Transverse Kerker Scattering for Angstrom Localization of Nanoparticles

Angstrom precision localization of a single nanoantenna is a crucial step towards advanced nanometrology, medicine, and biophysics. Here, we show that single nanoantenna displacements down to few angstroms can be resolved with sub-angstrom precision using an all-optical method. We utilize the tranverse Kerker scattering scheme where a carefully structured light beam excites a combination of multipolar modes inside a dielectric nanoantenna, which then, upon interference, scatters directionally into the far field. We spectrally tune our scheme such that it is most sensitive to the change in directional scattering per nanoantenna displacement. Finally, we experimentally show that antenna displacement down to 3 Å is resolvable with a localization precision of 0.6 Å.

Figure 1

Kerker scattering and Huygens dipole. (a) Zero backscattering. A $y$-polarized plane wave with $\mathbf{k}=k_z{\widehat{\mathbf{e}}}_z$ excites transverse electric and magnetic dipoles $p_y$ and $m_x$ of equal amplitudes and phases, resulting in a Huygens dipole and zero backscattering. Inset: polar plot of the resulting total far field intensity of the scattered light (black) and the individual emission patterns of $p_y$ (red) and $m_x$ (blue) in the meridional $xz$ plane. (b) Directional scattering perpendicular to the propagation direction of the beam (transverse Kerker scattering). A tightly focused azimuthally polarized beam can induce longitudinal magnetic $m_z$ and transverse electric $p_y$ dipoles, resulting in transverse Kerker scattering [inset similar to (a)].

Figure 2

(a) Scattering cross section ${\sigma }_{\mathrm{sca}}$ of a silicon nanoantenna (156 nm core diameter with 6 nm oxide shell) in free space calculated via Mie theory. The nanoantenna predominantly supports MD and ED modes along with a much weaker MQ mode. The relative phase between MD and ED scattering coefficients is shown below. Around $\lambda \approx 535\mathrm{nm}$ and $\lambda \approx 630\mathrm{nm}$, as indicated by dotted black circles and corresponding gray regions, ED and MD scattering coefficients are $\pi /2$ out of phase. (b) Theoretical field intensity and relative phase (inset) distributions for tightly focused azimuthally (top) and radially (bottom) polarized beams. For an azimuthally (radially) polarized beam, the transverse electric field ${\mathbf{E}}_t$ ($E_x{\widehat{\mathbf{e}}}_x+E_y{\widehat{\mathbf{e}}}_y$) is azimuthally (radially) polarized, the transverse magnetic field ${\mathbf{H}}_t$ ($H_x{\widehat{\mathbf{e}}}_x+H_y{\widehat{\mathbf{e}}}_y$) is radially (azimuthally) polarized, and the longitudinal field component is purely magnetic (electric), with $E_z=0$ ($H_z=0$). All intensity plots are normalized to the maximum value of the total field intensity $I_{\mathrm{tot}}={|\mathbf{E}|}^2+{|\mathbf{H}|}^2$ (Gaussian units). (c) Polar plots of the resulting far field intensities of the scattered light in the meridional $xz$ plane for a nanoantenna positioned at $(x,y)=(50,0)\mathrm{nm}$ in the focal plane. Transverse Kerker scattering is highly directional for azimuthal (radial) polarization at ${\lambda }_{\mathrm{azim}}=535\mathrm{nm}$ (${\lambda }_{\mathrm{rad}}=630\mathrm{nm}$) but weaker for ${\lambda }_{\mathrm{azim}}=630\mathrm{nm}$ (${\lambda }_{\mathrm{rad}}=535\mathrm{nm}$).

Figure 3

Experimental implementation. (a) Tight focusing of a collimated input beam (radially or azimuthally polarized) using a microscope objective of numerical aperture (NA) of 0.9 onto a silicon nanoantenna sitting on a glass substrate (scanning electron micrograph image in inset), which can be precisely positioned by a piezo stage. The forward scattered light is collected using an oil-immersion-type microscope objective of $\mathrm{NA}=1.3$. The back-focal plane (BFP) intensity distribution (for $\mathrm{NA}\in [0.98,1.3]$) is captured onto a CCD camera. (b)–(g) Exemplary BFP images showing symmetric scattering for an on-axis position (b)–(c), and transverse Kerker directionality for off-axis positions (d)–(g). The strength of directional scattering strongly depends on the wavelength as shown in (d)–(g) with maximum directionality observed at ${\lambda }_{\mathrm{azim}}=545\mathrm{nm}$ (d) and ${\lambda }_{\mathrm{rad}}=630\mathrm{nm}$ (e). (h) Sensitivity ($S_x$) or change in directivity $D_x$ per nm—$|S_x|$ is plotted for all wavelengths along with the numerically calculated results for comparison (radial and azimuthal polarization in red and blue). Inset: plot of the measured directivity $D_x$ versus the antenna’s $x$ position for ${\lambda }_{\mathrm{rad}}=630\mathrm{nm}$ (in red) and ${\lambda }_{\mathrm{azim}}=545\mathrm{nm}$ (in blue) where $|S_x|$ is strongest.

Figure 4

Differential BFP images $\mathrm{\Delta }\tilde{I}$ for relative displacements of less than 1 nm for an azimuthally polarized excitation beam with ${\lambda }_{\mathrm{azim}}=545\mathrm{nm}$ ($S_x\approx S_y\approx -0.025{\mathrm{nm}}^{-1}$). In each BFP image, $\mathrm{\Delta }\widetilde{I_j}$ values from the outlined regions are used to plot the histograms shown in the centers; dash-outlined regions ($j=1$, 3) correspond to $\mathrm{\Delta }x$ and solid ones ($j=2$, 4) to $\mathrm{\Delta }y$. Gaussian fits to the $\mathrm{\Delta }\widetilde{I_j}$ distributions (histogram) are used to estimate the relative displacements (peak to peak) and the precision of estimation (standard error of the mean). The Gaussian peaks are clearly distinguishable for a relative displacement down to 3 Å, with a precision of 0.6 Å.