Optimization of Purcell-enhanced microcavities with the cylindrical finite-difference time-domain algorithm

Quantum dots embedded in annular bullseye microcavities have been reported to be efficient single-photon sources. However, the proposed structures thus far often involve gratings of fixed periodicities, which may not be optimal considering the fact that Bessel functions are nonperiodic. In this paper, we present an optimization scheme for chirped annular microcavities where Purcell factors larger than 80 can be achieved through astute selection of the optimization function.

Figure 1

Yee grid in cylindrical coordinates displaying the arrangement of the field components.

Figure 2

Schematic of a (suspended) chirped bullseye cavity. $\theta$ represents the angle of emission.

Figure 3

Four different structures attained from the PSO algorithm. Structure A (blue; first) was attained with $w_1=1, w_2=0, w_3=0, w_4=0$, and $Q_{\text{max}}=600$. Structure B (green; second) was attained with $w_1=0.5, w_2=0.5, w_3=0, w_4=0$, and $Q_{\text{max}}=600$. Structure C (purple; third) was attained with $w_1=0.1, w_2=0.5, w_3=0.2, w_4=0.2$, and $Q_{\text{max}}=600$. Structure D (red; fourth) was attained with $w_1=0.05, w_2=0.5, w_3=0.2, w_4=0.25$, and $Q_{\text{max}}=300$.

Figure 4

Figure showing the steady-state fields (with normalized intensities) of structure A (a–c), structure B (d–f), structure C (g–i), and structure D (j–l). (a), (d), (g), (j) Emission field profiles in the $x\text{−}z$ plane plotted in the logarithmic scale for clarity. (b), (e), (h), (k) Far-field profiles in the horizontal plane. (c), (f), (i), (l) In-plane dipole emission at the position of the dipole source.

Figure 5

Plots summarizing data and FOMs obtained from the four structures. (a) Purcell factor as a function of energy (in eV). (b) Collection efficiency against acceptance angle. (c) Comparison of $Q$ factors attained. (d) Comparison of the normalized effective mode volume $V_{\text{eff}}^{\prime }=V_{\text{eff}}/{({\lambda }_0/n)}^3$. (e) Comparison of the average angle of emission calculated with the expression for $q_4$ in Eq. (16) (neglecting the $\pi /2$ at the denominator). (e) Comparison of the confinement factor calculated with Eq. (15).

Figure 6

A Pareto front (black outlines) interpolation based on multiple runs of the PSO algorithm. Each individual iteration is represented by different symbols as shown in the legend. The optimized structures from both the main text and an existing periodic structure [34] are included as well.