Numerical investigations based on the boundary element method (BEM) have been carried out to two-dimensional (2-D) silver dimer nano-antennas of various geometries. The near-field and far-field properties are mainly determined by the local geometry at the gap and the global shape of the antenna shafts respectively. A hybrid dimer antenna, which mixes the geometry ingredients of the rod dimer and the bowtie, benefits in both near and far field. Using a microcavity representation, the resonance in dimer nano-antennas is explained in a common and semi-analytical manner. The plasmonic enhancement and the wavelength mismatching in the optical dimer antenna are naturally embodied in this model. The quality factor of the resonance, which can be influenced by the wavelength and the geometry, is discussed intuitively. The understanding presented in this work could guide the future engineering of the optical dimer antenna.
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Exploiting localized surface-plasmon resonances (LSPR) in metallic nano-particles has opened the possibility to amplify the effects of a wide range of light-material interactions under weak incident illumination [1–7]. The typical examples include the single-molecule Raman scattering , the single-molecule fluorescence [2,3], the emission of quantum dots , the white light continuum generation , and the two-photon luminescence [6,7]. The localized and enhanced electric field around the metallic nano-particles originates from the resonant excitation of the collective conduction electrons which results in an accumulation of charges around the so-called hot spots. Such metallic nanostructures are termed as optical nano-antennas  in the sense that the coupling between the surface current and the incident/re-emitted light is involved.
The near field around the metallic nano-particles can be enhanced even further via the mutual coupling between closely-spaced particles [1,9]. Dimer is one example of such ensemble structures. Two canonical dimer structures, which have given rise to many studies over the past five years, are the rod dimer [5,7] and the bowties [4,6,9]. In this paper, we introduce a hybrid geometry that consists of a mixture of the rod dimer and the bowtie. We discuss the influence of the geometry of the dimer nano-antenna to the optical properties.
To describe the optical response of a dimer antenna, a conventional method is to solve the polarization of the single particle , and then to evaluate the mode splitting induced by the coupling between these two particles . However, except the numerical method, there is not universal method giving out the polarization for arbitrary nano-particle. Furthermore, when two particles are very close, representing them as single polarization (either dipolar or higher-order polar) is not accurate any more. An alternative approach is to analogize the nano-antenna as a nano-circuit. The concept of nano-impedances in the optical domain has been proposed [12,13]. But, in contrast to the RF and microwave ones, the optical antennas are usually excited as a whole; applying the nano-impedance concept is not convenient. Till now, the nanometer-scaled transmission line, which can locally excite optical nano-antennas, is still a challenge.
In this paper, we adopt the ‘microcavity’ representation to understand the dimer nano-antenna. The mode excited inside the microcavity is assumed to be determined by the local cross-section vertical to the antenna axis. To take into account the geometrical modification of the nano-antenna, the phase variation along the antenna axis is summed up based on the local mode. Thus, the microcavity picture allows us to discuss the geometry tunability of the dimer antennas with different geometries in a common way. Previously, such discussions can only be implemented for a specific geometry (either the rod dimer  or the bowtie ). Moreover, in this representation, the uniqueness of the optical nano-antenna, i.e. the plasmonic enhancement and the mismatching between the effective wavelength and the radiative wavelength , can be conveniently illustrated.
The paper is structured as follows: Section 2 describes the structures of the dimer antennas and explains the microcavity representation. Section 3 presents the near-field and the far-field optical properties of three types of dimer antennas. With respect to the dipolar resonance, the numerically simulated results of various dimer antennas are compared with each other. The analytical calculations based on the microcavity representation exhibit an agreement with the simulation. Section 4 discusses the quality factor of the dimer nano-antennas. By using the microcavity picture, an insight to the influence of the geometry modification to the Q factor is presented. To the best of our knowledge, it is the first time that a common explanation is proposed to the tunability of the optical dimer antenna of various geometries.
2. Geometries and microcavity representation of dimer antennas
Our work is mainly related to two dimensional (2-D) dimer antennas, whose simple configuration enables an easy analytical treatment. However, all the understanding obtained here can be verified in the 3-D. The analytical modeling to 3-D dimer nano-antennas is just more tedious.
