Systematic study of spontaneous emission in a two-dimensional arbitrary inhomogeneous environment

The spontaneous emission (SE) of the excited atoms in a two-dimensional (2D) arbitrary inhomogeneous environment has been systematically studied. The local density of states, which determines the radiation dynamics of a point source (for 3D) or a line source (for 2D), in particular, the SE rate, is represented by the electric dyadic Green’s function. The numerical solution of the electric Green’s tensor has been accurately obtained with the finite-difference frequency-domain method with the proper approximations of the monopole and dipole sources. The SE of atoms in photonic crystal and plasmonic metal plates has been comprehensively and comparatively investigated. For both the photonic crystal and plasmonic plates systems, the SEs depend on their respective dispersion relations and could be modified by the finite-structure or finite-size effects. This work is important for SE engineering and the optimized design of optoelectronic devices.

Figure 1

The five-point stencil for the FDFD method. ${\Delta }_x$ and ${\Delta }_y$ are, respectively, the spatial steps along the $x$ and $y$ directions. $\Phi =E_z$ for a TM wave, and $\Phi =H_z$ for a $\mathrm{TE}$ wave. $\epsilon =n_c^2$ is the relative permittivity in the discretized region, and $n_c$ is the complex dielectric constant of the optical material.

Figure 2

The imaginary parts of the electric DGFs in 2D free space along the $y=0$ direction. The line source is located at $(x=0,y=0)$. (a) The imaginary part of the $E_z$ field generated by a $z$-polarized monopole source. The analytical solution is $0.25J_0\left(k\right|x\left|\right)$ and $\mathrm{Im}[G_{\mathit{zz}}({\mathbf{t}}_0,{\mathbf{t}}_0)]=0.25$. (b) The imaginary part of the $E_x$ field generated by an $x$-polarized dipole source. The analytical solution is $J_1(k|x|)/4k_0\left|x\right|$ and $\mathrm{Im}[G_{\mathit{xx}}({\mathbf{t}}_0,{\mathbf{t}}_0)]=0.125$. (c) The relative errors of the DGFs at the singularity points as a function of points per wavelength (PPW) plotted on the log-log scale.

Figure 3

(a) The normalized SER of the atoms, which are located at the air and surrounded by the $4\times 4$ or $20\times 20$ dielectric rods. (b) The dispersion curve of the infinite PC structure with dielectric rods for the TM mode. The SER is calculated by $\mathrm{Im}[G_{\mathit{zz}}({\mathbf{t}}_0,{\mathbf{t}}_0)]$.

Figure 4

(a) The normalized SER of the atoms, which are located at the dielectric medium and surrounded by the $4\times 4$ or $20\times 20$ air rods. (b) The dispersion curve of the infinite PC structure with air rods for the TE mode. The SER is calculated by the summation of $\mathrm{Im}[G_{\mathit{xx}}({\mathbf{t}}_0,{\mathbf{t}}_0)]$ and $\mathrm{Im}[G_{\mathit{yy}}({\mathbf{t}}_0,{\mathbf{t}}_0)]$.

Figure 5

The normalized SER as a function of the distance between the atoms and the gold plates. The transition frequency of the atoms is set to be $510\hspace{0.5em}\mathrm{nm}$.

Figure 6

The normalized SER as a function of the transition frequency of the atoms. The distance $d$ is set to be $30\hspace{0.5em}\mathrm{nm}$.

Figure 7

(a) $\mathrm{Im}[E_y]$ for ${\mathrm{â}}_y$ dipole source at $510\hspace{0.5em}\mathrm{nm}$; (b) $\mathrm{Im}[E_y]$ for ${\mathrm{â}}_y$ dipole source at $800\hspace{0.5em}\mathrm{nm}$; (c) $\mathrm{Im}[E_x]$ for ${\mathrm{â}}_x$ dipole source at $510\hspace{0.5em}\mathrm{nm}$; (d) $\mathrm{Im}[E_z]$ for ${\mathrm{â}}_z$ monopole source at $510\hspace{0.5em}\mathrm{nm}$.