1. Introduction

Optical antennas are fundamental nanostructures that convert far-field optical radiation into near-field localized energy, and vice versa [1, 2]. They enable the control and manipulation of optical fields at subwavelength scale. Traditional nanoantennas rely on metallic structures, in which the surface plasmon polaritons (SPPs) greatly enhance the optical electric response [3–5]. However, metallic nanoantennas also suffer from inevitable Ohmic losses because of the plasmonic excitations [6]. In the last few years, high refractive index dielectric nanoantennas have drawn a lot of attentions for their relatively low losses and optical magnetism formed by the induced circular displacement currents inside the antennas [7, 8]. Optical magnetism provides an extra dimension to control and engineer the far-field scattering pattern. The combination of the electric mode and magnetic mode may yield enhanced or suppressed forward/backward scattering [9–16]. Forming two antennas into a dimer could also bring about some intriguing effects, such as hotspots and asymmetric scattering [17–23]. Besides, the excitation field could also bring about extra field polarization, amplitude and phase distribution to control the scattering properties of the antennas. Gittes et al. proposed a method to determine the lateral position by monitoring the intensity shift caused by the interference between the scattering field by the particle and unscattered field in the back focal plane [24]. Recently, Barreda et al. observed a switching effect controlled by the incident polarization [17]. M. Neugebauer et al. proposed a method to achieve distinct lateral directionality by exciting longitudinal mode inside a single silicon antenna with tightly focused radially polarized light [25]. The resultant directionality relies on the antenna’s position relative to the focus, which can be applied for position detection. Similarly, Xi et al. introduces a far-field deep subwavelength position detection method based on the interaction of focused azimuthally polarized beam with metallic dimer antennas [26].

In this paper, we present a comprehensive analysis of the far-field scattering of dielectric nano-dimer antenna excited precisely by a linearly polarized Gaussian beam. We find that a simple Gaussian beam is sufficient to obtain prominent lateral directivity. Similarly, the resultant directivity is determined by the relative position of the dimer center to the beam center. In turn, by detecting the lateral scattering intensity contrast ratio, one can also determine the dimer’s displacement relative to the incident field. Moreover, unlike its metallic counterpart which only responds to the incident field with polarization perpendicular to the dimer axis [26], our dielectric dimer antenna can respond to incident polarization both perpendicular and parallel to the dimer axis.

The whole paper is arranged as follows: Section 2 gives a full depiction of the interaction between the incident inhomogeneous field and the nano-dimer antennas. A semi-analytical method, called dipole-dipole interaction model, is applied to give a relation of the scattering far-field to the translational displacement. Section 3 makes a detailed analysis of the far-field scattering radiated from nano-dimer antennas formed by two coupled dielectric nano-cubes. Section 4 presents a numerical demonstration when the nano-dimer antenna is illuminated by a Gaussian beam. Finally, a summary and conclusion of the study will be drawn.

2. Electric/magnetic dipole-dipole interaction model

Let us first consider a nano-dimer antenna formed by two identical particles embedded in a uniform host medium. The host medium is assumed to be air throughout the whole paper. The dimer is illuminated by a monochromatic wave along +z axis. The incident field is inhomogeneous and related to its spatial coordinates: E = E(r). Under dipole approximation, each individual particle can be regarded as a combination of an induced electric dipole and an induced magnetic dipole. The induced electric dipole moment ${\mathbf{p}}_i=\left(p_i^x,p_i^y,p_i^z\right)$ and magnetic dipole ${\mathbf{m}}_i=\left(m_i^x,m_i^y,m_i^z\right)$ are strongly related to its local electric field ${\mathbf{E}}_{loc}\left({\mathbf{r}}_i\right)=\left(E_{loci}^x,E_{loci}^y,E_{loci}^z\right)$ and magnetic field ${\mathbf{H}}_{loc}\left({\mathbf{r}}_i\right)=\left(H_{loci}^x,H_{loci}^y,H_{loci}^z\right)$:

