A neural network approach to kinetic Mie polarimetry for particle size diagnostics in nanodusty plasmas

TL;DR

The paper tackles automated retrieval of nanoparticle radius $a$ and complex refractive index $n$ from kinetic Mie polarimetry data in nanodusty plasmas. It introduces HERMiNe, a hybrid 1D-CNN/LSTM network that maps the evolution of ellipsometric data $(\\Psi,\\Delta)$ to $(\\mathrm{Re}(n),\\mathrm{Im}(n))$, enabling direct inference of radius via the reference curve. Training on 247,500 synthetic $\\Psi$-$\\Delta$ curves with Mie theory, and validated by Monte Carlo error estimation and non-linear growth tests, demonstrates robustness and uncertainty quantification. Applied to experimental data, HERMiNe achieves comparable accuracy to CRAS-Mie fitting but with less spread and a roughly 7x speedup, paving the way for automated, real-time imaging of nanoparticle growth in nanodusty plasmas.

Abstract

The analysis of the size of nanoparticles is an essential task in plasma technology and dusty plasmas. Light scattering techniques, based on Mie theory, can be used as a non-invasive and in-situ diagnostic tool for this purpose. However, the standard back-calculation methods require expertise from the user. To address this, we introduce a neural network that performs the same task. We discuss how we set up and trained the network to analyze the size of plasma-grown amorphous carbon nanoparticles (a:C-H) with a refractive index n in the range of real(n) = 1.4-2.2 and imag(n) = 0.04i-0.1i and a radius of up to several hundred nanometers, depending on the used wavelength. The diagnostic approach is kinetic, which means that the particles need to change in size due to growth or etching. An uncertainty analysis as well as a test with experimental data are presented. Our neural network achieves results that agree with those of prior fitting algorithms while offering higher methodical stability. The model also holds a major advantage in terms of computing speed and automation.

Figures (8)

• Figure 1: Left: Definition of the ellipsometric angles $\Psi$ and $\Delta$ on the polarization ellipse. The oscillating electric field vector $E(t)$ has a component parallel ($\parallel$) and perpendicular ($\perp$) to the scattering plane. The phase difference between those is depicted by $\Delta$, whereas the ratio of the corresponding amplitudes ($E_{0,\parallel},E_{0,\perp}$) gives $\tan(\Psi)$. Right: Exemplary $\Delta(\Psi)$ curves for a variety of color-coded refractive indices $n$, calculated over a range of particle sizes $a$ from Mie theory using a wavelength of $\unit[663]{nm}$. For each curve either the imaginary or real part of $n$ is held constant. The dashed black lines represent hypersurfaces of constant $a$. The curve sections plotted with continuous lines indicate the training domain of our network. 0
• Figure 2: Network Layer Graph. The network consists of 1D convolutional, long short-term memory (LSTM), dropout and fully connected layers of depicted unit quantity. The kernel size of the convolutions is shown as bold numbers, the dropout layer's probability is given in percentage. As a sequence, the $\Delta(\Psi)$ curve is first fed through three differing branches. The last LSTM layer group outputs one scalar value per unit, which are added element-wise along the branches and further reduced afterwards. The final outputs are the predicted components of the complex refractive index $n$, which, in turn, unambiguously define a $\Delta(\Psi(a))$ curve as reference.
• Figure 3: Network parameter domain and training conditions.
• Figure 4: Monte Carlo error estimates for refractive index and particle radius on each point of the trained parameter space. (a) Exemplary analysis of the Monte Carlo method for the refractive index $1.7+i0.07$: The gray dots depict the ensemble of simulated curves as their radius distributions $a_\mathrm{MC}$ and respective reconstructed values $a_\mathrm{rec}$ from the neural network predictions. The continuous red line marks the identity, whereas the black dashed lines mark the $\unit[95]{\%}$ confidence interval around it. The mean deviation of the ensemble is indicated by the blue dot and dash line. (b) Standard error $\delta$ of the mean for real (Re) and (c) imaginary (Im) part of refractive index prediction. (d) Mean ensemble deviation of reconstructed particle radius. (e) Width of the $\unit[95]{\%}$ confidence interval (CI) of the reconstructions.
• Figure 5: Network's Response to simulated non-linear particle growth. Left: Saturating progressions of particle radiusF over time. The color-coded scatter plots show the original simulated courses. Black lines represent the smoothed recalculation from the network's prediction. The part of the data sequence which was fed into the network ends with the vertical mark. Right:$\Delta(\Psi)$ curves for each course calculated from Mie theory with $1.7+i0.08$ and $\unit[1]{deg}$ Gaussian noise, plotted on top of each other. The colors match those on the right side. As the final particle size gets larger and larger, the $\Delta(\Psi)$-series progresses further. The black continuous lines show the refractive indices estimated by the network, depicted as what the corresponding $\Delta(\Psi)$ curves would look like.