Optimizing the chiral Purcell factor for unidirectional single-photon emitters in topological photonic crystal waveguides using inverse design

We present an inverse-design approach to significantly improve the figures of merit for chiral photonics with quantum emitters in topological photonic crystal slab waveguides. Beginning with a topological waveguide mode with a group index of approximately 10 and a maximum forward or backward Purcell factor at a chiral point of less than 0.5, we perform optimizations of the directional Purcell factor. We use a fully three-dimensional guided-mode expansion method to efficiently calculate waveguide band-dispersion properties and modes, while automatic differentiation is employed to calculate the gradient of objective functions. We present two example improved designs: (i) a topological mode with an accessible group index of approximately 30 and a maximum unidirectional Purcell factor at a chiral point greater than 4.5, representing a nearly 10-fold improvement to the Purcell factor, and (ii) a slow-light mode well away from the Brillouin-zone edge with a group index greater than 350 and a maximum unidirectional Purcell factor at a chiral point greater than 45.

Figure 1

(a) Schematic of the topological edge-mode waveguide. The direction of propagation $x$ and relevant geometric parameters are indicated. (a) Schematic (top-down view) of the photonic crystal. The rhombic unit cell is outlined. Magnitude of the Berry curvature for the mode below the band gap $\mathrm{\Omega }\left(\mathit{k}\right)$ (in units of $[a^2/4{\pi }^2]$) for the (b) upper and (c) lower crystals shown in (a) for slab thickness $h=170$ nm, $a=250$ nm, $r_1=28$ nm, and $r_2=62.5$ nm. The outer dashed black line is the edge of the first Brillouin zone, and the inner dashed line's vertices are the $K$ and $K^{\prime }$ points.

Figure 2

(a) Band structure for the initial design [21] with $a=266$ nm. The (NT) guided mode is drawn in solid purple, the second guided mode is drawn in dashed purple, and bulk modes are in black. The NT mode's SM bandwidth is highlighted in turquoise. Additional guided modes existing at the secondary interface at the edge of the GME unit cell have been removed here, as well as in Figs. 3 and 4 (see Appendix pp1). (b) The group index for the NT mode is drawn in purple, and the maximum chiral forward Purcell factor is drawn with a solid blue line. Frequencies at which the NT mode is above the light line are highlighted in gray. The following fields are calculated at the circles in (a) and (b), where $n_g=8.5$. The $xy$-plane cross sections are at the center of the slab. (c) Forward Purcell factor $P^f$ for $\mathit{\mu }={\mathit{\sigma }}_{-}$, (d) the associated chiral directionality $C$, and (e) the associated chiral forward Purcell factor ${\tilde{P}}^f$. The red dashed box in (c) encloses the region shown in (d) and (e).

Figure 3

Summary of design A. (a) Band structure. (b) Group index and the maximum chiral forward Purcell factor for the NT mode. The following fields are calculated at the circles in (a) and (b), where $n_g=30.2$. (c) Forward Purcell factor $P^f$ for $\mathit{\mu }={\mathit{\sigma }}_{-}$, (d) the associated chiral directionality $C$, and (e) the associated chiral forward Purcell factor $\widetilde{P^f}$. The holes which were not allowed to be modified during the optimization are drawn in gray, while the holes which were are drawn in white.

Figure 4

Summary of design B. (a) Band structure. (b) Group index and the maximum chiral forward Purcell factor for the NT mode. The following fields are calculated at the circles in (a) and (b), where $n_g=360$. (c) Forward Purcell factor $P^f$ for $\mathit{\mu }={\mathit{\sigma }}_{-}$, (d) the associated chiral directionality $C$, and (e) the associated chiral forward Purcell factor $\widetilde{P^f}$. The holes which were not allowed to be modified during the optimization are drawn in gray, while the holes which were are drawn in white.

Figure 5

(a) Band structure for the unit cell in (b). The modes existing at guide 1 are drawn in purple, the modes existing at guide 2 are drawn in green, and the bulk modes are drawn in black. (b) GME unit cell containing the waveguide which the optimization started from [21] and another waveguide [31].