Multimode lasing in wave-chaotic semiconductor microlasers

We investigate experimentally and theoretically the lasing behavior of dielectric microcavity lasers with chaotic ray dynamics. Experiments show multimode lasing for both $\mathsf{D}$-shaped and stadium-shaped wave-chaotic cavities. Theoretical calculations also find multimode lasing for different shapes, sizes, and refractive indices. While there are quantitative differences between the theoretical lasing spectra of the stadium and $\mathsf{D}$-cavity, due to the presence of scarred modes with anomalously high-quality factors, these differences decrease as the system size increases, and are also substantially reduced when the effects of surface roughness are taken into account. Lasing spectra calculations are based on steady-state ab initio laser theory, and indicate that gain competition is not sufficient to result in single-mode lasing in these systems.

Figure 1

Top view SEM images of (a) a $\mathsf{D}$-cavity with radius $R=200\mu \mathrm{m}$ and (b) a stadium cavity with $L=238\mu \mathrm{m}$. (c), (d) Perspective SEM images of the curved sidewall of a $\mathsf{D}$-cavity, highlighting its verticality and low surface roughness.

Figure 2

Lasing spectra integrated over a $2-\mu \mathrm{s}$-long pump pulse with $500\mathrm{mA}$ pump current for (a) a $\mathsf{D}$-cavity with $R=100\mu \mathrm{m}$, (b) a stadium with $L=119\mu \mathrm{m}$, and (c) an ellipse with $a=127\mu \mathrm{m}$ and $b=254\mu \mathrm{m}$. The insets illustrate the geometry of the cavities.

Figure 3

Spectrochrongram of (a) a $\mathsf{D}$-cavity with $R=200\mu \mathrm{m}$ and (b) a stadium with $L=238\mu \mathrm{m}$ during a $500-\mu \mathrm{s}$-long pulse measured with $1-\mu \mathrm{s}$ time resolution. The emission spectra show no discernible change over the course of $10\mu \mathrm{s}$. The pump current was $800\mathrm{mA}$ in both cases. (c) Emission spectrum of the $\mathsf{D}$-cavity during the time interval 460–$461\mu \mathrm{s}$ and (d) of the stadium during the time interval 450–$451\mu \mathrm{s}$, showing multiple lasing peaks.

Figure 4

(a) Calculated spectrum ($Q$ factors and wavelengths of passive cavity modes) for the stadium resonator, with $2L=10\mu \mathrm{m}$ (black circles), and $\mathsf{D}$-cavity resonator, with $R=4.2\mu \mathrm{m}$ (red stars). The spectra contain 200 resonances each in a wavelength window centered at $\lambda =1\mu \mathrm{m}$. The stadium data show many resonances with higher-$Q$ factors than the typical resonances, whereas the $\mathsf{D}$-cavity data do not have any significant outliers. (b) Distribution of the $Q$ factors of 1000 resonances for the stadium resonator and the (c) $\mathsf{D}$-cavity resonator. The insets show the spatial field distributions of the resonances that support the first two lasing modes. The green lines highlight the scarring by the double-diamond unstable periodic orbit and the triangle unstable periodic orbit two of the modes.

Figure 5

(a) Resonance SPA-SALT results for the lasing intensities of the stadium resonator, with $2L=10\mu \mathrm{m}$, as a function of the pump strength. There are two lasing modes that turn on within a factor of 10 of the first lasing threshold. The lasing interactions prevent more modes from turning on in this interval. (b) Resonance SPA-SALT results for the lasing intensities of the $\mathsf{D}$-cavity resonator, with $R=4.2\mu \mathrm{m}$, as a function of the pump strength. There are eight lasing modes that turn on within a factor of 10 of the first lasing threshold. The lasing interactions are not strong enough to prevent the first seven lasing modes from turning on within a factor of 2.0 of the first lasing threshold. The gain spectrum width used in the simulations is 50 nm.

Figure 6

Normalized SALT interaction coefficients [Eq. (11)] for (a) stadium and (b) $\mathsf{D}$-cavity. (a) For the stadium, the interaction between modes No. 1 and No. 4 is higher than the interaction between modes No. 1 and the other lasing modes, Nos. 2,3,5,6, due to their strong spatial overlap (as shown in the inset, top mode No. 1, bottom mode No. 4). (b) For the $\mathsf{D}$-cavity, the interactions are overall more uniform than for the stadium. Modes which have similar spatial patterns form two groups: modes $1,6,7$ and modes 2,3,4,5,8. The insets show the spatial distribution of the electric field for the lasing mode No. 1 (top) and lasing mode No. 6 (bottom). This pair has the highest SALT interaction coefficient for the $\mathsf{D}$-cavity. The cavities have identical index ($n=3.5$) and identical surface area, corresponding to a stadium length of $2L=10\mu \mathrm{m}$.

Figure 7

Resonance SPA-SALT results for the subthreshold intensities of the (a) stadium and the (b) $\mathsf{D}$-cavity as a function of the pump. The black dashed lines show the intensity variation neglecting modal interactions and intersect the $x$ axis at the noninteracting thresholds. The colored lines show the subthreshold intensities of the same modes when including the effects of interactions. The resonance SPA-SALT approximations ensure that the subthreshold intensities are linear between adjacent thresholds, with discontinuities of both their value and slope at the interacting thresholds, which are marked by the colored arrows. The dashed vertical lines serve as a guide for the eye and connect the values of the subthreshold intensities immediately before and after a new threshold is reached. The gain curve width used in the simulations is 50 nm. The cavities have identical index ($n=3.5$) and identical surface area, corresponding to a stadium length of $2L=10\mu \mathrm{m}$.

Figure 8

Dependence of the $Q$-factor distributions on the size of the stadium resonator. (a) $2L=10\mu \mathrm{m}$. The inset shows the electric field distribution of the highest-$Q$ mode. (b) $2L=20\mu \mathrm{m}$. The inset shows the normalized standard deviation of the three $Q$-factor distributions as a function of cavity size. (c) $2L=60\mu \mathrm{m}$. The inset shows the spatial distribution of the electric field of the highest-$Q$ mode. For (a), (b) and (c), the red dashed vertical line marks the $Q$ factor of the long-living ray trajectories. (d) Dependence of $Q$ factors on size $L$. Indicated are the highest-$Q$ factor $Q_{MAX}$ (black solid line), the average of the 10 highest-$Q$ factors ${〈Q〉}_{10}$ (blue solid line), and the average of all $Q$ factors ${〈Q〉}_{ALL}$ (green solid line). The linear extrapolations of $Q_{max}$ and ${〈Q〉}_{all}$ from their values at $L=5\mu \mathrm{m}$ are shown as black and green dashed lines, respectively. The red dashed line indicates the extrapolation of $Q_{max}$ according to square-root scaling.

Figure 9

$Q$-factor distributions for the (a) stadium, (b) $\mathsf{D}$-cavity, and (c) ellipse resonators with surface roughness. Each distribution contains 1000 resonances centered at $\lambda =1.0\mu \mathrm{m}$. The random boundary roughness in each of the simulations is drawn from the same distribution, and the average of the deviation of the boundary deformations in the normal direction is $\sigma =30\mathrm{nm}$. The insets show the spatial patterns of the modes with the highest-$Q$ factors for each distribution. The scarring effect observed for the highest-$Q$ modes of cavities with smooth boundaries does not persist for this level of surface roughness. The cavities have identical index ($n=3.5$) and identical surface area, corresponding to a stadium length of $2L=10\mu \mathrm{m}$.