Sub-nanometer 3D morphometric precision of polarisation-resolved wide-field optical extinction microscopy determines the roundness of individual gold nanospheres

TL;DR

The study expands polarization-resolved extinction microscopy by adding radial polarization to the condenser back focal plane, enabling direct probing of NP axial geometry for 3D morphometry. By measuring extinction cross-sections across six wavelengths and both linear and radial polarisations, the authors solve a multi-parameter inverse problem using a Rayleigh–Gans ellipsoid model with a size-dependent surface damping term and a rotated polarizability tensor. The results on 30 nm ultra-uniform gold nanospheres achieve sub-nanometer precision in the ellipsoid semiaxes and sub-5% precision in aspect ratios, with an estimated diameter accuracy improved further when retardation is included via Mie theory. The Olmon permittivity dataset with surface damping $g=1.8$ provides the best fits, highlighting the method’s sensitivity to nanoscale material properties and its potential as a high-throughput, TEM-complementary morphometry tool, capable of probing nonlocal and retardation effects at the nanoscale.

Abstract

Quantitative polarisation-resolved optical extinction microscopy of individual plasmonic nanoparticles has recently been introduced as a powerful tool to characterise the nanoparticle's morphology with a precision comparable to electron microscopy, while using a simple optical microscope [Nanoscale 12, 16215 (2020)]. Here we extend the technique by adding measurements for radial polarisation in the condenser back focal plane, probing plasmonic resonances polarised in axial direction. The combined linear and radial polarisation measurements provide a significantly enhanced precision of the retrieved 3D morphology, as we show on defect-free ultra-uniform gold nanospheres of 30 nm nominal diameter characterised by transmission electron microscopy. The measured cross-sections are quantitatively described by an ellipsoid model, determining the three semi-axes and rotation angles by fitting the measurements. Evaluation the distribution of the fit error across the set of measured particles, the material permittivity dataset and surface damping parameter g providing the best fit are found to be the single crystal dataset by Olmon et al. [Phys. Rev. B 86, 235147 (2012)] and g = 1.8, respectively. The precision of the retrieved aspect ratio is below 5%, and all three ellipsoidal semi-axes are determined with an impressive precision of 0.25 nm. Notably, corrections to the Rayleigh-Gans ellipsoid model due to retardation are significant even though the particle diameters are more than an order of magnitude smaller than the wavelength, and taking them into account improves the accuracy to below a nanometer.

Figures (4)

