1. Introduction

Optical nanoantenna is the fundamental building block for the future nanophotonic devices due to their remarkable ability of the local field enhancement and the highly efficient interconversion between near-field energy and far-field radiation [1,2]. For example, impelled by applications of optical sensing [3], light-emitting devices [4] and photodetectors [5], tremendous research effects have been devoted to the directivity control of scattered (emission) light with nanoantennas [6,7]. Until now, various configurations of metallic nanoantennas have been proposed to tune the directional scattering by carefully molding the geometric parameters such as shape, size, angle and gap dimension [8–13]. However, these designs are generally complex because it needs to satisfy both the rigorous phase matching and the retardation requirements. In contrast, all-dielectric nanoantennas with high refractive index have attracted considerable interest in the last decade, owing to their low optical loss in the visible and near-infrared ranges, a wealth of electromagnetic modes, and the excellent compatibility with semiconductor device technologies [14–17]. Particularly, it has been demonstrated that the excellent unidirectional scattering can be achieved simply by using an individual dielectric nanoparticle under the Kerker’s condition [16,18–22]. In this scenario, the simultaneously excited electric and magnetic dipoles have the same amplitude and phase in scattering coefficient, resulting in the zero-backward radiation (the first Kerker condition). On the other hand, these two dipoles with comparable strength oscillate out-of-phase, leading to minimum forward scattering (the second Kerker condition) [23,24]. In addition, such dielectric nanoparticles (Kerker type nanoantennas) enjoy the merits of the robustness and broadband in unidirectional scattering by using core-shell structure and shape engineering [25–27].

When the Kerker type nanoantenna is located at the focal plane of a tightly focused cylindrical vector beam (CVB) exemplified by radially polarized beam (RPB) and azimuthally polarized beam (APB). The longitudinal electric and magnetic dipoles can be selectively excited by RPB and APB, respectively [28]. Moreover, the transverse Kerker scattering can be realized through the interference between the longitudinal and transverse dipolar scattering when the nanoparticle is placed at a specific position in the focal plane [29–31]. More interestingly, as the Kerker type nanoantenna is moved in the inhomogeneous focal field of CVB, this dipolar scattering interference can be well tuned in terms of phase and amplitude, giving rising to a displacement related scattering directivity. In this sense, CVB illuminated nanoantenna can serve as a displacement sensor by monitoring the directivity of the scattering [29,30]. Recent works have proven that this displacement sensor possesses a high resolution even down to a few angstroms [29,32]. Besides the high precision, the displacement sensor with tunable measuring range is also highly demanded especially in applications of the single molecule tracking and the nanoscopy [33–35]. Unfortunately, few attentions have been paid to the measuring range of the nanoantenna-based displacement sensor excepted for some theoretical works [36,37]. Also, these approaches to tune measuring range of displacement sensor mainly rely on carefully adjusting geometrical parameters of the nanoantenna, rendering a great challenging for the fabrication accuracy of the nanoantenna.

To address this issue, we propose and demonstrate a facile way to change the measuring range of displacement sensor using a CVB excited Si nanoantenna. This Si nanoantenna shows a good linearity with a lateral resolution of 4.5 nm in displacement sensing. It is found that the measuring range of this displacement sensor is extended by twice via merely switching the excitation from APB to RPB. Our method avoids the sophisticated nanoantenna fabrication processes and multi operation wavelengths, showing a large flexibility. We envision that our results may find applications in super-resolution microscopy and optical nanometrology.

2. Principle and theory of transverse scattering

To achieve a considerable transverse scattering, it is necessary to introduce electric and magnetic dipoles with perpendicular dipole moments, one of which is parallel to the optical axis [38]. To this end, a tightly focused CVB is preferably used as the excitation. As the considerable magnetic (electric) component in the focal field of APB (RPB), a longitudinal magnetic dipole m_z (electric dipole p_z) can be excited when the dielectric nanoantenna locating at the focal spot (the origin of Cartesian coordinate system) of the illumination [29]. When the nanoantenna is slightly deviated away from the focal spot, the transverse electric dipole p_y (magnetic dipole m_y) will be induced due to the nontrivial lateral electromagnetic components of the illumination [39,40], as illustrated in Fig. 1(a). In particular, when the nanoantenna approaches to a position that two dipolar scatterings have the same amplitude and phase (Huygens dipole), the interference results in a transverse unidirectional (Kerker) scattering, as illustrated by the black line in Fig. 1(b) [29,36].

