Quasi-static method and finite element method for obtaining the modifications of the spontaneous emission rate and energy level shift near a plasmonic nanostructure

Yun-Jin Zhao; Meng Tian; Xiao-Yun Wang; Hong Yang; Heping Zhao; Yong-Gang Huang

doi:10.1364/OE.26.001390

1. Introduction

The modifications of the spontaneous emission rate and the energy level shift of a quantum emitter near a plasmonic nanostructure have attracted considerable attention in the recent years for both fundamental and applicative purposes [1–27]. Great progresses have been made in this field including large single-molecule fluorescence enhancement [7–10], surface plasmon enhanced LEDs [11–13], single-molecule sensing [14–16], plasmonic nanolaser [17–20], enhanced Raman scattering [21–27], and so on.

According to quantum electrodynamics [28–31], both the spontaneous emission and the energy level shift of a quantum emitter are arising from the interaction with its electromagnetic environment. Surface plasmonic nanostructure is one promising platform to tailor these two properties because it can generate surface plasmon resonances at the interfaces of metals and dielectrics and can squeeze the optical fields down to the nanoscale, which can greatly enhance its interaction with quantum emitter. For small emitter-surface distance, the strength of interaction can ever overcome radiative and nonradiative losses and the system enters to the strong coupling regime which leads to many novel quantum phenomena [32–35].

Theoretically, the modifications of the spontaneous emission rate and the energy level shift of a quantum emitter for the emergence of a plasmonic nanostructure can be expressed in terms of the classical Green tensor, which can be evaluated by many different methods. For nanostructures with high symmetric, such as metal nano-sphere, metal surface, metal nano-wire and so on, semianalytic method can be used to construct the Green tensor [36–38]. However, for general nanostructures, numerical method is demanded. Finite difference time domain (FDTD) method and finite element method (FEM) are considered to be the most representative methods of full-vector [39–43]. Compared to FDTD technique, in which rectangular subdomain is used and the electromagnetic fields are assumed to be invariant in one meshgrid, FEM method can be applied to more complex and sophisticated nanostructure. More flexible discretization strategy by using different mesh shapes can be adopted, and more importantly, the dramatic change of the electromagnetic fields of the plasmonic near the surface can be handled properly. However, it needs a significant amount of memory and time. Furthermore, the performance of the variational procedure used in a FEM calculation needs to be examed carefully since the real part for the Green tensor with equal space arguments is divergent. For emitter-surface distance down to nanometer scale, quasi-static approximation method is usually adopted [39,44], where retardation is neglected and a lot of the electrostatic methods such as image method can be applied [45]. This can greatly simplify the computation. However, the questions about under what circumstances and to what extent the quasi-static approximation can be applied are unclear and need to be investigated.

In this work, we propose the answers to the above questions. By the quasi-static method and FEM method, we investigate the modifications of the spontaneous emission rate and the energy level shift when an emitter is near two typical kinds of metal naonstructures. One is a metal-air planar interface, which supports surface plasmon polaritons (SPP). The other is a metal nano-sphere, which supports localized surface plasmon polaritons (LSPP). In the above two cases, analytic results are presented as a reference. We also take into account the emitter-surface distances on the effectiveness of the FEM method and the quasi-static method. We first describe the model and present the analytical method, quasi-static method and the FEM method used in this paper. Next, we make a comparison between the results by FEM or quasi-static and the analytical ones, when the emitter is located above a metal-air planar interface for two different emitter-surface distances. Then, we turn to show their performances when LSPP exists. Results from both methods for a nanosphere dimmer are demonstrated.

2. Models and methods

We consider two different kinds of plasmonic nanostructures, as shown in Figs. 1(a) and 1(b). The metal is chosen to be silver with a permittivity given by the Drude model $ε_{\infty} - ω_{p}^{2} / (ω^{2} - i γ ω)$ , where ε_∞ = 6, ω_p = 7.89eV, γ = 0.051eV [46]. The permittivity of the air is given by ε₁ = 1. A two-level quantum emitter is located in the air at a distance l away from the surface. The transition dipole moment is d = dn where d is the strength and n represents the unit vector along dipole moment. In all the calculations, d = 24D (≈ 0.5e nm) [46] and the orientation n is normal to the metal surface for the sake of simplicity. The radius for the silver sphere is a = 20nm.