The three types of 2-D dimer antennas considered in this work, i.e. the rod dimer, the bowtie, and the hybrid dimer, are depicted in Fig. 1(a) . The hybrid dimer possesses one bowtie-like gap and two rod-like shafts. In Fig. 1(a), all dimers have a gap of g = 20nm, all the bowtie-like apexes have an angle of α = 60°, and all the corners are rounded off with a radius of curvature of 5nm. The environment is the vacuum, and the antennas are made of silver. Compared with gold, silver is preferred due to its weaker dissipation in the visible and near infrared.
The dimer antennas are illuminated by a plane wave linearly polarized along the antenna axis (x-axis). Under the normal illumination (k-vector is along the z-axis), a longitudinal charge/current oscillation can be excited as Fig. 1(b) illustrates. Whatever the antenna has a gap, the oscillating current corresponding to the dipolar resonance is co-directional in the whole antenna body. The nodes of the current only appear at the extremities. The introduction of the gap forces the current to suddenly vanish in the middle and leads to a high charge accumulation at the gap. Limited by the symmetry, under the normal illumination, the next higher-order resonance which can be excited is the octupolar one as Fig. 1(b) shows.
The plasmonic nature of the noble metal allows the excitation and the propagation of a transversely confined wave in an insulator-metal-insulator guide. This wave is called as Surface Plasmon Polariton (SPP), whose field is confined and enhanced at the metal-dielectric interface . In our simplified model, the transverse field profile of the SPP is locally determined and is an eigenmode of the waveguide with the local cross section. When the local SPP approaches the extremities of the guide, it is reflected to the reverse direction. Note that, the SPP specified here is the short-range SPP. The long-range SPP is hardly reflected at the extremities , especially when the metal layer becomes very thin.
In a dimer antenna, the SPP wave cannot directly propagate across the gap. It ‘jumps’ to the other segment with the assistance of the charge piled-up at the gap. This mechanism can be named as the ‘capacitive coupling’. In contrast, a dimer having a bridge in the gap supports a continuous propagation of the SPP , which should be regarded as the ‘inductive coupling’. In some bridged dimer antenna, the capacitive and inductive couplings may coexist and can lead to two resonant branches. The one associated with the inductive coupling will red-shift to the infinite if the bridge becomes very thin . Altogether, the SPP wave globally propagates along the axis of the dimer antenna. Its lateral field confinement and the longitudinal reflection at the extremities can be visualized as a microcavity.
The interest of above approximation resides in the fact that some local SPP modes can be analytically calculated. In particular, for a free-standing silver slab, the short-range 2-D SPP corresponds to the asymmetric transverse magnetic (TM) mode, whose propagation constant, βx, obeys the eigen-function:21]Fig. 1(b), the SPP mode propagates along the x axis, and A stands for the normalization coefficient.
In the optical domain, the modal index of the SPP, Re(βx)/k 0, varies with the wavelength and shows a peak corresponding to the plasmonic resonance. At this wavelength, the field in the vicinity of the silver-air interface is highly confined and enhanced, and the effective wavelength of the SPP, λ 0·k 0/Re(βx), is highly mismatched with the radiative wavelength, i.e. the free-space wavelength λ 0. On the other hand, as the width of the silver slab decreases, both the propagation constant, βx, and the near-field amplitude, |Ez(z = ±t/2)|, of the SPP mode increase monotonously [22,23], which is called as superfocusing effect. The wavelength and dimension dependences of the SPP are very important for understanding the optical properties of the dimer nano-antennas and will be addressed later.
According to the electric field expressions of the SPP mode presented by Eqs. (2)a and 2b, the charge and current densities on the silver-air interface can be given out as and . Here, Dz and Ex are the normal electric displacement and the tangential electric field of the SPP mode respectively. The contributions from the valence and polarization charges are both taken into account in the evaluation of the quantities, q and I .
