Optimal conditions for detecting optical dichroism at the nanoscale by electron energy-loss spectroscopy

Abstract

The emergence of optical circular dichroism in chiral nanoscale and molecular systems provides not only a way for analyzing the sample chirality itself but also additional degrees of freedom in manipulating light. Such manipulation can be reached even at the nanoscale level; however, probing and understanding the properties of optical fields well below the diffraction limit requires an adequate technique. Electron energy-loss spectroscopy (EELS) with orbital angular momentum (OAM)-based electron state sorting has been suggested as a suitable candidate, but to date, no conclusive experiments have been performed. We, therefore, theoretically explore the emergence of dichroism in EELS for a canonical single-twist helix nanostructure and present a detailed analysis of the optimal parameters to obtain a robust signal. Our work offers novel insights into the interpretation and volatility of the OAM-resolved EELS signal, which can inspire and guide future experimental efforts.

Figures (11)

• Figure 1: Overview of proposed experiment: (a) A scheme of the setup. Electrons from the source (1) in the superposition of different VEB states (2) pass through an OAM pre-filter (3). A VEB with a well-defined OAM is prepared (5) and the states with different-than-chosen OAM are sorted out (4). A VEB passing near the sample can excite a plasmon (6) and scatter (7). After the interaction, the beam is in a statistically mixed state (8). An OAM sorter (9) is used to separate the mixed states. The electrons are further dispersed in energy by the EELS spectrometer (10). (b) Pseudo-charge density and phase of the projected eigenpotentials $w_n$ corresponding to the first three modes of a silver helix with thickness $2 \xi=20\,\mathrm{nm}$, pitch $d=100\,\mathrm{nm}$, and radius $R_0=50\,\mathrm{nm}$. (c) Calculated OAM-resolved EELS image for the sample from (b) illuminated by a non-vortex Gaussian beam ($\ell_\mathrm{i}=0$, $\mathrm{s}_0=10$ nm, $\alpha_\mathrm{f}=5$ mrad). The image is smeared in the (vertical) OAM axis by a Gaussian to illustrate the finite resolution of the OAM sorter. (d) A cut-through for direct comparison between spectra obtained for final OAMs $\pm \hbar$ and $\pm 2\hbar$, respectively.
• Figure 2: Relative dichroism dependence on the parameter $t_n(\omega_n,d,v)$ for the first three modes (each column corresponds to one mode). The upper row (a1--a3) is calculated for $\Delta\ell=1$ while the lower row (b1--b3) for $\Delta\ell=2$. Notice the similarities in panels (a1) and (b1--b3) and a qualitatively different behaviour in (a2) and (a3). The values of $t_n$ for maximal and minimal $\mathcal{RD}$ are displayed in (a2, a3), and (b2). Shaded regions depict relevant $t_n$ intervals for the first three modes studied in Fig. \ref{['fig:Fig2']}(b1--d3).
• Figure 3: Dichroism for an infinitesimally thin silver single-twist helix [$R_0=50$ nm, a thickness of $\xi=10$ nm was used to determine the normalisation constant $\sigma_n$ from MNPBEM simulation, pitch is $d=100$ nm in (a) and varies in (b1--d3)] perfectly aligned with an electron beam [waist $\mathrm{s}_0=$10 nm, energy is 30 keV in (a) 20--60 keV in (b--d)]. We consider the collection of all electrons, thus $Q_\mathrm{c}\rightarrow\infty$. (a) EEL spectra for $\ell_\mathrm{i}=0$ and positive ($\Gamma_0^1$; yellow) and negative ($\Gamma_0^{-1}$; green) OAM exchange with corresponding absolute dichroism ($\mathcal{AD}_0^1$; dashed black) and relative dichroism ($\mathcal{RD}_0^1$; maroon). (b) Absolute dichroism for the first three modes as a function of the helix pitch and electron energy for the OAM transition $0\rightarrow\pm1$. The contour step is $5\cdot10^{-4}\,\mathrm{eV}^{-1}$, and the blue rectangles indicate the values of the absolute dichroism in (a). (c) Same as (b) for $\ell_\mathrm{i}=1$ and the transition in OAM $1\rightarrow\pm2$. (d) Same as (c) but for the OAM transition $1\rightarrow\pm3$. The contour steps are $5\cdot10^{-7}$, $5\cdot10^{-5}$ and $5\cdot10^{-5}\,\mathrm{eV}^{-1}$ respectively. Ticks in colorbars mark the contour levels.
• Figure 4: Absolute dichroism $\mathcal{AD}_{0}^{1}$ for an infinitesimally thin silver single-twist helix [$R_0= 50$ nm, $\xi=10$ nm, $d=100$ nm in (c,d) and $d=0$ nm in (e)] probed by a 30 keV Gaussian electron beam with $\mathrm{s}_0=10$ nm, and $\alpha_\mathrm{f}=5$ mrad when the electron beam is scanned or tilted. (a) Scheme of the "scanning" simulation, where the beam's trajectory is parallel to the helix axis, impinging at position ${\bf R}_\mathrm{b}=(x_\mathrm{b}, y_\mathrm{b})$. $\mathcal{AD}$ is plotted in the $20\times20$ nm computation area in (c1--c3) for the first three modes. The contour steps are $\{1;2;0.5\}\cdot10^{-4}\,\mathrm{eV}^{-1}$ respectively. (b) Scheme of the "tilting" simulation for the beam tilt represented in the spherical coordinate system by angles $(\theta;\varphi)$. The electron beam passes through the origin of the coordinate system. Each "pixel" in the circular "lid" represents a unique impinging direction of the beam. The results are computed for polar angles $\theta$ ranging from $0\degree$ to $5\degree$ with a step $1\degree$. The azimuthal dependence is divided into 12 parts. (d1--d3) $\mathcal{AD}_0^1$ for the first three modes when tilting the beam. (e1--e3) Same as (c1--c3) but for a "C-shaped" structure ($d=0$ nm) exhibiting extrinsic dichroism.
• Figure S1: Extended calculations related to Fig. 4 of the main text: absolute and relative dichroism for an achiral thin silver "C-shaped" object with radius $R_0=50$ nm (pitch $d=0$ nm) exposed to a tilted beam. The tilt is defined in the same way as in Fig. 4(c). (a1--a3) show the dependence for the incident Laguerre-Gaussian beam with $\ell_\mathrm{i}=1$ and $\Delta\ell=1$. (b1--b3) Same as (a1--a3) for $\ell_\mathrm{f}=3$ and $\Delta\ell=2$. The lower row shows relative dichroism for $\Delta\ell=1$ [(c1--c3)] and $\Delta\ell=2$ [(d1--d3)]. In all cases, the electron beam has parameters $\mathrm{s}_0=10$ nm, energy 30 keV, and we consider a collection angle $\alpha_\mathrm{f}=5$ mrad.