Fig. 1 Schematic diagrams. An emitter ‘A’ is placed near (a) a silver-air interface or (b) a silver nano-sphere. For simplicity, the dipole moment of the emitter ‘A’ is normal to the surface. ε₁ and ε₂ are the permittivities of air and silver, respectively. The distance between the emitter and the surface of metal is l. The radius for the sphere is a.

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According to quantum electrodynamics, the spontaneous emission rate and the energy level shift can be expressed by the classical Green tensor at the weak coupling regime [39,46],

\begin{array}{l} Γ (r; ω) = \frac{2 d^{2} ℑ [n \cdot G (r; r; ω) \cdot n]}{ℏ ε_{0}}, \\ Δ (r; ω) = - \frac{d^{2} ℜ [n \cdot G (r; r; ω) \cdot n]}{ℏ ε_{0}}, \end{array}

where the Green tensor in a medium with local response permittivity ε(r, ω) and permeability μ = 1 is described by the equation

\nabla \times \nabla \times G (r, r^{'}; ω) - ε (r, ω) k_{0}^{2} G (r, r^{'}; ω) = k_{0}^{2} I δ (r - r^{'})

(Readers interested in nolocal response medium can refer to [47–50]. In this work, the medium is considered to be local which is similar to the case of Ref. [32–35,51]). Here, k₀ = ω/c, ω is the frequency and c represents speed of light in vacuum.

As pointed out previously [40,42], the real part of the Green tensor with equal space arguments is divergent and the scattered Green tensor can be used to take place of the total Green tensor where the homogeneous-medium contribution is attributed to the definition of the transition frequency of the quantum emitter. The scattered parts of the analytical Green tensor are shown in Eq. (5) [for metal-dielectric planar interface shown in Fig. 1(a)] and Eq. (6) [for metal-sphere structure shown in Fig. 1(b)] in Appendix A and Appendix B, respectively. The following parts of this section are devoted to the scattered part of the Green tensor by quasi-static method and FEM method.

2.1. Quasi-static method

Since we are interested in the case of short emitter-metal distance, the metal surface takes great effect in the modification of the spontaneous emission rate and the energy level shift. Usually, this is treated by the quasi-static approximation where the electromagnetic field is considered in its static limits (the wave vector k → 0). For a planar interface, such as shown in Fig. 1(a), image dipole method is usually adopted. In this case, the scattered Green tensor for the boundary effect can be written as $G_{static} (r^{'}, r) = G_{s} (r^{'}, - r) = \pm \frac{ε_{2} - ε_{1}}{4 π ε_{1} (ε_{2} + ε_{1}) {| r + r^{'} |}^{3}} (\frac{3 (r + r^{'}) (r + r^{'})}{{| r + r^{'} |}^{2}} - I)$ , where ± correspond to the dipoles polarized normal and parallel to the interface, respectively, r is the position of the emitter and −r is the position of its image. In our model, the relevant component of the scattered Green tensor can be clearly written as [39]

G_{static}^{z z} (r, r) \equiv z \cdot G_{static} (r, r) \cdot z = \frac{ε_{2} - ε_{1}}{16 π ε_{1} (ε_{2} + ε_{1}) l^{3}},

where l is the distance between the emitter and the silver surface.

In the case of metal-sphere shown in Fig. 1(b), this can be resolved by solving the electrostatic equation and the radial term of the scattered Green tensor can be written as [35]

G_{static}^{r r} (r, r) = \frac{1}{4 π ε_{1} a^{3}} \sum_{n = 1}^{\infty} \frac{{(n + 1)}^{2}}{{(1 + ξ)}^{2 n + 4}} \frac{ε_{2} - ε_{1}}{ε_{2} + \frac{n + 1}{n} ε_{1}},

where ξ = l/a, l is the distance between the emitter and the surface of the sphere, and a is radius for the sphere.

For nanostructure with arbitray shape, we should resort to numerical method. We use the MNPBEM tool box with the quasi-static solver to obtain the quasi-static results for the nanosphere dimmer [44].