3. Lateral far-field scattering of a dielectric nano-dimer antenna

Consider a nano-dimer consisting of two identical silicon nano-cubes located at ${\mathbf{r}}_1=(-\frac{d}{2},0,0)$ and ${\mathbf{r}}_2=(\frac{d}{2},0,0)$, respectively, as shown in Fig. 1(a). d = 200nm is the distance between the two cube centers and a = 100nm is the side length of the nano-cube. The permittivity of silicon used here is obtained from Palik [27]. Due to the isotropy of the cube, its polarizabilities can be written as scalar: $\overline{\overline{\alpha }}=\alpha I$, where I is the identity matrix. Meanwhile, for a dielectric cube, both electric polarizability α_ee and magnetic polarizability α_mm exist. The exact polarizability components, shown in Fig. 1(b), are retrieved using the scattering far-field method [28]. The analysis can also be applied for spherical particles due to its isotropy. While for a general geometrical shape, we need to consider all the components of its polarizability tensor.

 

Fig. 1 (a) Schematic of the structure. The origin of the coordinate system is chosen at the center of the gap of the dimer and the x-axis is chosen to be aligned with the dimer axis. The dimer consists of two coupled silicon nano-cubes. The distance between the two cube centers is d = 200nm. The side length of the nano-cube is a = 100nm. (b) Electric (blue solid/dashed line for amplitude/phase part) and magnetic (green solid/dashed line for amplitude/phase part) polarizability of an individual cube as a function of the wavelength.

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3.1. Electric polarization of the incident wave perpendicular to the dimer axis

Now, consider a wave illuminating along z axis with the electric field polarized along the y axis and magnetic field along the x axis (dimer axis), as shown in Fig. 2(a). Due to the existence of both α_ee and α_mm, y-polarized electric dipoles and x-polarized magnetic dipoles are directly induced by the incident field. Meanwhile, additional z-polarized magnetic dipoles are induced by the scattering fields from nearby y-polarized electric dipoles. When the beam is tightly focused, the longitudial components of the incident fields, Ez and Hz, also need to be taken into consideration. All the dipole moments can be deduced from Eqs. (1)–(3):

 

Fig. 2 (a) Schematic representation of the induced dipoles of the silicon dimer antenna when the electric polarization of the incident wave is perpendicular to the dimer axis. (b) Lateral scattering intensity contrast ratio V as ρ varies from 0 to +1 when φ = 0. (c) Contrast ratio V as φ varies from −π to +π when ρ = 1. (d) Contrast ratio V at λ = 528nm.

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As we know, the x-polarized dipoles do not contribute the far-field scattering along x-axis. Thus, the lateral scattering is the interference of two pairs of orthogonal dipoles perpendicular to the x axis, that is:

Obviously, V = 1 means that the electric field is totally scattered towards +x direction; V = −1 means that the electric field is totally scattered towards −x direction; and V = 0 means equal scattering. Once the structure is fixed, under weak focusing when the longitudinal components Hz can be ignored, V is determined only by the relative incident field $\frac{E_{inc2}^y}{E_{inc1}^y}=\rho e^{i\phi }$, where ρ is the amplitude ratio and φ is the phase difference. Figures 2(b) and 2(c) depict the relation between the lateral intensity contrast ratio V and wavelength λ from 400 nm to 800 nm when there is only amplitude ratio (φ = 0) or phase difference (ρ = 1), respectively. We can achieve unidirectional lateral scattering either by solely adjusting the phase difference or by solely adjusting the amplitude ratio between the incident fields. Interestingly, by adjusting the phase difference we can get unidirectional scattering within a wide spectrum while by adjusting the amplitude ratio the unidirectional scattering can be achieved only at some particular wavelength. Here, for our structure, the +x unidirectional scattering can be achieved only at λ = 528 nm when there is no phase difference between the incident fields. This provides the possibilities for a Gaussian beam to achieve unidirectional scattering. Figure 2(d) presents the contrast ratio V in a complex plane at λ = 528 nm.

