Background-Free 3D Nanometric Localization and Sub-nm Asymmetry Detection of Single Plasmonic Nanoparticles by Four-Wave Mixing Interferometry with Optical Vortices

Single nanoparticle tracking using optical microscopy is a powerful technique with many applications in biology, chemistry, and material sciences. Despite significant advances, localizing objects with nanometric position precision in a scattering environment remains challenging. Applied methods to achieve contrast are dominantly fluorescence based, with fundamental limits in the emitted photon fluxes arising from the excited-state lifetime as well as photobleaching. Here, we show a new four-wave-mixing interferometry technique, whereby the position of a single nonfluorescing gold nanoparticle of 25-nm radius is determined with 16 nm precision in plane and 3 nm axially from rapid single-point measurements at 1-ms acquisition time by exploiting optical vortices. The precision in plane is consistent with the photon shot-noise, while axially it is limited by the nano-positioning sample stage, with an estimated photon shot-noise limit of 0.5 nm. The detection is background-free even inside biological cells. The technique is also uniquely sensitive to particle asymmetries of only 0.5% ellipticity, corresponding to a single atomic layer of gold, as well as particle orientation. This method opens new ways of unraveling single-particle trafficking within complex 3D architectures.

Popular Summary

Many complex dynamics and interactions in chemical and biological processes occur at the nanometer scale in under a millisecond. Despite significant advances, rapid tracking of nano-objects in three dimensions with high precision and in an environment that scatters light remains challenging. Fluorescent tags are often used to achieve the necessary optical contrast, but they suffer from fundamental limits in the emitted light. We report on a conceptually new way of precisely and rapidly measuring the 3D position of a single nano-object in a scattering environment.

Our method exploits the strong optical absorption and scattering of a single gold nanoparticle, which is detected in a nonlinear way using a sequence of short optical pulses that generate transient resonant four-wave mixing. This provides a sensitive and specific particle detection that does not rely on fluorescence and is completely background-free. We demonstrate, conceptually and experimentally, that a position precision of better than 20 nm in plane and 1 nm axially, ultimately limited by photon shot noise, is possible from single-point acquisition on a one-millisecond time scale using single gold nanoparticles with radii between 15 nm and 30 nm. The technique is also sensitive to particle asymmetries down to the single atomic layer and to particle orientation.

Our technique paves the way to a new form of single-particle tracking, where not only the particle position but also its asymmetry and orientation are detected, revealing unprecedented information about the particle’s complex dynamics while moving and interacting within a disordered 3D environment.

Figure 1

Four-wave-mixing interferometry epidetected dual-polarization resolved. (a) Sketch of pump pulses that are amplitude modulated (AM) at ${\nu }_m$ and probe pulses that are frequency shifted by ${\nu }_2$, using acousto-optical modulators (AOMs). Pulses are coupled into an inverted microscope equipped with a high NA microscope objective (MO). Pump and probe beams are adjusted to be circularly polarized at the sample by $\lambda /4$ and $\lambda /2$ wave plates. Circular polarizations are transformed into horizontal ($H$) and vertical ($V$) linear polarization by the same wave plates, and both components are simultaneously detected through their interference with a frequency-unshifted reference linearly polarized at 45°. BP: Balanced photodiodes. WP: Wollaston prism deflecting beam out of drawing plane. ($P$)BS: (Polarizing) Beam splitter. $P$: Polarizer. Inset: Amplitude ($A$) and phase ($\mathrm{\Phi }$) of the reflected probe and FWM field, measured by a multichannel lock-in, where $+$ ($-$) refers to the co-(cross-)polarized component relative to the incident circularly polarized probe. (b) Calculated field distribution in the focal plane of a 1.45 NA objective for an incident field left circularly polarized. Here, $A_{+}$ $(A_{-})$ is the co-(cross-)polarized amplitude, and ${\mathrm{\Phi }}_{+}$ (${\mathrm{\Phi }}_{-}$) is the corresponding phase.

Figure 2

Nanometric localization using optical vortices. Calculated amplitude (phase) components of the reflected probe field and FWM field, $A_{2r}^{\pm }$ and $A_{\mathrm{FWM}}^{\pm }$ (${\mathrm{\Phi }}_{2r}^{\pm }$ and ${\mathrm{\Phi }}_{\mathrm{FWM}}^{\pm }$), respectively, as a function of particle position in the sample focal plane $(x,y)$, and in a section along the axial direction $(x,z)$ through the focus, where $+$ refers to the co-polarized component and $-$ to the cross-polarized component relative to the left-circularly polarized incident probe. The calculation assumes a perfectly spherical gold NP in the dipole approximation. The inset shows a sketch of how the amplitude and phase of the FWM field ratio and the phase of the co-polarized FWM field can be used to locate, in 3D, the spatial position of the NP relative to the focus center. The linear grey scale is from $-\pi$ to $\pi$ for all phases, and from $m$ to $M$ for field amplitudes, as indicated. The amplitude ratios of reflected probe and FWM are shown on a logarithmic grey scale over 5 orders of magnitude.