2.2. Finite element method

For the FEM method, the Green function is obtained by calculating the electric fields of an oscillating electric point dipole, which is modeled by an electric line current in the limiting case of when the length approaches zero while maintaining the product between the current and its length (see “Electric Point Dipole” in the help documentation of COMSOL Multiphysics). The electric field at r′ of a electric point dipole d at r is E(r′) = G(r′, r) · d /ε₀. Therefore, the component of the scattered Green tensor is [39,52]

G_{FEM}^{n n} (r, r) \equiv n \cdot G (r, r) \cdot n = ε_{0} E_{s} (r) / d,

where E_s(r) = E_total(r) − E₀(r). Here, E_total(r) and E₀(r) are the components of the electric fields along the dipole direction, which are calculated with and without the nanostructure, respectively, at the dipole location. To calculate the right scattered electric field E_s(r), we should adopt the same simulation domain and the same mesh for calculating E_total(r) and E₀(r), since both terms are divergent in theory and are mesh and domain dependent in the procedure of FEM. It should be noted that we adopt the oscillating electric point dipole source other than the oscillating current source with finite size, since point source can be used to model dramatic change of the electromagnetic field more precisely. We have numerically proved this. Furthermore, curved mesh elements supplied by COMSOL Multiphysics can be used to approximate a boundary much better.

For the planar metal surface model, the simulation domain is chosen to be cuboid and it is sphere for the nanosphere model. Perfectly matched layers (PML) are used to simulate the unbounded or infinite domains. Because of the symmetry of our model, the simulation domain is reduced to one quarter. The size of the simulation domain is of the order of 2λ³. Since there is dramatic change for the radiated field near the dipole location and around the metal surface, more fine mesh element is needed. A nonuniform mesh is employed with a maximum element size of 0.2nm in the dipole neighborhood. To this end, a small domain about 10nm³ with the point dipole located at its center is used. For the nanosphere model, maximum element size is 5nm inside the silver sphere. In the case of planar metal surface model, maximum element size is λ/30 at the planar metal interface. In both models, the maximum element size elsewhere is set to λ/6 and ten layers of PML were used. We have checked that these parameters provide accurate numerical convergence for both models done in this work.

3. Results and discussions

In all the calculations, the spontaneous emission rate is normalized by the spontaneous emission rate $Γ_{0} = d^{2} ω^{3} \sqrt{ε_{1}} / 3 π ℏ ε_{0} c^{3}$ in homogeneous medium. To demonstrate the ability of both the FEM method and the quasi-static method, we show the results for emitter located above the silver-air interface in Fig. 2, where SPP may take effect and for emitter located above a silver sphere in Fig. 3, where LSPP may come into play. In both models, as an example, we consider two different emitter-surface distances and change the transition frequency over a wide range.

Fig. 2 The modified spontaneous emission rate and the energy level shift. The red solid line, dashed blue line and the black circles are the results by the methods of analytical, quasi-static and FEM, respectively. (a) – (d) are for short emitter-interface distance l = 2nm. (a) Normalized spontaneous emission rate Γ/Γ₀ with Γ₀ and (b) energy level shift. The insets are for frequency away from ω_sp. There are no visible differences for both the quasi-static method and the FEM method. (c) and (d) are for Γ_{R_error} ≡ (Γ_i − Γ_analy)/Γ_analy and Δ_error ≡ Δ_i − Δ_analy, respectively, where the subscript i represents the quasi-static (Eq. (2)) or the FEM method (Eq. (4)) and subscript analy is the analytic (Eq. (5)). The inset in (d) shows a slight blue-shift for Γ/Γ₀ by the quasi-static method to explain the relative large error around ω_sp. (e) – (f) are for longer emitter-interface distance l = 5nm.

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At short emitter-interface distance, for example l = 2nm shown in Figs. 2(a)–2(d), both the FEM method and the quasi-static method work well for the calculation of the spontaneous emission rate and the energy level shift. There are no visible differences between the numerical results and the analytical results [shown in Figs. 2(a) and 2(b)]. The insets are the results for frequency away from the SPP resonance frequency ( $ω_{sp} = ω_{p} / \sqrt{1 + ε_{\infty}} \approx 2.982 e V$ [39]), from which we can see that errors are small. To show these minor differences, we define Γ_{R_error} ≡ (Γ_i − Γ_analy)/Γ_analy and Δ_error ≡ Δ_i − Δ_analy, where Γ_i and Γ_analy are the numerical and analytical spontaneous emission rate, respectively and similarly, Δ_i and Δ_analy represent the numerical and analytical energy level shift respectively. From Fig. 2(c), we find that Γ_{R_error}| < 2% for both methods. Further, we find that FEM method works more better than the quasi-static method. For Δ_error shown in Fig. 2(d), numerical errors are less than 0.055meV for both methods when the frequency (below 2.5eV) is away from ω_sp. The maximal errors occurred at frequency around ω_sp where SPP produces a drastic modification of the energy level shift. Although the quasi-static method is subject to greater errors compared to the FEM method, the errors |Δ_error| are small.

