Metallo-dielectric hybrid antenna for high Purcell factor and radiation efficiency

Xianghao Zeng; Wenhai Yu; Peiju Yao; Zheng Xi; Yonghua Lu; Pei Wang

doi:10.1364/OE.22.014517

1. Introduction

Optical antennas, a technology that can enhance light-matter interactions, have attracted attention for recent years [1–4]. Different nanostructures designed as optical antennas hold promise for high directivity light emission [5–8], molecular fluorescence enhancement [9–11] and high efficiency radiation [12]. For metal nanoantennas such as metallic nanoparticles (MNPs), the enhancement of spontaneous emission, described by the Purcell factor [13, 14] can be quite high thanks to localized surface plasmons excited at metallic interfaces. However, for small distances from source to metal surface, the nonradiative decay rate is enhanced. As a result, the radiation efficiency is low. To avoid the problem, dielectric antennas without metal loss are proposed [15, 16]. But the side effect of low Purcell factor arises.

To overcome the dilemma, a hybrid nanoantenna, i.e., a combination of metallic and dielectric nanoparticles, is introduced in the present paper. The antennas composed of a silver dimer and two dielectric spheres are investigated by a method based on the dyadic Green’s functions. The radiation enhancement factor is introduced to assess this system with spontaneous enhancement and radiation efficiency treated equivalently. The dependence of the Purcell factor and the radiation efficiency of this system on different parameters, such as the gap of the silver dimers, the radius of silver and dielectric spheres and the distance from source to sphere is investigated. For single peak, Purcell factor above 2000 and radiation efficiency above 50% are achieved. For broadband operation, Purcell factor above 900 and radiation efficiency above 55% are achieved.

2. Method

Our investigations are based on the rigorous dyadic Green’s function G, which is defined by:

\nabla \times \nabla \times G (r - r^{'}) - \frac{ω^{2}}{c^{2}} ε (r) G (r - r^{'}) = I δ (r - r^{'})

where I is the unit dyad. Together with the boundary conditions of G → 0 when |r − r′| → ∞, the Green’s function of a multisphere system can be expanded in spherical coordinates as [17]:

\begin{array}{l} G^{(0 s)} (r, r^{'}) = - δ_{s 0} \frac{1}{k_{s}^{2}} \hat{r} \hat{r} δ (r - r^{'}) + j \frac{k_{s}}{4 π} \sum_{v m n} c_{m n} [δ_{s 0} F_{v, m n}^{(\tilde{1})} (k_{0} r) F_{v, - m n}^{(\tilde{3})} (k_{0} r^{'}) \\ + \sum_{i} F_{v, - m n}^{(3)} (k_{0} r_{i}) A_{v, m n}^{(i)} (r^{'})] \end{array}

\begin{array}{l} G^{(is)} (r_{i}, r_{i}^{'}) = - δ_{si} \frac{1}{k_{s}^{2}} {\hat{r}}_{i} {\hat{r}}_{i} δ (r_{i} - r_{i}^{'}) + j \frac{k_{s}}{4 π} \sum_{v m n} c_{m n} [δ_{si} F_{v, m n}^{(\tilde{1})} (k_{i} r_{i}) F_{v, - m n}^{(\tilde{3})} (k_{i} r_{i}^{'}) \\ + F_{v, - m n}^{(1)} (k_{i} r_{i}) C_{v, m n}^{(i)} (r^{'})] \end{array}

where G^(0s) and G^(is) are dyadic Green’s functions of the background region 0 and the interior of sphere i, excited from source region s. The coordinate’s origin of field point r and source r′ is arbitrary, while those with subscript i is with respect to spherical center O_i.

F_{v, m n}^{(i)}

is the spherical harmonics of

M_{m n}^{(i)}

or

N_{m n}^{(i)}

, which are the solutions of the wave function ∇ × ∇ × F − k²F = 0 in homogenous space. The coefficient c_mn = (−1)^m(2n + 1)/n(n + 1) = c_−mn. The detailed derivation of the Green’s function is in ref. [17] and [18]. In the numerical calculation, the maximum order of m and n is taken as 25. The maximum error is 1.1% which is obtained by increasing the maximum order of m and n.

The Purcell factor F_p is a key parameter evaluating spontaneous emission defined as the ratio of the spontaneous decay rate F = γ/γ₀ [19, 20]. With the dyadic Green’s function obtained, the partial local density of states (LDOS) ρ and the spontaneous decay rate γ can be derived as follows [21]:

γ = \frac{2 ω}{3 \bar{h} ε_{0}} {| μ |}^{2} ρ (r_{0}, ω)

ρ (r_{0}, ω) = \frac{6 ω}{π c^{2}} [n_{μ} . Im {G (r_{0}, r_{0}; ω)} \cdot n_{μ}]

where r₀ is the position vector of the emitter, μ is the dipole moment, and n_μ is its unit vector. The Purcell factor can thus be expressed in terms of the Green’s function as:

F_{p} (r_{0}, ω) = \frac{n_{μ} \cdot Im {G (r_{0}, r_{0}; ω)} \cdot n_{μ}}{n_{μ} \cdot Im {G (r_{0}, r_{0}; ω)} \cdot n_{μ}}

where G₀(ω) is the Green’s function in homogeneous space and n_μ · Im{G₀(r₀, r₀; ω)} · n_μ = ω₀/(6πc).