One more issue to be addressed before starting the next section is that, in our simplified model, we only consider the propagation of the local SPP. The important coupling from the propagating SPP to the radiative wave in a tapered waveguide is ignored. To estimate it, the adiabaticity criterion should be discussed [23, 25].
3. Optical properties of 2-D dimer antennas with different geometries
The boundary element method (BEM)  is used for the numerical simulation of 2-D dimer nano-antennas. BEM has the advantages of consuming less computational resource, not needing artificial boundary for the computation domain, and being convenient to solve both the near-field and far-field quantities. The BEM equations adopted in our work are from the references  and . The experimentally measured dielectric function of the vacuum-evaporated silver film  is used. The variations of the permittivity of the silver due to the scattering of the free-electron on the interface  and the non-local effects  are both ignored, considering the antenna size under study. The numerical error of the simulation is controlled to be less than 2% by adjusting the mesh size.
Figures 2(a) , 2(b) and 2(c) respectively plot the intensity near-field enhancement in the gap, the normalized scattering cross section (SCS) and the normalized absorption cross section (ACS) of a silver rod dimer antenna as a function of the wavelength, which ranges from 300 to 900nm, and the antenna length, which ranges from 60 to 420nm. The rod width is 40nm. The intensity near-field enhancement is defined as |E 1/E in|2, where E 1 is the electric field in the middle of the gap and E in is the incident field. The normalization to the SCS and the ACS is implemented by dividing the antenna length which represents the physical cross section of the nano-antenna in the 2-D. In Figs. 2(a) and 2(b), the dipolar resonance, which red-shifts with the increase of the antenna length, is manifest. As the antenna length increases to 330nm, the octupolar resonance emerges at a short wavelength as Fig. 2(b) illustrates. On the other hand, for the absorption cross section, significant metal dissipation only appears at the wavelength shorter than 380nm (Fig. 2(c)), where the interband transition gives rise to a large imaginary part of the dielectric constant of the silver. In the long wavelength region (λ0 > 380nm), the scattering overwhelms the dissipation.
Additionally, although Fig. 2 just gives out the results of the rod dimer antenna, the demonstrated characteristics, i.e. the red-shift of the resonant wavelength with the antenna length and the domination of the scattering in the wavelength region greater than 380nm, are also observed in the other two types of silver antennas, the bowtie and the hybrid dimer.
3.1 Rod dimers
Based on the simulated results, Fig. 3(a) plots the resonant wavelengths of the rod dimer as a function of the antenna length. The resonant wavelengths are extracted from the spectra associated with the near-field enhancement, λEnh, and the SCS, λSCS, respectively. It is found that λEnh is longer than λSCS. Actually, the red shift of the near-field associated resonant wavelengths (λEnh) compared with the far-field associated resonant wavelengths (λSCS) has already been reported not only for the dimer antennas  but also for the single metallic nanoparticle . In order to understand this, we plot in Fig. 3(a) the peak wavelength of the current, λCurrent, and the peak wavelength of the charge, λCharge, deduced respectively from the simulated surface charge and current densities.
In any nano-particle, the charge and current densities obey the conservation theorem, . At a monochromatic frequency, this is equivalent to j0 ~ ωρ 0/d ∝ ρ 0/(λ 0 d), where ρ 0 and j 0 are the maximum charge and current densities in the particle respectively, and d is a length parameter determined by the gradient of the current. Since we consider the situation close to the dipolar resonance, the opposite charges are distributed around the extremities of the particle; the parameter d is approximate to the particle size and is nearly constant. According to the charge conservation, at λCharge, ρ 0 reaches its peak value; while the peak value of j0 appears at a shorter wavelength, λCurrent.
The relation of λEnh ≈ λCharge > λCurrent ≈ λSCS is clearly manifest in Fig. 3(a). The near-field and far-field properties of the nano-antenna exhibit different resonant conditions. To a 2-D dimer antenna, the broad resonant peak of the near-field enhancement (Fig. 2(a)), which renders the charge accumulation at the gap, is modified and blue-shifted by the factor of 1/λ0 in the charge conservation formula [j0 ∝ ρ 0/(λ 0 d)] to address the current flow in the antenna body, which is finally embodied in the spectra of the far-field radiation (Fig. 2(b)). In Fig. 3(a), when the length of the rod dimer antenna is equal to 300nm, the λEnh and the λSCS are respectively 778nm and 557nm (~71.5% of the former). Notice that this big deviation is due to the broad spectrum of the 2-D antenna. For a 3-D nano-antenna, the deviation is much smaller like the refs . and  present.
