Enhanced Spontaneous Emission at Third-Order Dirac Exceptional Points in Inverse-Designed Photonic Crystals

We formulate and exploit a computational inverse-design method based on topology optimization to demonstrate photonic crystal structures supporting complex spectral degeneracies. In particular, we discover photonic crystals exhibiting third-order Dirac points formed by the accidental degeneracy of monopolar, dipolar, and quadrupolar modes. We show that, under suitable conditions, these modes can coalesce and form a third-order exceptional point, leading to strong modifications in the spontaneous emission (SE) of emitters, related to the local density of states. We find that SE can be enhanced by a factor of 8 in passive structures, with larger enhancements $\sim \sqrt{n^3}$ possible at exceptional points of higher order $n$.

Figure 1

Inverse-designed 2D square lattices comprising refractive indices $=\{2,3,1.82\}$ (upper, middle, and lower schematics) materials in air (white regions), with periodicity $a=\{1.05,0.6,1\}\lambda$. Lower right: Band structure of the lattice with refractive $\mathrm{index}=2$ (upper schematic), revealing a Dirac point induced by the presence of an accidental third-order degeneracy (D3) of monopolar ($M$), dipolar ($D$), and quadrupolar ($Q$) modes (upper insets). A schematic of the Brillouin zone (BZ) denoting high-symmetry $\mathbf{k}$ points ($Y$, $\mathrm{\Gamma }$, $X$, $M$) is also shown. Because of the lack of $C_{4v}$ symmetry, the dispersions along the $X$ and $Y$ directions differ.

Figure 2

(a) Real and (c) imaginary eigenfrequencies as a function of $k_x$ and $k_y$ in the vicinity of a third order exceptional point (EP3) of the structure described in Fig. 1, located at ${\mathbf{k}}^{\mathrm{EP}3}\approx \{7,1.8\}\times {10}^{-5}(2\pi /a)$ (red dot). The blue contours denote regions of second-order exceptional points where two of the three modes coalesce. The plots in (b) and (d) show the corresponding band structures along the $\mathbf{k}$ lines marked by red arrows. (e) Contour plot showing the enhanced Petermann factor (PF) associated with one of the modes in the vicinity of the EP3, and (f) corresponding enhancement along the direction shown by the red arrow, for all three modes.

Figure 3

(a) Local density of states (LDOS) at the center of the unit cell of the structure in Fig. 1, evaluated at either ${\mathbf{k}}^{\mathrm{EP}3}\approx \{7,1.8\}\times {10}^{-5}(2\pi /a)$ (red curves) or $\mathbf{k}=\{7,1.8\}\times {10}^{-2}(2\pi /a)\gg {\mathbf{k}}^{\mathrm{EP}3}$ (blue curves). (b) Maximum (eightfold) LDOS enhancement associated with an EP3, computed via the reduced $3\times 3$ Hamiltonian model of (2). (c)–(f) LDOS profiles evaluated at either ${\omega }_{\mathrm{EP}3}$ or at the nondegenerate frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$, corresponding to the EP3 and far-away points described in (a). Note that the LDOS is evaluated only in air regions since the LDOS within a lossy medium formally diverges [53].