3.2. Electric polarization of the incident wave parallel to the dimer axis

For an x-polarized incident beam as shown in Fig. 3(a), the induced dipoles are x-polarized electric dipoles, y-polarized magnetic dipoles as well as z-polarized electric dipoles:

 

Fig. 3 (a) Schematic representation of the induced dipoles of the silicon dimer antenna when the electric polarization of the incident wave is parallel to the dimer axis. (b) Lateral scattering intensity contrast ratio V as ρ varies from 0 to +1 when φ = 0. (c) Contrast ratio V as φ varies from −π to +π when ρ = 1. (d) Contrast ratio V at λ = 482nm.

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Figures 3(b) and 3(c) depict the relation between the lateral intensity contrast ration V and wavelength λ from 400 nm to 800 nm when there is only amplitude ratio (φ = 0) or phase difference (ρ = 1), respectively. Similarly, for x polarized incident fields, the +x unidirectional scattering can be achieved only at λ = 482 nm when there is no phase difference between the incident fields. Figure 3(d) presents the contrast ratio V in a complex plane at λ = 482 nm.

4. Dimer antenna illuminated by a Gaussian beam

Let’s consider that the dimer is illuminated by a Gaussian beam whose electromagnetic field is related to its spatial coordinates. Assuming propagation in the +z direction, for a fundamental Gaussian beam, the electric field at focal plane can be calculated using Richard-Wolf diffraction integral [29]. Assuming the dimer is initially located at the center of the beam waist, the contrast ratio is zero for ρ = 1 and φ = 0. When the beam is moving towards +x axis with a small displacement Δx as shown in Fig. 4(a), the incident field amplitude ratio then become a function of Δx while φ is always zero. The asymmetric field amplitude distribution leads to asymmetric lateral scattering and the lateral contrast ratio can therefore be expressed as a function of Δx. This encodes the lateral displacement into the far-field scattering intensity contrast ratio which means the lateral displacement can be determined by detecting the far-field contrast ratio. Figures 4(b) and 4(c) show the relation between contrast ratio V and lateral displacement Δx under x-polarized excitation at 482 nm and under y-polarized excitation at 528 nm, respectively. It can be seen that the contrast ratio V is in approximate proportion to lateral displacement Δx. Here, we define sensitivity $S=\left|\frac{\partial V}{\partial \mathrm{\Delta }x}\right|$. A high sensitivity means that a small displacement change will cause drastic change in the far-field contrast ratio which is always preferable for experimental detecting. For x-polarized excitation, the sensitivity can reach 5.0 × 10⁻³nm⁻¹ when the beam waist is 400 nm. While for y-polarized excitation, the sensitivity can reach 6.6 × 10⁻³nm⁻¹. As the beam waist increases, the sensitivity decreases. However, for y-polarized excitation, the sensitivity is always higher than that for x-polarized excitation. To be more detailed, the sensitivity can be decomposed as two parts: $S=\left|\frac{\partial V}{\partial \rho }\right|\left|\frac{\partial \rho }{\partial \mathrm{\Delta }x}\right|$, where the first item is determined by the dimer and the second item is related to the incident field. Figure 5 shows the relation between ρ and Δx as well as V and ρ. We know that the partial derivative means the slope of a curve. From Fig. 5(a), it can be seen that $\left|\frac{\partial \rho }{\partial \mathrm{\Delta }x}\right|$ increases as ω₀ decreases, thus leading to higher sensitivity for smaller beam waist. Similarly, it can be seen from Fig. 5(b) that $\left|\frac{\partial V}{\partial \rho }\right|$ is bigger for y-polarized beam than that for x-polarized one. That accounts for why y-polarized beam has higher sensitivity than x-polarized one does for the same beam waist.

 

Fig. 4 (a) Schematic illustration of the configuration, the Gaussian beam is initially focusing on the center of the dimer, then moves towards +x direction with a displacement Δx. Contrast ration as a function of Δx for (b) x-polarized incident Gaussian beam at 482 nm and (c) y-polarized incident Gaussian beam at 528 nm, respectively.