The total power P_t dissipated by the dipole can be calculated through the LDOS [22]

P_{t} = \frac{π ω^{2}}{12 ε_{0}} {| μ |}^{2} ρ (r_{0}, ω)

The radiative power P_r can be calculated by integrating the Poynting vector over a spherical surface. The radius of the sphere is taken as 15μm in this letter. Comparing the radiative power with the total power, we have the radiation efficiency η = P_r/P_t.

After calculating the Purcell factor and radiation efficiency, the radiation enhancement factor F_r is introduced in order to reach a balanced assessment of both the spontaneous enhancement and radiation efficiency. F_r is defined as the product of the Purcell factor and the radiation efficiency F_r = F_p × η, which can also be expressed as F_r = P_t/P₀ × η = P_r/P₀. Here, P₀ is the power dissipated in vacuum space and the ratio of P_t/P₀ indicates the Purcell factor.

3. Silver dimer

Metallic dimers have high Purcell factor and radiation efficiency [23, 24]. Here, dimers formed by twin silver spheres with a dipole placed in the middle of the gap, as shown in the inset of Fig. 1(a), are revisited in the context of radiation enhancement factor. The dielectric constant of silver is a fitted polynomial to the data in ref. [25]. Firstly, dimers with gap length of 10 nm are calculated for the dependence of Purcell factor, radiation efficiency and radiation enhancement on the sphere radius and the results are plotted respectively in Figs. 1(a)–1(c). The peak at 340 nm wavelength in Fig. 1(a), which barely changes with different radii, is the local mode representing the intrinsic property of silver. From Fig. 1(b), it is clearly shown that the radiation efficiencies at the local mode are less than 1%. So this mode is unsuitable for radiation enhancement. The other peaks in Fig. 1(a) represent the radiation modes. The radiation efficiencies of these radiation modes is increasing as the radius increases [Fig. 1(b)], but the Purcell factor is decreasing simultaneously [Fig. 1(a)]. The blue dashed line in Fig. 1(c) shows a maximum value of the radiation enhancement factor of 703. The corresponding Purcell factor and radiation efficiency are 1260 and 55.8%, respectively, shown by the blue lines in Figs. 1(a) and 1(b). In this case, the radius of the silver sphere is 35 nm.

Fig. 1 (a)–(c) Purcell factors, radiation efficiencies and radiation enhancement factors for different radii of silver dimers with fixed gap of 10 nm; (d)–(f) the same parameters for different gap lengthes of silver dimers of 35 nm radius.

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Then, the Purcell factor, radiation efficiency and radiation enhancement factor with different gap lengthes at the radius of the spheres fixed as 35 nm are plotted in Figs. 1(d)–1(f), respectively. Although pulling the silver spheres away from the dipole makes the radiation efficiency increase as shown in Fig. 1(e), the Purcell factor drops dramatically [Fig. 1(d)]. As a result, the radiation enhancement factor also drops [Fig. 1(f)]. So, the silver dimer of 35 nm radius spheres with 10 nm gap is chosen for further investigation in the next step.

4. Hybrid 4-sphere antenna

Now, the hybrid antenna shown in the inset of Fig. 2(a) is introduced by tailoring the silver dimer with two dielectric spheres. The refractive index of the dielectric spheres is 3.4. The radius of the dielectric spheres is increased from 60 nm to 80 nm, while the distance from the sphere surface to the dipole is fixed at 160 nm. The corresponding Purcell factor, radiation efficiency and radiation enhancement factor are plotted in Figs. 2(a)–2(c). As the radius of the dielectric sphere increases, it can be seen in Fig. 2(a) that the peak value of Purcell factor of the radiation mode is decreasing and a redshift occurs at the same time. The radiation efficiencies do not increase significantly as the radius increases [Fig. 2(b)], but are generally larger than the ones of the silver dimers [Fig. 1(b)]. From Fig. 2(c), it is shown that the radiation enhancement factor reaches a maximum value when the radius of the dielectric spheres is 65 nm. And when the radius is larger than 70 nm, the radiation enhancement factor keeps a high value for a wide range of bandwidth. For example, the antenna with dielectric spheres of 75 nm radius, represented by the blue dashed line, has a radiation enhancement factor above 500 for nearly 100 nm bandwidth (from 400 nm to 500 nm). And as shown by the blue lines in Figs. 2(d) and 2(e), the Purcell factor is above 900 and radiation efficiency is above 57% in the broadband regime.

Fig. 2 (a)–(c) The parameters as in Fig. 1 for different radii of dielectric spheres with 160 nm distance; (d)–(f) the same parameters for different distances of dielectric spheres with 75 nm radius.