Although both FEM method and quasi-static method work well for emitter in close proximity to the metal interface, the quasi-static method produces relative large errors and drastic change for frequency around ω_sp. On closer inspection, we find that there is a slight blue shift to higher frequency [shown in the inset of Fig. 2(d)] for the normalized spontaneous emission rate Γ/Γ₀ by the quasi-static method, which leads to the above phenomena. It originates from the omission of retardation effect. With the increase of the emitter-interface distance, Γ/Γ₀ by the quasi-static approximation will deviate more from the exact one. This can be clearly seen in Figs. 2(e)–2(h), where we consider a longer distance, such as l = 5nm. For example, the inset of Fig. 2(e) shows that the results by the quasi-static method deviate much from the analytical solutions and Γ_{R_error} are larger than 10% [see Fig. 2(g)] for frequency away from ω_sp, comparing to the results shown in the inset of Fig. 2(a) and in Fig. 2(c). For frequency around ω_sp, the blue shift [see the inset in Fig. 2(h)] is about 1.55meV, which is larger compared to 0.3meV in the case of l = 2nm [see the inset in Fig. 2(d)]. For the energy level shift [shown in Fig. 2(f) and the inset], both methods work well. Besides, the FEM method also performs even better than the quasi-static method which can be clearly seen in Fig. 2(h). For the quasi-static method, the errors Δ_error are less than 0.02meV and the relative errors Δ_error/|Δ_analy| are less than 1.8% compared to 10% for Γ_{R_error} when frequency (below 2.5eV) is away from ω_sp.

From the above results, we can conclude that FEM method can be able to exactly calculate the spontaneous emission rate and the energy level shift for both short and long emitter-interface distances. However, the quasi-static method can be applied only for short emitter-interface distance. For longer emitter-interface distance, it also shows good performance for calculating the energy level shift but not for the spontaneous emission rate when the transition frequency is away from ω_sp. For frequency around ω_sp, both Γ(r ; ω) and Δ(r ; ω) show a minor blue shift to higher frequency (about 1.55meV for l = 5nm in this model).

Then, we turn to the case for an emitter above a silver sphere. The results are shown in Fig. 3. Similar to the case of silver-air interface, the results obtained by the FEM method are in good agreement with the analytical ones. However, for the quasi-static method, both the spontaneous emission enhancement and the energy level shift deviate from the analytical results for frequency around 2.76eV, which is near the resonance frequency of the dipole mode of the LSPP. These can be clearly seen in the insets of Fig. 3 and we find that the deviation increases with the emitter moving away from the sphere, for example, see the insets of Figs. 3(a) and 3(c) for the spontaneous emission rate and the insets in Figs. 3(b) and 3(d) for the energy level shift. Furthermore, for frequency away from the dipole mode, for example, around the higher order modes, excellent agreement is found between the quasi-static method and the analytical method.

Fig. 3 The modified spontaneous emission rate and the energy level shift for emitter around a silver nano-sphere. The red solid line, dashed blue line and the black circles are the results by the methods of analytical, quasi-static and FEM, respectively. (a) and (b) are for short emitter-interface distance l = 2nm. (a) Normalized spontaneous emission rate Γ/Γ₀ with Γ₀ and (b) energy level shift. The insets are for frequency around the dipole mode of the LSPP. (c) and (d) are for longer emitter-interface distance l = 5nm.