Using the microcavity representation, the resonance in the rod dimer antenna can be viewed in an analytical way. As the resonance is built up, the counter-propagating SPP wave forms a standing wave. The corresponding surface charge and current densities can be analytically expressed as,Eqs. (3a) and (3b) are valid in the region of B < |x| < L/2-Δ. The definitions of Δ and B are illustrated in Fig. 3(b). Owing to the boundary continuity of the electromagnetic fields, the illumination to the ends of the silver rod generates both the propagating wave (SPP) and evanescent wave along the waveguide (x-axis). The evanescent waves are confined near the extremities and the gap (the ‘singular region’); whereas, the SPP wave propagates out of the ‘singular regions’ and dominates in the so-called ‘shaft region’. Δ and B define the boundaries of the ‘shaft region’ as Fig. 3(b) illustrates.
Comparing the simulated surface charge/current distributions, and , with the Eq. (3a) in the middle part of the rod shaft yields the fitted quantities q 0 and I 0. With these, the simulated phase variation is given out as . Near the boundaries of the so-called ‘shaft region’, the calculated phase variation, ϕ(x) from Eq. (3b), and the simulated phase variation, ϕ'(x), diverge. According to this divergence, the parameters B and Δ can be evaluated.
Figure 3(c) plots the curves of ϕ(x) and ϕ'(x). Also shown is the normalization factor, , which is supposed to be equal to the unity within the ‘shaft region’. In Fig. 3(c), the agreement between the calculated and the simulated phase variations together with the coincidence of the simulated normalization factor Nor(x) with the unity are simultaneously obtained in the rod shaft region, which corroborates the microcavity picture. Moreover, it is interesting to mention that, albeit the gap separation of the rod dimer influences the boundary parameter B in Eqs. (3), it has little influence to the parameter Δ. Detailed discussions to this character will be presented elsewhere.
3.2 Bowtie dimers
With respect to the bowtie antenna, the mismatch between the radiative wavelength and the effective wavelength of the local SPP gradually decreases, as the SPP approaches the extremities. Within the angular spectrum representation , the reduction of the wavelength mismatch means that more energy of the oscillating current in the nano-antenna can be emitted to the far field. This increased radiative damping lowers the excitation efficiency of the current along the antenna and therefore decreases the intensity of the far-field radiation. The normalized SCS of the bowtie antenna is presented in Fig. 5(b) and is found to be worse than those of the rod and the hybrid dimer antennas.
However, on the other hand, the sharp gap apexes of the bowtie intensify the charge concentration and increase the near-field enhancement. Figure 4(a) schematically illustrates how the superfocusing of the SPP gives rise to a localized and enhanced field at the gap. A simulation to the electric field around a bowtie antenna is shown in the inset of Fig. 4(a), where a plane wave is illuminated from the left.
To elucidate the effect of the bowtie structure in the near-field, Fig. 4(b) plots the intensity near-field enhancement of various dimer antennas at their respective resonant wavelength. The curve of the bowtie, which demonstrates a bump at the edge of the metal dissipation region (λ0 ~ 380nm), is much higher than that of the rod dimer antenna, which increases monotonically with the antenna length. The bump-like feature in Fig. 4(b) seems strongly related to the sharp gap apex, since it reappears in the hybrid dimer structure (see the next section).
To describe a resonant bowtie as a microcavity, we have to calculate the propagation constant (Eq. (1) and the eigen-mode (Eqs. (2) of the local SPP in all the cross-sections. The Eqs. (3) of last section have to be modified as:Eq. (2) and are the normal and the tangential components of the electric field at the slab interface respectively. t(x) is the slab width.
