Polarization-controlled directional scattering for nanoscopic position sensing

Controlling the propagation and coupling of light to sub-wavelength antennas is a crucial prerequisite for many nanoscale optical devices. Recently, the main focus of attention has been directed towards high-refractive-index materials such as silicon as an integral part of the antenna design. This development is motivated by the rich spectral properties of individual high-refractive-index nanoparticles. Here we take advantage of the interference of their magnetic and electric resonances to achieve strong lateral directionality. For controlled excitation of a spherical silicon nanoantenna, we use tightly focused radially polarized light. The resultant directional emission depends on the antenna's position relative to the focus. This approach finds application as a novel position sensing technique, which might be implemented in modern nanometrology and super-resolution microscopy set-ups. We demonstrate in a proof-of-concept experiment that a lateral resolution in the Ångström regime can be achieved.

The material properties of Si and SiO 2 are adopted from the database of Palik [4]. In general, Si has a high refractive index and a small extinction coefficient in the visible regime (e.g. n Si = 3.85 − 0.02i at λ = 652nm). In order to adapt the simulation to the experiment, we consider a tightly focused linearly polarized Gaussian beam as a source (maximum NA = 0.9), and collect only the light scattered in the forward direction into the far-field with a polar collection angle set to NA ∈ [0.95, 1.3] (see sketch in Supplementary Fig. 1b). In the investigated spectral regime, a Si nanosphere of the chosen size placed on a substrate supports three pronounced resonances, the magnetic dipole (λ MD ≈ 660 nm), the electric dipole (λ ED ≈ 540 nm) and the magnetic quadrupole (λ MQ ≈ 515 nm) [3]. For this reason, the scattering spectrum (see red line in Supplementary Fig. 1c) can be described, in first approximation, by the sum of three individual Lorentzian curves, fitted to the simulation.
The weak contribution of the magnetic quadrupole (dashed green line) can be neglected for wavelengths above 600 nm. Therefore, only electric (dashed gray line) and magnetic (dashed blue line) dipole resonances need to be considered for the chosen excitation wavelength of 652 nm. Now, we consider the relative phases of the magnetic and electric dipole moments with respect to their corresponding excitation fields. Following the Lorentz oscillator model [5], the relative phases of the electric (φ ED ) and magnetic (φ M D ) dipole moments depend on the excitation wavelength. Supplementary Fig. 1d shows φ ED (gray line), φ M D (blue line), and the phase difference ∆φ = φ M D − φ ED (dashed black line). We find two wavelengths with ∆φ = π/2. However, we choose the wavelength 652 nm, which is close to λ MD and guarantees a sufficient overlap between both types of dipoles (see dashed vertical and horizontal red lines). As mentioned above, the magnetic quadrupole can be neglected for the chosen wavelength. Another advantage of this choice of wavelength close to the magnetic resonance is the much higher efficiency, with which the magnetic dipole mode can be excited in comparison to its electric counterpart. This leads to comparable scattering signal strengths from both induced magnetic and electric dipole moments (even though the electric field is much stronger than the magnetic field close to the optical axis) and, therefore, stronger asymmetry upon interference. Consequently, an enhanced position dependence of the directionality is realized.

SUPPLEMENTARY NOTE 2
Negligibility of the transverse electric dipole As mentioned in the main manuscript, we expect the electric and magnetic dipole moments to be proportional to the respective local field vectors, p ∝ E and m ∝ H. This includes the transverse electric dipole moments p x ∝ E x and p y ∝ E y . However, for several reasons we can neglect the influence of the transverse electric dipole moments in first approximation.
First of all, even at the rim of the region of linearity, roughly 50 nm away from the optical axis, the longitudinal electric field is still stronger than the transverse ones by a factor of 4 (see Fig. 1d in the manuscript). Second, with the given excitation wavelength close to the magnetic dipole resonance of the antenna, we expect to excite the transverse magnetic dipole moment with a higher efficiency than the transverse electric dipole moment (see Sup-  Supplementary Figs. 2b, e and h. The ratios between these power values for p x , m y and p z equals to 1 : 6.4 : 14.5. We conclude that the influence of transverse electric dipole moments can indeed be neglected in first approximation. In particular, its influence on the directivity parameters is very small.

SUPPLEMENTARY NOTE 4 Estimation of the resolution
In order to estimate the resolution of our position sensing experiment, we compare two post-selected nearly identical far-field images (see difference image in Supplementary   Fig. 3) and calculate the average intensity differences for all four regions, ∆I 1 = 11 · 10 −3 , ∆I 2 = −10 −3 , ∆I 3 = −8 · 10 −3 , ∆I 4 = −10 −3 . The corresponding standard deviations (σ i ≈ 25 · 10 −3 for i ∈ [1,4], see histograms plotted as insets in Supplementary Fig. 3) and the number of pixels in each region (1050 pixels), yield an uncertainty of the mean intensities of ±10 −3 for each intensity value. These results indicate that a relative shift of the antenna's position of ∆x = −2 ± 0.2 nm and ∆y = 0 ± 0.2 nm was measured with an uncertainty in theÅngström regime. Hence, the two almost identical back focal plane images used for this estimation correspond to two antenna positions, which were different by only 2 ± 0.2 nm.