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The above phenomena can be understood as follows. In the quasi-static method, the zeros of the denominator for the scattered Green tensor $G_{static}^{r r} (r, r; ω)$ are the roots of $ε_{2} + \frac{n + 1}{n} ε_{1} = 0$ for n = 1, 2, 3, ..., which represent the resonance frequency of mode n. In the static case, the resonance frequency is independent of the radius of the sphere. However, if we take into account retardation, the resonance frequency will be dependent on the radius of sphere. This can be clearly seen in the denominator of R^V in Eq. (7). The real parts of the zeros of the denominator of R^V are the resonance frequencies of the LSPR. For a = 20nm, we find that the resonance frequency obtained by the quasi-static method deviates much from the analytical ones for the first few modes and are nearly the same for higher order modes. For example, the real parts of the first three roots by the quasi-static method are $ω_{1}^{quasi} = 2.7894 e V$ , $ω_{2}^{quasi} = 2.8809 e V$ , $ω_{3}^{quasi} = 2.9135 e V$ and the results by the analytical method are $ω_{1}^{analy} = 2.7562 e V$ , $ω_{2}^{analy} = 2.8749 e V$ , $ω_{3}^{analy} = 2.9111 e V$ . Their difference $δ ω_{n} = ω_{n}^{quasi} - ω_{n}^{analy}$ for n = 1, 2, 3 are 0.0332eV, 0.006eV, 0.0024eV. For n ⩾ 4, they are less than 0.0013eV and decreasing with n increasing. These findings are consistent with the results shown in Fig. 3(c), where the first peak is 2.791eV for the quasi-static method and is different from the analytical 2.756eV. Their difference is 0.035eV, which is approximately equal to δω₁. With the emitter moving away from the sphere, the contribution from higher order modes are decreasing more rapidly than those of lower order modes since the electromagnetic fields of higher order modes decrease more quickly. This can be clearly seen from Eq. (8) where the radial part of the vector function $h_{n}^{(1)} (x) \propto x^{- (n + 1)}$ for small x. Thus, the contribution originated from the dipole mode, which presents the largest δω_n, increases with the increasing of distance for frequency around 2.76eV. This leads to the more obvious differences between the quasi-static results and the analytical ones [see the insets in Figs. 3(a) and 3(c)].

At the end of this section, we apply the theory to the case of a nanosphere dimmer [38]. Schematic diagram is shown in Fig. 4. A perfect electric dipole emitter with parallel orientation is located at the center of silver nanosphere dimer of radius a = 20nm. The dimmer gap is L. All the other paremeters such as the permittivity of the silver and the dipole moment are the same as before. The permittivity of the homogeneous background is ε₁ = 1. For the quasi-static method, we use the MNPBEM tool box with the quasi-static solver [44]. For the FEM, mesh settings are the same as in the sphere model.

Fig. 4 Schematic diagram. A perfect electric dipole emitter with parallel orientation is located at the center of silver nanosphere dimer of radius a = 20nm and gap L.

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The modified spontaneous emission rate and the energy level shift are shown in Figure 5. Similar to the case of one sphere model, blue shift by the quasi-static method also exists for frequency around the resonance frequency of radiative mode. For frequency away, results from the quasi-static method agree well with those by FEM. Besides, we see that there is a red shift for the lowest mode of the dimmer (dipole mode) and the closer the two spheres, the larger the shift is [see Figs. 5(a) and 5(c)]. This agrees well with the theory of plasmon hybridization [53]. In addition, extremely large modifications of the spontaneous emission rate and energy level shift can be observed. For example, in the case of L = 2nm, Γ/Γ₀ ≈ 7 × 10⁵ can be achieved with frequency at the dipole mode of the dimmer. The energy shift can reach 0.46eV for ω = 2.36eV, which is comparable to the transition frequency. According to Eq. 1, Γ and Δ are propotional to the square of the dipole moment strength d². With a larger dipole moment, the system may come into the ultrastrong coupling regime. Besides, a bound state may be formed if the level shift is larger than the transition frequency which leads to a negative eigen energy of the system [43,54].

Fig. 5 The modified spontaneous emission rate and the energy level shift for emitter with parallel orientation located at the center of nanosphere dimmer. The dashed blue line and the black circles are the results by the methods of quasi-static and FEM, respectively. (a) and (b) are for longer dimmer gap L = 4nm. (a) Normalized spontaneous emission rate Γ/Γ₀ with Γ₀ and (b) energy level shift. (c) and (d) are the same as (a) and (b) except for a shorter dimmer gap L = 2nm.

