Analysis of the Purcell effect in photonic and plasmonic crystals with losses

(a) A metal membrane with uniform metal-dielectric interfaces and the electrical field cross-section of a SPP mode. (b) A plasmonic crystal fabricated on top of a metal membrane.

(a) SPP’s electromagnetic field with a momentum k at a uniform metal (z ≤ 0) -dielectric (z > 0) boundary with an area l ². $E_k^z$ is an out-of-plane component of the electric field, and $E_k^{//}$ and $H_k$ are in-plane components of the electric and magnetic fields. $E_k^z$ and $E_k^{//}$ are parallel to z-direction and k, respectively. An emitter with an electric dipole μ lies at the distance of z_A from the metal surface (positioned at z = 0). (b) Time-evolution of $E_k^z$ and $E_k^{//}$ oscillating in the z-k plane. e_k is a unit vector composed of the amplitudes of $E_k^z$ and $E_k^{//}$ .

(a) Distributions of the SPP spontaneous emission enhancement spectrum i(υ,k) by classical electrodynamics analysis (left-side panel), following References [13–16] and by quantum electrodynamics analysis, f(υ,k) (right-side panel). The electric dipole lies normal to the Au surface at a distance of 10 nm from it. ω_p and τ_p are set 1.21 × 10¹⁶ sec⁻¹ and 1.05 × 10⁻¹⁴ sec, respectively. (b) Spectra of f(υ,k) and i(υ,k) at k = 0.1 nm⁻¹ along the dashed lines in Fig. 3(a), showing an excellent match between the two analyses. The Δυ _k shows a spectral half-width. (c) Distributions of i(υ,k) above the light line. The distances of μ from the Au surface are set to 200 nm and 400 nm.

Purcell enhancement factors at various frequencies from quantum analysis - $F^{\text{sp}}$ (black lines) and classical analysis - $I^{\text{sp}}$ (red lines) for an exciton z_A = 10 nm and 30 nm away from the Au/InP boundary, estimated by summing up f(υ,k) and i(υ,k) over k ≤ 0.3 nm⁻¹, respectively. The values of f(υ,k) and i(υ,k) for z_A = 10 nm are plotted in Fig. 3(a). Blue lines show the Purcell enhancement factor $F_{\text{non-ab}}^{\text{sp}}$ for non-absorbing media. The electric dipole lies normal to the Au surface. Non-SPP modes were ignored in the calculation of these plots.

(a) 1D-array of high-Q resonators, composed of defects in 2D-photonic crystal with a periodic permittivity ε(r). The defects are aligned in x-direction with a lattice vector R e _x . An exciton lies at the position r _A in a defect. (b) A single defect in a 2D-photonic crystal with a permittivity ε ⁰(r).

(a) The plasmonic crystal consists of hexagonally arranged InP-filled holes in the Au membrane. The thickness of the Au layer is 20 nm, and the periodicity of the crystal a and radius r of the InP holes are determined by a = 450 nm and r/a = 0.2. (b) Dispersion diagrams of antisymmetric modes in the plasmonic crystal, obtained by FDTD simulation (dots). Dashed lines show the dispersion branches of the unpatterned InP/Au/InP structure.

(a) Field patterns on surface of Au layer in the plasmonic crystal, belonging to the upper edge (top figures) and the lower edge (bottom figures) of the plasmonic band gap shown in Fig. 6(b), obtained by FDTD simulation. The field of each band edge consists of three orthogonal modes (monopoles in the left figures and dipoles in the right figures). At the band edge, three dispersion branches overlap so close that their eigen-modes could not be separated by our FDTD simulation. (b) The sum of the density of photonic states ${\displaystyle {\sum }_j{\displaystyle {\sum }_K{D}_{K,j}(\upsilon )}}$ for the plasmonic crystal (red line) and unpatterned structure (black line). For computation, we take the summation over the region K ≤ 0.0005 nm⁻¹, considering the degeneracy of each branch. The Q-factor for the plasmonic crystal and unpatterned structure equals 96 and 98 by FDTD simulation, respectively. In the estimation, the upper branches were ignored because of the large leakage loss.

Values of $1/(1+{\Theta }_k)$ , $W_{Ez}/W$ , $W_{E//}/W$ , and $W_H/W$ plotted for different k’s for a SPP mode at Au/InP interface, where $W_{E//}$ ≡ $\frac{1}{2}{\displaystyle \iiint \partial (\omega \epsilon )/\partial \omega |E_k^{//}{|}^{2}dr}$ , $W_{Ez}$ ≡ $\frac{1}{2}{\displaystyle \iiint \partial (\omega \epsilon )/\partial \omega |E_k^z{|}^2}dr$ , $W_H$ ≡ $\frac{1}{2}{\displaystyle \iiint {\mu }_0|H_k{|}^{2}dr}$ , and W is a k-mode’s total field energy. The lines show the values estimated by analytically solving Maxwell’s equations [21], and the marks by FDTD. The index and plasma frequency at Au/InP interface are set n = 3.2, and $2\pi c/{\omega }_p$ = 156 nm, respectively.

(a) An InP/Au/InP structure with an electric dipole lying on the dielectric side (top), and electric field components parallel to the Au surface of an antisymmetric and symmetric modes (bottom). The Au membrane is 10 nm thick. The electric dipole lies normal to the Au surface at a distance of 10 nm apart from it. (b) The distribution of $i^{layer}(\upsilon ,k)$ estimated for the structure shown in Fig. 9(a). The ${\upsilon }_{v=0}$ is a frequency with $v^g=0$ for the antisymmetric modes, and ${\upsilon }_p\equiv {\omega }_p/{(1+n^2)}^{1/2}$ . (c) Dissipation spectrum $i^{layer}(\upsilon ,k)$ at k = 0.1 nm⁻¹ along the dashed line in Fig. 9(b). The $i^{layer}(\upsilon ,k)$ below the light line is expressed by the sum of two Lorentzian spectra, $i^{layer}(\upsilon ,k)$ = $i^{anti}(\upsilon ,k)$ + $i^{sym}(\upsilon ,k)$ , corresponding to SE into the antisymmetric and symmetric modes. The Δυ _k is a half-width of the spectrum.

(a) Quality factor ${\upsilon }_k^0/\Delta {\upsilon }_k$ of symmetric and antisymmetric modes plotted with different k’s, estimated by fitting $i^{layer}(\upsilon ,k)$ at RT (shown in Fig. 9(c)) and 77 K with Lorentzian spectra. The τ_p is set 1.05 × 10⁻¹⁴ sec at RT and 3.72 × 10⁻¹⁴ sec at 77 K, respectively, and ω_p = 1.21 × 10¹⁶ sec⁻¹. The circles show the quality factors, estimated by $\upsilon /2k^{″}{v}^g$ for antisymmetric modes at RT [8]. (b) Purcell enhancements for the symmetric and anti-symmetric mode, $I^{sym}$ and $I^{anti}$ , at RT and 77 K, estimated by summing up $i^{sym}$ and $i^{anti}$ shown in Figs. 9(b) and 9(c) over k ≤ 0.3 nm⁻¹, respectively.