Methods In this manuscript, we only consider the case of weak QE-field coupling regime. In this check details regime, the SE decay lifetimes for both homogeneous and inhomogeneous environment are calculated

by the formula [32–34] (1) where ω is the angular frequency, c is the speed of light in vacuum, is the unit vector of the dipole moment stands for the imaginary part of Green’s tensor, and is the position of the QE. Notice that the SE lifetime depends on the dipole orientation. As is known that the quantity in vacuum equals , where is a unit tensor. We can easily deduce the SE lifetime τ vac(ω) = [ω 3 d 2/(3πℏϵ 0 c 3)]- 1 of QE embedded in vacuum according this website to Equation 1. Then, the normalized orientation-dependent SE lifetime could be defined as . To evaluate the difference degree of the lifetime orientation distribution, we define the anisotropic factor as (2) The Green tensor in Equation 1 satisfies

(3) where ϵ is the relative permittivity. It could be calculated from the electric field of a dipole source as [35, 36] (4) where is a dipole source at position . The whole elements of the Green tensor could be attained after setting the dipole source with x, AZD8931 in vitro y, and z polarizations in turn. Results and discussion In this paper, the dielectric constant of the gold nanorod is obtained by fitting the experimental data from Johnson and Christy with piecewise cubic interpolation [37]. The nanorod is placed upon the SiO2 substrate with refractive index of 1.5. Other parts are set as vacuum. We consider rectangular, cylinder, and capsule nanorods in the simulations. The corresponding schematic diagrams of the structures are shown in Figure 1a,b,c, respectively. The cross sections of each structure at x = 0 plane are shown in Figure 1d,e,f, respectively. The width of the rectangular nanorod is a = 20 nm, much the length is L = 120 nm, and the height is h = 20 nm. The diameter of the cylinder

nanorod is d = 20 nm and the length is also L = 120 nm. The capsule nanorod is modified from the cylinder shape nanorod by changing the two ends into a half-sphere shape. The total length of the capsule-shaped nanorod is still L = 120 nm. We perform the simulations by the Finite Element Method with the help of the software COMSOL Multiphysics. The coordinate origin is set at the center of the nanorod, and the nanorod is placed along the x axis. We adopt the perfectly matched layer (PML) for the absorption boundary. Figure 1 Schematic diagrams of the gold nanorod structures. (a) Rectangular, (b) cylinder, and (c) capsule nanorods. (d, e, f) The cross sections corresponding to (a, b, c), respectively. In order to calculate for the plasmonic resonance frequency, we consider a planewave normal incident with x polarization as , where k 0 is the wave number in vacuum.

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