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4. Conclusions

We have tested the applicability and accuracy of the quasi-static method and the finite-element method in the investigation of the modifications of the spontaneous emission rate and the energy level shift for a quantum emitter near a plasmonic nanostructure. To this end, a silver-air planar interface and a silver nano-sphere have been selected, where SPR or LSPR may take effect and the analytical results are presented as a reference. The modifications of the spontaneous emission rate and the energy level shift are expressed by the scattering photon Green tensor. For the FEM method, the scattering Green tensor are expressed by the difference of the electric fields of an oscillating electric point dipole by COMSOL Multiphysics software with PML boundary conditions. It has been proved to be able to exactly resolve the divergence of the real part for the Green tensor with equal space arguments. Although the FEM method needs a significant amount of memory and time, we have proved that FEM method is an accurate and general method. For frequency away from the radiative mode, we have found that the quasi-static method can be applied more effectively for calculating the energy level shift than the spontaneous emission rate. But for frequency around, we have observed a blue shift to higher frequency for both and this shift increases with the increasing of emitter-silver distance. Applying the theory to the nanosphere dimmer, we have seen similar phenomenon where more obvious shift by the quasi-static method are shown. Besides, we have found extremely large modifications of the spontaneous emission rate and energy level shift.

Appendix A: green function for planar stratified media

For a silver-air planar interface, the scattered part of the Green tensor can be written by [36]

\begin{matrix} G_{planar}^{(s)} (r, r^{'}, ω) = \frac{i}{4 π} \sum_{p = e, o} \sum_{n} \int_{0}^{\infty} \frac{(2 - δ_{n, 0})}{λ h_{1}} [R^{A} {\bar{M}}_{λ n p} (h_{1} r) {\bar{M}}_{λ n p} (h_{1} r^{'})] \\ + [R^{B} {\bar{N}}_{λ n p} (h_{1} r) {\bar{N}}_{λ n p} (h_{1} r^{'})] d λ, \end{matrix}

when both of the source point r and the field point r′ are inside the air. Here, R^A and R^B are the reflection coefficients and are given by

R^{A} = \frac{h_{1} - h_{2}}{h_{1} + h_{2}}, R^{B} = \frac{k_{2}^{2} h_{1} - k_{1}^{2} h_{2}}{k_{2}^{2} h_{1} + k_{1}^{2} h_{2}} .

The vector functions are defined by

\begin{array}{l} {\bar{M}}_{λ n e} (h_{1} r) = [- \frac{n J_{n} (λ r)}{r} \sin n ϕ \hat{r} - \frac{\partial J_{n} (λ r)}{\partial r} \cos n ϕ \hat{ϕ}] e^{i h_{1} z}, \\ {\bar{M}}_{λ n o} (h_{1} r) = [\frac{n J_{n} (λ r)}{r} \cos n ϕ \hat{r} - \frac{\partial J_{n} (λ r)}{\partial r} \sin n ϕ \hat{ϕ}] e^{i h_{1} z}, \\ {\bar{N}}_{λ n e} (h_{1} r) = \frac{1}{\sqrt{λ^{2} + h_{1}^{2}}} [i h_{1} \frac{\partial J_{n} (λ r)}{\partial r} \cos n ϕ \hat{r} - \frac{i h_{1} n}{r} J_{n} (λ r) \sin n ϕ \hat{ϕ} + λ^{2} J_{n} (λ r) \cos n ϕ \hat{z}] e^{i h_{1} z}, \\ {\bar{N}}_{λ n o} (h_{1} r) = \frac{1}{\sqrt{λ^{2} + h_{1}^{2}}} [i h_{1} \frac{\partial J_{n} (λ r)}{\partial r} \sin n ϕ \hat{r} + \frac{i h_{1} n}{r} J_{n} (λ r) \cos n ϕ \hat{ϕ} + λ^{2} J_{n} (λ r) \sin n ϕ \hat{z}] e^{i h_{1} z} . \end{array}

Here, J_n(λr) is the Bessel functions of the first kind,

k_{i} = ω \sqrt{ε_{i}} / c

, and

h_{i} = \sqrt{k_{i}^{2} - λ^{2}}

, with ε₁ and ε₂ corresponding to the permittivities of the background and metal respectively.