As discussed before, the phase variation can be analytically integrated according to Eqs. (4) or be derived according to the simulated surface charge and current distributions and , . Figure 4(c) plots the two phase variations and the normalization factor . The results show that the microcavity representation is still applicable in the ‘shaft region’ of the bowtie antenna where the two ϕ(x)’s agree with each other and the normalization factor is equal to the unity. Figure 4(c) also shows that the divergence from the microcavity picture becomes serious on the boundary of the shaft region close to the gap (|x| ~B). In the bowtie antenna, the coupling from the propagating SPP to the radiation, which has been ignored in our treatment, becomes important.
3.3 Hybrid dimers
Following above logic, a hybrid dimer, which mixes the geometry ingredients of the rod dimer and the bowtie, is proposed (Fig. 1(a)). It possesses advantages from both sides: the thin antenna shafts lead to the big wavelength mismatch and the high excitation efficiency of the current; while the sharp gap apexes benefit the charge concentration and the near-field localization and enhancement. The hybrid dimer antenna exhibits a compromise between the bowtie and the rod dimer in both the near-field and far-field.
Figure 5a plots the simulated electric fields around a rod and a hybrid dimer antenna. The near-field localization in the hybrid dimer is much better than that in the rod dimer and is alike to that in the bowtie (compare with the inset in Fig. 4(a)). The resonant intensity of the near-field enhancement of the hybrid dimer antenna, which is much stronger than that of the rod dimer, has already been presented in Fig. 4(b), where the two curves corresponding to the bowtie and the hybrid dimer antennas both exhibit a hoist at short wavelength.
With respect to the far field property, Fig. 5(b) plots the normalized SCS of the three types of dimer antennas at their resonant wavelengths. The intensities of the far-field radiation of the rod and the hybrid dimers are stronger than that of the bowtie.
Furthermore, as the carrier of the oscillating current, the antenna shafts play the main role in terms of not only the total radiated power but also the directional radiation property. Figure 5(c) plots the normalized radiation patterns of the three types of dimer antennas. The curves corresponding to the rod and the hybrid dimer antennas are completely overlapped. While, the asymmetric radiation pattern of the bowtie antenna is because of the non-negligible phase retardation between the front and the back surfaces which causes unequal current distributions.
4. Q factor
In the radio frequency and microwave, antenna acts as a passive transducer transferring energy from a localized electric circuit to a propagating wave and vice versa; whereas, in the optical domain, antenna can store a substantial amount of energy in the vicinity. To understand the resonant character of the optical antenna, Q factor is a useful quantity. Its behavior is worth to be emphasized.
The Q factor can be defined, from the time-domain viewpoint, as the ratio between the stored energy inside the cavity and the dissipated energy per cycle, or, from the frequency-domain viewpoint, as the ratio of the resonant frequency and the full width at half maximum (FWHM) of the spectrum . Generally, the overall Q factor should take into account the contributions from the material dissipation and the radiative dissipation, 1/Q = 1/Qnr + 1/Qr, where Qnr and Qr render the energy loss due to the material dissipation and the radiation respectively. However, our study concentrates in the wavelength region out of the interband transition of the silver. We just discuss the influence from the radiative dissipation in the context of the mismatch between the effective wavelength and the radiative wavelength.
Figure 6(a) gives out the Q factors of the three types of 2-D dimer antennas as a function of their resonant wavelength. The Q factors are obtained from the SCS spectra. The lengths of the antennas are chose to be 100, 150, 200, 250, and 300nm, respectively. Firstly, as the resonant wavelength increases, the Q factor decreases. This is because the wavelength mismatch effect fades away at the long wavelength. In the infrared region, the dielectric constant of the silver approaches -∞, the optical antenna gradually loses its capability of storing energy and is eventually converted to a non-plasmonic antenna like a RF one. Secondly, in Fig. 6(a), the Q factors of the rod dimer are greater than those of the bowtie. Compared with the latter, the effective wavelength of the local SPP in the rod dimer antenna is more mismatched with the radiative wavelength, which inhibits the radiative dissipation and increases the Q factor. The high-Q characteristic of the rod dimer antenna is inherited by the hybrid dimer antenna.