Figure 3

Background-free FWM detection of single NPs in heterogeneous environments. Top panels: Fixed HeLa cells that have internalized gold NPs of 20 nm radius imaged by (a) differential interference contrast (DIC) microscopy, (b) co-circularly polarized reflection amplitude $A_{2r}^{+}$, (c) cross-circularly polarized reflection amplitude $A_{2r}^{-}$, and (d) co-circularly polarized FWM amplitude $A_{\mathrm{FWM}}^{+}$. FWM was acquired with a pump-probe delay time of 0.5 ps, pump (probe) power at the sample of $30\mu \mathrm{W}$ ($15\mu \mathrm{W}$), 2-ms-pixel dwell time, pixel size in plane of 95 nm and $z$ stacks over $6\mu \mathrm{m}$ in 250-nm $z$ steps. FWM is shown as a maximum intensity projection over the $z$ stack, while the reflection is on a single $(x,y)$ plane (scanning the sample position). Bottom panels: Single $(x,y)$-plane image of a 25-nm-radius gold NP in 5% agarose gel, via (e) the co-circularly polarized reflection amplitude $A_{2r}^{+}$, (f) cross-circularly polarized reflection amplitude $A_{2r}^{-}$, and (g) phase ${\mathrm{\Phi }}_{2r}^{-}$, (h) co-circularly polarized FWM amplitude $A_{\mathrm{FWM}}^{+}$, and (i) phase ${\mathrm{\Phi }}_{\mathrm{FWM}}^{+}$. FWM was acquired with 0.5-ms-pixel dwell time and pixel size in plane of 38 nm. Grey scales are linear from $-\pi$ to $\pi$ for all phases and from $m$ to $M$ for field amplitudes, as indicated.

Figure 4

FWM of a single NP attached on glass. We show the amplitude and phase as a function of NP position in the $(x,y)$ plane at different axial positions $z$. The linear grey scale is from $-\pi$ to $\pi$ for phases and from $m$ to $M$ for field amplitudes. The amplitude ratio is on a logarithmic scale over 4 orders of magnitude. Experiment: The top inset shows a sketch of the sample and a TEM image of a typical NP from the batch used. The pump (probe) power at the sample was $18\mu \mathrm{W}$ ($9\mu \mathrm{W}$), with a 3-ms-pixel dwell time and 0.5-ps pump-probe delay time; the pixel size in plane (axial) was 17 nm (75 nm). Calculations assume particle asphericity in plane with semi-axes $a=30.135\mathrm{nm}$ and $b=30\mathrm{nm}$ (see text).

Figure 5

Position localization precision. Difference between the set position of the nanoparticle in 3D and the deduced position using the FWM field amplitude and phase with the coordinate reconstruction parameters as described in the text. The color bar gives the values of this difference in nanometers. The top panels show maps in the $(x,y)$ plane, and the bottom panels are axial $(x,z)$ maps. The cross shows the position of the focus center. (a) Spherical particle without shot noise. (b) Spherical particle with relative shot noise of 0.07% as in the experiment in Fig. 4. (c) Spherical particle with relative shot noise of 0.7%. (d) Asymmetric particle with semi-axis $a=30.135\mathrm{nm}$ and $b=c=30\mathrm{nm}$, as in the simulations in Fig. 4.

Figure 6

Rotational averaging of nanoparticle asymmetry. We show single 25-nm-radius gold NP freely rotating while encaged in an agarose gel pocket (see sketch). (a) Experimental $xy$ scan of the cross- and co-circularly polarized FWM field in the focal plane, using a 1.27 NA water-immersion objective. We show the linear grey scale from $-\pi$ to $\pi$ for phases and from $m$ to $M$ for field amplitudes, with $M$ given relative to the maximum $A_{2r}^{+}$. The amplitude ratio is on a logarithmic scale. The pump (probe) power at the sample was $70\mu \mathrm{W}$ ($10\mu \mathrm{W}$) with pump (probe) filling factor 2.15 (0.97). Measurements were performed with 0.5-ms-pixel dwell time, 0.5-ps pump-probe delay time, and 13-nm pixel size in plane. Data are shown as spatial averages over an effective area of $3\times 3$ pixels. (b) Time traces of the retrieved-particle-position coordinates in 3D from the measured FWM amplitude and phase (symbols) compared with the coordinates from the scanning piezoelectric sample stage (lines). Traces are binned to an equivalent 1 ms acquisition time per point. After subtracting the stage coordinates, histograms of the difference between consecutive points are shown, and the corresponding localization precisions are indicated (see text). (c) Calculated FWM field ratio assuming a polarization tensor that projects the longitudinally polarized field component into the $(x,y)$ plane (see text).

Figure 7

In situ calibration and nanoparticle tracking. (a) Power spectrum of the FWM field ratio showing the 25 Hz oscillation imposed on the sample stage for calibration, as sketched. (b) Retrieved position coordinates versus time of a single 25 nm radius NP in agar (2 ms per point). (c) Axial-position time traces of a single gold NP while jumping between two gel pockets. (d) Zoom of top trace in panel (c), showing the axial position and FWM field ratio dominated by the NP asymmetry; the corresponding in-plane ellipticity and orientation are shown in panel (f). (e) Analysis of axial-position time traces acquired at 0.2 ms per point (see text).