 

Fig. 1. Principle of transverse Kerker scattering. (a) Tightly focused APB (RPB) induced longitudinal and transverse dipoles (m_z and p_y for APB, p_z and m_y for RPB) in a Si nanoantenna, which has a lateral displacement along + x axis. (b) Transverse Kerker scattering (black line) resulted from the interference between the m_z dipole (blue line) and the p_y (red line) dipole under APB illumination [the p_z (blue line) and the m_y (red line) under RPB illumination] in a Si nanoantenna.

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We use a Si nanoparticle with a diameter of 204 nm to examine the transverse scattering because it can simultaneously support electric and magnetic dipole modes with the comparable oscillation strength. According to the Mie theory, we calculate the scattering cross section of the Si nanoparticle in free space as shown by the blue curve in Fig. 2(a). This scattering spectrum is recognized as the contributions from the electric dipole (ED), magnetic dipole (MD) and magnetic quadrupole (MQ) modes through the multipole expansion. Since the MQ contribution to the scattering is small especially beyond the wavelength λ=600 nm, the particle can be approximately regarded as the electric and magnetic dipoles. Thus, the induced dipolar moments can be expressed as p∝a₁E and m∝b₁H. Here, E and H are the local electromagnetic fields, and a₁ and b₁ are Mie scattering coefficients of the ED and MD [30,41], which define the amplitude and phase of induced dipole moments. Figure 2(b) plots the amplitude ratio (|b₁|/|a₁|) between coefficients b₁ and a₁, which suggests that |b₁|/|a₁| = 0.33 at λ=633 nm. To clearly analyze the interference of electric and magnetic dipoles, we also depict the relative phase between coefficients b₁ and a₁ shown in Fig. 2(c), which reveals that the magnetic and electric dipoles have a π/2 phase difference at λ=633 nm. This phase difference can compensate the relative phase between the longitudinal and transverse components. In this case, the interference between two induced dipoles and the underlying scattering directivity are only determined by the relative amplitude of two dipolar moments.

 

Fig. 2. (a) Scattering cross section of a silicon nanoparticle (diameter of 204 nm) calculated with Mie theory. (b) Amplitudes of scattering coefficients a₁ and b₁, and the ratio |b₁|/|a₁|. (c) Relative phase difference between scattering coefficients a₁ and b₁. The phase difference between ED and MD scattering coefficients is π/2 at the labeled wavelength λ≈ 633 nm. (d) Normalized amplitudes of E_y, ZH_z, and the radio |E_y|/|ZH_z| under the illumination of tightly focused APB (NA=0.85, λ = 633 nm). (e) Normalized amplitudes of ZH_y, E_z, and the radio |ZH_y|/|E_z| under the illumination of tightly focused RPB (NA=0.85, λ = 633 nm).

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The inhomogeneous focal field of CVB, serving as a position-related driving force, provides an opportunity to tune relative amplitude of two dipolar moments. The intensity distribution of tightly focused CVB in the focal field can be calculated by the vector diffraction theory with the same parameters as in their experiment counterparts discussed later [40]. The tightly focused APB produces a longitudinal magnetic and a transverse electric dipole. Noting that the cylindrical symmetry of the CVB in the focal plane, we only take the variation of focal field along x axis as an example. Figure 2(d) plots the driving forces of these two dipoles, i.e. E_y and ZH_z, as well as their ratio |E_y|/|ZH_z| as a function of nanoparticle position along x axis. Here, Z=$\sqrt {\mu /\varepsilon }$ is the impedance of the background medium with the permeability (permittivity) of μ (ε). It is found that the ratio |E_y|/|ZH_z| is approximated to increase linearly within the region of 0 ∼ ±150 nm. On the other hand, the tightly focused RPB excites a longitudinal electric dipole and a transverse magnetic dipole. The ratio |ZH_y|/|E_z| also shows a good linearity within the same region as that of APB [see Fig. 2(e)]. Such that the amplitude ratio of two dipolar moments can be adjusted linearly by moving the particle in the focal plane of CVB, resulting in a position dependent dipolar scattering interference (directivity). In other words, this scattering directivity can be used to monitor nanometric displacement of Si nanoparticle in the focal plane of CVB.

