Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides

We develop a self-consistent finite-element method to quantitatively study spontaneous emission from emitters in nanoscale proximity of plasmonic waveguides. In the model, it is assumed that only one guided mode is dominatingly excited by the quantum emitter, while the cross section of the plasmonic waveguide can be arbitrary. The fraction of the energy coupled to the plasmonic mode can be calculated exactly, which can be used to determine the efficiency with which single optical plasmons are generated. We apply our numerical method to calculate the coupling of a quantum emitter to a cylindrical metallic nanowire and a square metallic waveguide, and compare the cylindrical metallic nanowire with previous work that employs quasistatic approximation. For the cylindrical metallic nanowire we observe good agreement with the quasistatic approximation for radii below 10 nm, but for increasing radius the spontaneous emission $\beta$ factor and the plasmonic decay rate deviate substantially, by factors of up to 5–10 for a radius of $\sim 100\text{nm}$, from the values obtained in the quasistatic approximation. We also show that the quasistatic approximation is typically valid when the radius is less than the skin depth of the metals at optical frequencies. For the square metallic waveguide we estimate an optimized value for the spontaneous emission $\beta$ factor up to 80%.

Figure 1

(Color) Different emission channels involved in the decay process of a quantum emitter (red dot) coupled to a plasmonic waveguide. In the radiation channel the photons are traveling in free space. In the plasmonic channel the plasmonic modes are excited and guided by the metallic nanowire. In the nonradiative channel, electron-hole pairs are generated.

Figure 2

Dispersion relation versus radius for the cylindrical gold nanowire with the background medium of PMMA $\left(n=1.414\right)$ at the wavelength of $1\mu \text{m}$. Inset (a) shows the waveguide structure. Inset (b)–(g) show electric field orientation of the possible eigenmodes supported by the waveguide.

Figure 3

Dispersion relation versus side length of square plasmonic waveguides with background medium of PMMA $\left(n=1.414\right)$ at a wavelength of $1\mu \text{m}$. The metal core is gold. Inset (a) shows the waveguide structure. Inset (b)–(e) show electric field orientation of the possible eigenmodes supported by square plasmonic waveguides.

Figure 4

(Color) A single quantum emitter coupled to a metallic nanowire. The gray transparent region represents the perfectly matched layers, the mode matching boundary condition is applied on the top and the bottom of the structure. The quantum emitter is implemented by an electric line current.

Figure 5

Comparison of FEM-simulated results based on the dyadic Green’s function with the quasistatic approximation for the metallic nanowire.

Figure 6

The radius dependence of $\frac{{\gamma }_{pl,\text{FEM}}}{{\gamma }_{pl,quasi}}$ for four different metals. The skin depths of the four different metals are 16, 22.5, 35, and 40 nm, the corresponding relative optical permittivities are $-100+3.85i$, $-50+3.85i$, $-20+3.85i$, and $-15+3.85i$. The vacuum wavelength is $1\mu \text{m}$.

Figure 7

(Color) Distance dependence of the plasmonic decay rates and spontaneous emission $\beta$ factors for the square plasmonic waveguide.

Figure 8

Length-dependence study of the total decay rate for the metallic nanowire. The radius of the metallic nanowire is 20 nm, the distance of emitter to the wire edge is 30 nm.

Figure 9

(a) Length dependence of the total decay rate for the square plasmonic waveguide. The side length is 30 nm, the distance of the emitter to the edge of the square metal core is 20 nm. (b) Length dependence for the points from (a) (marked by ellipses) where $\text{real}\left(e^{j\left(\beta L_0-\varphi \right)}\right)=0$ holds approximately. (c) Illustration of the reflection of the normal magnetic field of the fundamental hybrid in the 3D model, with $r$ and $\theta$ being the reflection coefficient and phase shift, respectively.