Except above two characteristics, when we compare one rod dimer and one hybrid dimer having the same antenna length, it is found that the resonant wavelength of the hybrid dimer antenna is slightly bluer, and its Q factor is a little bit higher (see the red arrows from the solid squares to the hole squares in Fig. 6a). In Fig. 7 , we schematically plot the charge distribution in these two dimer antennas. The charge piled-up around the gap apexes of the hybrid dimer is distributed more apart compared with the rod dimer. Within the microcavity representation, this stands for a larger gap-induced phase shift (ϕB in Eq. (3b)) and leads to a shorter resonant wavelength for the hybrid dimer antenna. Consequently, the Q factor of the hybrid dimer antenna is slightly bigger than that of the rod dimer antenna owing to the wavelength mismatch effect at short wavelength.
Extending above discussion to the infrared, we simulate the Q factors of a 3-D rod and a 3-D hybrid dimer antenna by using finite-difference time-domain method . The lengths of the antennas are 150, 200, 250, 300, and 350nm respectively. (Note that, for a 2-D antenna, as the resonant wavelength enters the infrared region, the Q factor becomes very small and incomparable.) In Fig. 6(b), the aforementioned two tendencies, i.e. the shorter resonant wavelength and the bigger Q factor for the hybrid dimer antenna compared with the rod dimer antenna, remain in the 3-D, no matter that the resonance occurs in the visible or in the near infrared (~900nm).
We further modify the hybrid dimer structure by decreasing the gap separation from 20nm to 12nm (the star symbols in Figs. 6). As we expect according to the microcavity picture, such geometry modification causes the resonant wavelength slightly red-shift and the Q factor slightly decrease. These trends are both observed in the 2-D and in the 3-D (see the blue arrows from the hole squares to the hole stars in Figs. 6(a) and 6(b) respectively).
The Q factor can also be evaluated from the near-field enhancement spectra. However, in such case, the above discussion based on the microcavity picture, which stresses the global propagation of the SPP, is not applicable due to the strong influence from the local geometry of the dimer gap. Figure 6(c) plot the near-field associated Q factors of the 3-D rod and the 3-D hybrid dimer antennas. Although the blue shift of the resonant wavelength is still maintained, the Q factor of the hybrid dimer antenna can be higher or lower than that of the rod dimer antenna at different resonant wavelength (see the red and blue arrows in Fig. 6(c)). The discussion to the geometry tunability of the near-field associated Q factor is out of the range of this paper.
Numerical studies have been implemented to compare the near-field and the far-field optical properties of the rod dimer, the bowtie and the hybrid dimer antennas in the 2-D. The charge accumulation around the gap determines the near field property, whereas the global current flow determines the far field property. The resonant wavelengths associated with the near-field enhancement are longer than those associated with the far-field radiation, which reflects the conservation requirement of the electron charge. With regard to the geometry tunability of the dimer antenna, the near-field localization/enhancement and the far-field radiation are related with the tip geometry in the gap and the global shape of the shaft respectively. A hybrid dimer antenna possesses the confined/enhanced near-field and the intense far-field radiation simultaneously.
By using a microcavity representation, the resonance in various dimer antennas can be described in a simplified and semi-analytical manner. The local SPP, its propagation along the antenna axis, and its superfocusing towards the gap tip are rendered in the distributions of the surface charge and current. The influences of the plasmonic enhancement and the wavelength mismatch are naturally embodied in the eigen-mode expression of the local SPP. The resonance in a dimer antenna corresponds to the standing-wave of the surface charge and current distributions in the shaft region of the antenna. The Q factor of this resonance and its dependences with the wavelength and the geometry can be analyzed in an intuitive way. With this common understanding to the geometry tunability, we call for further efforts in the engineering of the optical nano-antennas.
This work was financially supported by the French National Research Agency (2007 “photohybrid” & Carnot funding) and the Région Champagne-Ardenne (projet emergence E2007-08052).
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