• Figure 1: Sketch of the microscope set-up and investigated sample. Under Köhler illumination of selectable wavelength range, the sample is imaged onto a camera, using a 1.34 NA condenser and a 1.45 NA objective. In the illumination a rotatable linear polariser or a radial polariser are used, the latter generating a significant axially-polarised illumination component in the focal plane. The sample consists of individual gold nanospheres deposited onto a glass coverslip, surrounded by silicone oil index-matched to glass.
• Figure 2: Measured properties of $N=51$ UGNSs of $D=30\,\mathrm{nm}$ nominal diameter. a) Asymmetry $\alpha_\Lambda$ versus cross-section $\sigma_\Lambda$ for $\Lambda=\{\textcolor{blue}{450\,\mathrm{nm}},\,\textcolor{cyan}{500\,\mathrm{nm}},\,\textcolor{green}{550\,\mathrm{nm}},\,\textcolor{orange}{600\,\mathrm{nm}},\,\textcolor{red}{650\,\mathrm{nm}},\,\textcolor{deepred}{700\,\mathrm{nm}}\}$, color-coded to reflect the center wavelength. Coloured crosses indicate the fitted parameters for linear polarisations using Eq. \ref{['eq:sinfit']}. Coloured vertical bars indicate the measured $\sigma_{\Lambda} (\gamma_\mathrm{R})$ for radial polarisation, at vertical positions chosen for clarity. A representative TEM imagePayneNS20PayneJCP21 of NPs from the same batch is shown on the right. The noise in $\alpha_\Lambda$ is shown as shaded areas for $\Lambda=\textcolor{blue}{450\,\mathrm{nm}}$ (bluish) and $\Lambda=\textcolor{deepred}{700\,\mathrm{nm}}$ (reddish), estimated as $(\hat{\sigma}_\mathrm{ext}/\sqrt{1.5})/\sigma_\Lambda$, corresponding to the lowest and highest noise versus $\Lambda$, respectively. See Sec. \ref{['SM-sec:optnoise']} for further details about the noise in the fit parameters. Solid vertical lines indicate the calculated $\sigma_\Lambda$ for a spherical gold NP for each $\Lambda$, taking into account averaging over filter ranges, for the sample-averaged TEM diameter, $27.94\,\mathrm{nm}$, and the OL permittivity dataset with $g=1.8$. The corresponding spectrum is given in the inset, showing additionally the case $g=0$. Dotted vertical lines indicate the same as solid, but calculated using Mie theory. The corresponding spectrum is shown by the dotted curve in the inset. Vertical colour bands indicate the filter ranges used in the experiment. The measurement error $\hat{\sigma}_\mathrm{ext}$, polarisation-averaged and FOV-averaged, are shown for each $\Lambda$, scaled by a factor $100$. b) Histograms of $\delta_\sigma$, for each $\Lambda$ as labelled, with gray bands showing $\pm\sqrt{2}\,\hat{\sigma}_\mathrm{ext}$, where the $\sqrt{2}$ factor accounts for the additional noise arising from the subtraction in $\delta_\sigma$.
• Figure 3: $\bar{S}$ and selected statistics of $S$ for $N=51$ UGNS of nominal $D=30\,$nm for varying surface damping parameter $g$ and different $\epsilon$ datasets as indicated by color. (a) Top: $\bar{S}$ versus $g$; bottom: $M\{\mathrm{AR}\}$, the median of the fitted average aspect ratio, AR $=\sqrt{bc}/a$. (b--j) Histograms of $S$, with (b--d) using the OL dataset with $g=\{0,\, 1.8,\, 3.6\}$, (e--g) JC dataset with $g=\{0,\, 0.3,\, 1.8\}$, (h--j) MP dataset with $g=\{0,\, 1.8,\, 3.6\}$. The values of $g$ shown are at $g=0$, the minimum of $\bar{S}(g)$ and a larger $g$.
• Figure 4: Morphometric results for $N=51$ nominally $D=30\,$nm UGNSs. (a) Retrieved ARs shown as filled circles with a color indicating the fitted diameter $D_V$. ARs retrieved using only $\sigma_\Lambda(\gamma_\mathrm{P})$ are shown as black crosses. Also shown are retrieved ARs for JC with $g=0.3$, (empty triangles), and for MP with $g=1.8$ (red circles). (b) Mean AR $\sqrt{bc}/a$ as a function of $D_V$, with the histogram showing the number distribution of $D_V$ (magenta filled circles). Data using only $\sigma_\Lambda$($\gamma_\mathrm{P}$) (cyan crosses) are also shown. The black filled squares indicate TEM measurements assuming $b=c$. Histograms are colored according to the corresponding data. Inset: pitch ($\theta$) and roll ($\phi$) angles, to show the out-of-plane orientation ($\theta\neq0$). The size of the circles in (a--b) is given by $S^-=1/(1+S)$, with larger size indicating lower fit error $S$. (c) Simulated effect of the measurement noise on the retrieved ARs for individual NPs indicated by the colored stars in (a--b). Results for 1000 realisations of Gaussian noise with $\hat{\sigma}_\mathrm{ext}$ standard deviation added to the measured extinction data are shown. Inset : $D_V$ histograms, providing a colour scale for the symbols. Results for OL, JC, and MP at $g$ of $1.8$, $0.3$, and $1.8$, respectively, with corresponding results labelled. Black symbols are results using only $\sigma_\Lambda(\gamma_\mathrm{P})$ for OL.