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Fig. 5 (a) Amplitude ratio as a function of lateral displacement Δx. (b) Contrast ratio as a function of ρ for x-polarized incident Gaussian beam at 482 nm (blue solid line) and y-polarized incident Gaussian beam at 528 nm (green solid line), respectively.

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To verify our model, a full-wave simulation is performed using Finite Difference Time Domain (FDTD) method. The dimer is located at the center of the simulation region. The surrounding medium is chosen to be air. The boundary is chosen as Perfect Matched Layer (PML). A non-uniform mesh is used throughout the whole simulation region and the minimum mesh size is 4nm. A broadband Gaussian beam illuminates the dimer and the monitors collects the data of the total fields. The scattering fields can be obtained by subtracting the incident fields from the total fields. Finally, a far-field transformation is carried out to get the far-field scattering pattern. Figures 6(a) and 6(b) show the far-field scattering pattern for x-polarized excitation and y-polarized excitation when beam waist is 400 nm, respectively. The different scattering pattern results from the interference of different polarized dipoles. It can be seen that when the lateral displacement Δx increases from 0 to 200 nm, the scattering pattern changes from symmetric to −x unidirectional scattering pattern. Also, when the displacement is fixed, Δx = 100nm, smaller beam waist will lead to a more asymmetric scattering pattern as shown in Figs. 6(c) and 6(d). Further, we plot the contrast ratio as a function of the displacement as shown Figs. 6(e) and 6(f). It can be seen that smaller beam waist leads to higher sensitivity and the sensitivity is higher for y-polarized beam than that of x-polarized beam, which agrees quite well with our analytical model.

 

Fig. 6 FDTD simulation results. Far-field scattering pattern under x-polarized excitation (a) and y-polarized excitation (b) for different lateral displacement when the beam waist is fixed at 400 nm. Far-field scattering pattern under x-polarized excitation (c) and y-polarized excitation (d) in xy-plane for different beam waist when the lateral displacement is fixed at 100 nm. Contrast ratio under x-polarized excitation (e) and y-polarized excitation (f) as a function of lateral displacement for different beam waist.

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It’s also very interesting to see the lateral contrast ratio and sensitivity when one particle is fixed at the center of the Gaussian beam while the other moves off the center towards +x direction. Figure 7 shows the lateral contrast ratio as a function of the gap. The same effect can also be observed: smaller beam waist leads to higher sensitivity and the sensitivity for y-polarized beam is larger than that of x-polarized beam.

 

Fig. 7 Contrast ratio as a function of the gap for (a) x-polarized incident Gaussian beam at 482 nm and (b) y-polarized incident Gaussian beam at 528 nm

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Theoretically, once the structure is fixed, we can always increase the sensitivity by reducing the beam waist. However, Gaussian beam with too small beam waist is hard to obtain experimentally while large beam waist will lose high sensitivity. In order to circumvent this problem, we further investigate Hermite-Gaussian beam: HG10 mode. The strong field gradient $\left|\frac{\partial \rho }{\partial \mathrm{\Delta }x}\right|$ make it possible to obtain large sensitivity. Similarly, the dimer is initially located at the center of the beam waist and the beam move towards +x direction with displacement Δx. Figure 8 shows the lateral contrast ratio as a function of the lateral displacement. It can be seen that the sensitivity is indeed enhanced and still keeps high even for larger beam waist.

 

Fig. 8 Contrast ratio as a function of Δx for x-polarized incident HG10 mode at 482 nm.

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5. Conclusion

To conclude, we analyzed the scattering properties of silicon dimer antennas under a Gaussian beam illumination using dipole-dipole interaction method. We also showed that the lateral far-field scattering contrast ratio is highly related to the incident field on each individual particle. Meanwhile, the contrast ratio is determined by the translational displacement relative to the dimer antennas. The detection sensitivity can be enhanced by using HG10 beam. This work will pave way to a novel position detection technique, which might be applied in nanoscale metrology and super-resolution imaging.