Appendix B: green function for sphere

Consider a sphere with permittivity ε₂ and radius a embedded in a homogeneous medium of permittivity ε₁. For both of the source point r and the field point r′ outside the sphere, the scattered part of the Green tensor can be written by [46]

\begin{array}{l} G_{sphere}^{(s)} (r, r^{'}, ω) & = \frac{i k_{1}^{3}}{4 π} \sum_{p = e, o} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} (2 - δ_{m, 0}) \frac{2 n + 1}{n (n + 1)} \frac{(n - m)!}{(n + m)!} \\ \times [R^{H} M_{n m p} (k_{1} r) M_{n m p} (k_{1} r^{'}) + R^{V} N_{n m p} (k_{1} r) N_{n m p} (k_{1} r^{'})] . \end{array}

Here, R^H and R^V are the centrifugal reflection coefficients, which are given by

R^{H} = \frac{k_{2} \partial τ_{2} τ_{1} - k_{1} \partial τ_{1} τ_{2}}{k_{2} \partial τ_{2} κ_{1} - k_{1} \partial κ_{1} τ_{2}}, R^{V} = \frac{k_{2} τ_{2} \partial τ_{1} - k_{1} τ_{1} \partial τ_{2}}{k_{2} τ_{2} \partial κ_{1} - k_{1} κ_{1} \partial τ_{2}},

and the vector functions are

\begin{array}{l} M_{nme} (k r) & = - \frac{m}{sin θ} h_{n}^{(1)} (k r) P_{n}^{m} (cos θ) sin m ϕ \hat{θ} - h_{n}^{(1)} (k r) \frac{d P_{n}^{m} (cos θ)}{d θ} cos m ϕ \hat{ϕ}, \\ M_{nmo} (k r) & = \frac{m}{sin θ} h_{n}^{(1)} (k r) P_{n}^{m} (cos θ) cos m ϕ \hat{θ} - h_{n}^{(1)} (k r) \frac{d P_{n}^{m} (cos θ)}{d θ} sin m ϕ \hat{ϕ}, \\ N_{nme} (k r) & = \frac{n (n + 1)}{k r} h_{n}^{(1)} (k r) P_{n}^{m} (cos θ) cos m ϕ \hat{r} + \frac{1}{k r} \frac{d (r h_{n}^{(1)} (k r))}{d r} \\ [\frac{d P_{n}^{m} (cos θ)}{d θ} cos m ϕ \hat{θ} - \frac{m}{sin θ} P_{n}^{m} (cos θ) sin m ϕ \hat{ϕ}], \\ N_{nmo} (k r) & = \frac{n (n + 1)}{k r} h_{n}^{(1)} (k r) P_{n}^{m} (cos θ) sin m ϕ \hat{r} + \frac{1}{k r} \frac{d (r h_{n}^{(1)} (k r))}{d r} \\ [\frac{d P_{n}^{m} (cos θ)}{d θ} sin m ϕ \hat{θ} - \frac{m}{sin θ} P_{n}^{m} (cos θ) cos m ϕ \hat{ϕ}], \end{array}

with τ_i = j_n(k_ia),

κ_{i} = h_{n}^{(1)} (k_{i} a)

,

\partial τ_{i} = \frac{1}{k_{i} a} \frac{\partial [k_{i} a j_{n} (k_{i} a)]}{\partial k_{i} a}

,

\partial κ_{i} = \frac{1}{k_{i} a} \frac{\partial [k_{i} a h_{n}^{(1)} (k_{i} a)]}{\partial k_{i} a}

. Here, j_n and

h_{n}^{(1)}

are the spherical Bessel functions and the spherical Hankel functions of the first kind,

P_{n}^{m}

is the associated Legendre function, and

k_{i} = ω \sqrt{ε_{i}} / c

.

Funding

National Natural Science Foundation of China (Grants No.11464014, 11347215, 11564013, 11464013); Hunan Provincial Innovation Foundation For Postgraduate (Grants No.CX2017B718).

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Quasi-static method and finite element method for obtaining the modifications of the spontaneous emission rate and energy level shift near a plasmonic nanostructure

Abstract

1. Introduction

2. Models and methods

2.1. Quasi-static method

2.2. Finite element method

3. Results and discussions

4. Conclusions

Appendix A: green function for planar stratified media

Appendix B: green function for sphere

Funding

References and links

Cited By

Figures (5)

Equations (10)

Optics Express