3. Experiment and results

The Si nanoparticles are fabricated on glass substrate (cover slip) using a single femtosecond laser pulse (800 nm, 35 fs, and 1 kHz), a more detailed description can be found in the literature [42,43]. The far-field scattering patterns of the particles are obtained with our home-built optical setup, as illustrated in Fig. 3(a). The CVB with the wavelength λ = 633 nm is generated by passing a Gaussian beam through a liquid crystal-based polarization converter (Thorlabs, WPV10L-633). The beam quality of CVB is then optimized by a Fourier-filtered before entering the illuminating objective (Leica, 50×, NA 0.85) [28]. The position of Si nanoparticle in the focal plane is controlled by a 3D piezo stage (Thorlabs, MAX311D) with an accuracy of 5 nm. The transmitted (forward scattered) light is collected by an oil-immersion-type objective (Nikon, 100×, NA 1.3). After passing a 4f optical system, the far-field pattern of the scattered light in the conjugated back focal plane (BFP, transverse k-space) of collection objective is imaged onto a CCD camera [44]. It is noted that the detection limit of far-field pattern in k-space is determined by the NA (1.3) of collection objective as sketched in Fig. 3(b). To eliminate the direct transmitted excitation, a circle-like beam dump is placed at the conjugated BFP to block the light within a solid angle corresponding to NA=0.9. The collected far-field pattern is representatively shown in Fig. 3(c), which is divided into 4 regions (labeled by the red solid lines with arc angle of Δϕ = 45°) to evaluate the scattering directivity. We denote the directivity as D_x= (I₁-I₃)/I_tot and D_y= (I₂-I₄)/I_tot with I_tot=(I₁+I₂+I₃+I₄)/2 representing the directional scattering along two perpendicular transverse directions. Here, I_l is the average intensity of the l th region in the BFP [29].

 

Fig. 3. (a) Optical setup for far-field scattering imaging in back-focal plane. (b) Sketch for the measuring range of the far-field scattering imaging. (c) Representative far-field scattering pattern imaging, where 4 regions within the red sold lines are for scattering directivity evaluation.

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We use the in-situ scattering spectrum measurement to confirm that the scattering pattern image is taken from the individual Si nanoparticle. Here, the CCD in Fig. 3(a) is replaced by a multi-mode fiber to deliver the scattered light of the nanoparticle to a spectrometer (Andor Shamrock, SR-500i). Figure 4(a) displays the measured scattering spectrum of Si nanoparticle (black curve) under the excitation of a x-polarized white light. This spectrum is further validated by a full-wave simulation based on finite difference time domain method (FDTD Solutions, Lumerical Inc.), where the refractive index of the substrate is set to be 1.52 and the permittivity of Si is taken from experimental data of Palik [45]. As can be seen, the simulated scattering spectrum of Si nanosphere [red curve in Fig. 4(a)] has an excellent agreement with the experimental one, implying the scattering signal is from a single Si nanoparticle. Moreover, the morphology and size of the Si nanoparticle are acquired by SEM image after the optical measurement. As shown in the inset of Fig. 4(a), the measured Si nanoparticle has a nearly spherical shape with diameter of 210 nm. Noting that the size of nanoparticle in our experiment is larger than that in simulation, this difference can be explained by a thin oxide shell created on Si nanoparticle during the fabrication process [42]. Also, it is found two pronounced resonant peaks locate at 620 nm and 785 nm of the scattering spectrum, which are assigned to be electric and magnet dipole modes, respectively [42]. The distributions of electric and magnetic fields at λ=633 nm are displayed in Figs. 4(b) and 4(c). As seen from Fig. 4(b), the electric field map presents a typical two-lobe distribution outside the particle, which corresponds to an electric dipole oriented along x axis. Also, a feeble circular distribution of electric field within the nanoparticle (see the white dotted line drawn to guide the eye) suggests a magnetic dipole oriented along y axis. This can be further verified from magnetic fields map shown in Fig. 4(c), where a strong spot appears at the same location as the circular-distributed electric field. Meanwhile, a weak magnetic quadrupole is found to superpose with the magnetic dipole at λ=633 nm, forming an extra weaker spot in Fig. 4(c). As the MQ is much weaker than ED or MD at λ=633 nm, its contribution to the far-field scattering can be neglected. In this case, the nanoparticle can be regarded as a combination of an ED and a MD.

 

Fig. 4. (a) Experimental (black curve) and simulated (red curve) scattering spectra of a Si nanosphere. The inset of (a) shows the SEM image of the Si nanosphere with the scalebar of 500 nm. (b) Electric and (c) magnetic field maps at λ=633 nm.

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Fig. 5. Representative back focal plane images of Si nanoparticle with different displacements along x direction at focal plane of (a)-(c) APB and (e)-(g) RPB. (d) and (h) Dependence of directivity D_x on lateral displacement Δx under APB and RPB excitations, respectively. The error bars represent standard errors of D_x.