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Next, the dependence of the Purcell factor, radiation efficiency and radiation enhancement factor on the distance from the the dielectric surface to the dipole is investigated with dielectric spheres radius fixed as 75 nm. The results are plotted respectively in Figs. 2(d)–2(f). When the distance is 50 nm, a radiation peak with Purcell factor of 3141 appears in Fig. 2(d) shown by the red line. But this antenna has quite low radiation efficiency, as shown by the red line in Fig. 2(e). As the distance increases to 150 nm, the Purcell factor of the radiation peak decreases [Fig. 2(d)] while the radiation efficiency increases [Fig. 2(e)]. As a result, the antenna with 100 nm distance has a maximum radiation enhancement factor above 1000 as the yellow dashed line shows in Fig. 2(f). When the distance is further increased to 250 nm, new peaks of radiation enhancement factor emerges as the blue and black lines show in Fig. 2(f). To optimize the maximum radiation enhancement factor, the dependence of the radiation enhancement factor on the distances around 100 nm is plotted in Fig. 3. It is clear in Fig. 3(a) that the highest value is achieved when the distance is 90 nm. The Purcell factor, radiation efficiency and radiation enhancement factor of this antenna are drawn in Fig. 3(b). The peak value of radiation enhancement factor is 1086 with corresponding Purcell factor of 2114 and radiation efficiency of 51.36%.

Fig. 3 (a) Peak radiation enhancement factors of antennas with different distances from the dielectric spheres to the dipole; (b) Purcell factor (blue solid line), radiation efficiency (green solid line) and radiation enhancement factor (blue dashed line) of antenna with 90 nm distance.

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For comparison, a dipolar emitter coupled to two dielectric sphere are also investigated. The structure is shown as the inset of Fig. 4(a). Because of the absence of loss, dielectric antennas have radiation efficiency of 100%. So Purcell factors of different sphere radii and distances drawn in Fig. 4 are our only concern. Obviously, the Purcell factors are much lower than those of the metallic antennas. In Fig. 4(a), the distance from the sphere surface to the dipole is 160 nm. When the sphere radius increases from 60 nm to 80 nm, the peak is red-shifted. The sphere radius is fixed at 75 nm in Fig. 4(b), while the distances are different. A sharp peak appears when the distance is 50 nm. As the distance increases, the peak value reduces and the peak becomes wider with redshift at the same time.

Fig. 4 (a) The Purcell factors for different radii of dielectric spheres with fixed distance of 100 nm; (b) The Purcell factors for different distances of dielectric spheres of 70 nm radius.

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To deeply understand the properties of broadband and high radiation enhancement factor tailored by the radiation modes, we choose different radiation peaks at the wavelength of 434 nm, 472 nm, 426 nm and 454 nm, marked by a, b, c and d in Figs. 2(c) and 2(f) respectively, and then plot their distributions of electric field amplitude in Fig. 5. Figures 5(a) and 5(c) have similar distributions of field amplitude, and it is more clearly seen in Fig. 5(c) that the electric field amplitude reaches a maximum of positive value (bright bar) where the dielectric spheres are located. Such field amplitude distribution enhances the radiation of the antenna. On the other hand, the field amplitude distribution of Fig. 5(b) resembles the one of Fig. 5(d). And in these situations, the field at the dielectric spheres has negative value (dark bar), which means a difference of phase with the situations of Figs. 5(a) and 5(c). But the absolute value of the field amplitude still reaches a maximum at the dielectric spheres, which also leads to radiation enhancement. As the peaks c and d show in Fig. 2(f), the radiation of the antenna is dominated mainly by one mode when the distance is 200 nm and 100 nm. But when the distance is 160 nm, the blue dashed line in Fig. 2(c) makes a platform with the peaks a and b because the two modes are enhanced at the same time.

Fig. 5 Spatial distributions of electric field amplitude at radiation peaks of antennas with different distances from the dielectric sphere surface to the dipole. (a) and (b) antenna with 160 nm distance; (c) antenna with 200 nm distance; (d) antenna with 100 nm distance.

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5. Conclusion

In summary, we have studied hybrid sphere antennas by use of existing formulation for the dyadic Green’s function. In order to characterize enhancement and radiation efficiency equally, the radiation enhancement factor is introduced. Due to the enhancement of resonant modes between dielectric spheres, radiation enhancement factor over 1000 is achieved with the hybrid antennas at single mode radiation. High efficiency radiation enhancement appears over a broad spectral range when two modes take effect together, which makes the antenna well suited for various exciting sources.

Acknowledgments

This work is supported by National Key Basic Research Program of China ( 2012CB921900, 2012CB922003), Key Program of National Natural Science Foundation of China ( 61036005) and National Natural Science Foundation of China ( 61177053, 61377053, 11274293)

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Metallo-dielectric hybrid antenna for high Purcell factor and radiation efficiency

Abstract

1. Introduction

2. Method

3. Silver dimer

4. Hybrid 4-sphere antenna

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (7)

Optics Express