vertical-Cavity Surface-Emitting Lasers* (Springer, 2013).

**33. **Y. K. Chembo, S. K. Mandre, I. Fischer, W. Elsässer, and P. Colet, “Controlling the emission properties of multimode VCSELs via polarization- and frequency-selective feedback,” Phys. Rev. A **79**, 013817 (2009). [CrossRef]

**34. **P. Moser, J. A. Lott, and D. Bimberg, “Energy efficiency of directly modulated oxide-confined high bit rate 850-nm VCSELs for optical interconnects,” IEEE J. Sel. Top. Quantum Electron. **19**, 1702212 (2013). [CrossRef]

**35. **J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature **385**, 45–47 (1997). [CrossRef]

**36. **C.-H. Chien, S.-H. Wu, T. H.-B. Ngo, and Y.-C. Chang, “Interplay of Purcell effect, stimulated emission, and leaky modes in the photoluminescence spectra of microsphere cavities,” Phys. Rev. Appl. **11**, 051001 (2019). [CrossRef]

**37. **S. Sekiguchi, T. Miyamoto, T. Kimura, G. Okazaki, F. Koyama, and K. Iga, “Improvement of current injection uniformity and device resistance in long-wavelength vertical-cavity surface-emitting laser using a tunnel junction,” Jpn. J. Appl. Phys. **39**, 3997–4001 (2000). [CrossRef]

**38. **K. Becker, I. Fischer, and W. Elsäßer, “Spatio-temporal emission dynamics of VCSELs: modal competition in the turn-on behavior,” Proc. SPIE **5452**, 452–461 (2004). [CrossRef]

**39. **A. Barchanski, T. Gensty, C. Degen, I. Fischer, and W. Elsäßer, “Picosecond emission dynamics of vertical-cavity surface-emitting lasers: spatial, spectral, and polarization-resolved characterization,” IEEE J. Quantum Electron. **39**, 850–858 (2003). [CrossRef]