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Subsequently, we investigate the far-field scattering pattern of the Si nanoparticle excited by the tightly focused CVB. We raster scanned the Si nanoparticle along x axis (x${\in}$[−500 nm, 500 nm]) with 15 nm step size and capture BFP images for each position. Figures 5(a)–5(c) show the BFP images of Si nanoparticle at 3 exemplary displacements in the focal plane of tightly focused APB. A broad ring is present in the BFP image when the nanoparticle is located at the center of the APB excitation (Δx=0) as shown in Fig. 5(b). This far-field scattering pattern is attributed to a longitudinal magnetic dipole that is produced by the significant longitudinal magnetic component of APB at the optical axis. When the nanoparticle has a displacement of ±100 nm according to the beam center, it is found a unidirectional scattering toward ± x direction, as shown in Figs. 5(c) and 5(a), respectively. The directional scattering is ascribed to the constructive (destructive) interference of the transverse scattering (±x direction) between two perpendicular magnetic and electric dipoles as illustrated in Fig. 1(a). Additionally, due to the good linearity of the ratio |E_y|/|ZH_z| around the optical axis [see Fig. 2(a)], the directivity D_x is almost in proportional to the absolute displacement |Δx| of the nanoparticle [29], serving as a nanometric displacement sensor. In Fig. 5(d), we plot the dependence of D_x on the displacement Δx of the Si nanoparticle under APB excitation. Each data point is an average of 10 measurement values of D_x. The standard error (SE) shown as error bars represents the stability of our measurement. Here, we fit the experiment data with linear regression model and the goodness of fit is evaluated by the R-squared value (R²) [46,47]. With the increasing of the displacement Δx, the directivity D_x deviates from the linear position-dependent relation, therefore, we discard the data with larger Δx to maintain a high goodness of fit (R²>0.98). For APB excitation, we use the linear dependence range of D_x on Δx with the R²=0.985 to determine the displacement measuring range, and denote the slope of the fitting line (S_x=0.0105 ± 0.00056 nm^-1) as the position sensing sensitivity of the nanoparticle. The resolution of the displacement sensor is defined as Δx_r=σ_Dx/k±σ_Dxσ_k/(k²-σ_k²) [32], where σ_Dx is the standard error of D_x, k and σ_k are the average and standard error of S_x, respectively. From Fig. 5(d), it is revealed that under the APB illumination, our displacement sensor has the measuring range of 90 nm and the resolution of Δx_r=3.50 ± 0.189 nm. Figures 5(e)–5(g) present the BFP images of Si nanoparticle at 3 representative displacements in the focal plane of RPB. An obvious ring pattern is also found in the BFP image when the particle is located at the focal spot of the RPB as shown in Fig. 5(f). This far-field scattering is from a longitudinal electric dipole that is induced by the considerable longitudinal electrical component of RPB. In addition, as seen from Figs. 5(e) and 5(g), the unidirectional transverse scattering is realized when the nanoparticle has an absolute displacement |Δx|=120 nm from the center of RPB. Notably, the origination of the dipoles induced by RPB are reversed comparing with that produced by APB [also see Fig. 1(a)]. The destructive (constructive) interference between the dipolar scatterings occurs in the direction of + x (-x) axis, leading to the scattering direction is opposite to the moving direction of the nanoparticle. Furthermore, we plot the directivity D_x against the displacement Δx of the Si nanoparticle under RPB excitation as shown in Fig. 5(h). The R² (0.992) of the linear fitting to the experimental data allow us to determine the measuring range of this displacement sensor to be 180 nm. It also shows that this RPB excited displacement senor has a resolution of Δx_r=4.43 ± 0.121 nm and a sensitivity of S_x=-0.0093 ± 0.00025 nm^-1, which are in the same order of magnitude as those of APB. Comparing Fig. 5(h) with Fig. 5(d), it is indicated that the measuring range of our proposed displacement sensor can be extended by twice through switching the excitation of Si nanoparticle from APB to RPB, providing a flexible tool for the optical nanometrology.

4. Conclusions

We have demonstrated a scheme to extend the measuring range of the nanometric displacement sensor by exciting a single Si nanoantenna with a cylindrical vector beam (CVB). It reveals that the tightly focused CVB can induce the transverse and longitudinal dipoles in the Si nanoparticle, leading to a transversely interfered dipolar scattering in the far-field. Moreover, the transverse scattering directivity can be well tuned by changing the driving force on the two dipoles through moving the position of the nanoparticle in the focal field of the CVB. As a result, this position related scattering directivity can be served as a displacement sensor with a 4.5 nm lateral resolution. Interestingly, by simply switching the excitation from APB to RPB, the measuring range of this nanometric displacement sensor can be extended by twice yet without losing the sensitivity and the resolution. Our present work is based on the interaction between a spherical nanoparticle and a CVB, but the principle is equally valid for particles with other shapes, such as the nanoring and the nanodisk. We envision our results may provide a solution toward the super-resolution microscopy and the optical nanometrology.