Orientation averaging of optical chirality near nanoparticles and aggregates

Atefeh Fazel-Najafabadi, Sebastian Schuster, and Baptiste Auguié

Phys. Rev. B 103, 115405 – Published 4 March 2021

Abstract

Artificial nanostructures enable fine control of electromagnetic fields at the nanoscale, a possibility that has recently been extended to the interaction between polarised light and chiral matter. The theoretical description of such interactions, and its application to the design of optimised structures for chiroptical spectroscopies, brings new challenges to the common set of tools used in nano-optics. In particular, chiroptical effects often depend crucially on the relative orientation of the scatterer and the incident light, but many experiments are performed with randomly oriented scatterers, dispersed in a solution. We derive new expressions for the orientation-averaged local degree of optical chirality of the electromagnetic field in the presence of a nanoparticle aggregate. This is achieved using the superposition $T$ -matrix framework, ideally suited for the derivation of efficient orientation-averaging formulas in light scattering problems. Our results are applied to a few model examples and illustrate several nonintuitive aspects in the distribution of orientation-averaged degree of chirality around nanostructures. The results will be of significant interest for the study of nanoparticle assemblies designed to enhance chiroptical spectroscopies, and where the numerically efficient computation of the averaged degree of optical chirality enables a more comprehensive exploration of the many possible nanostructures.

Received 17 December 2020
Revised 17 February 2021
Accepted 18 February 2021

DOI:https://doi.org/10.1103/PhysRevB.103.115405

Physics Subject Headings (PhySH)

Chirality Electromagnetism Near-field optics

Nanoparticles

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Atefeh Fazel-Najafabadi ^1,2,*, Sebastian Schuster ^1,2,3,†, and Baptiste Auguié^1,2,‡

¹School of Chemical and Physical Sciences, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
²The MacDiarmid Institute for Advanced Materials and Nanotechnology, P.O. Box 600, Wellington 6140, New Zealand
³School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand

^*atefeh.fazelnajafabadi@vuw.ac.nz
^†sebastian.schuster@sms.vuw.ac.nz
^‡baptiste.auguie@vuw.ac.nz

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Vol. 103, Iss. 11 — 15 March 2021

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Images

Figure 1
Schematic illustration of the light scattering problem under consideration. A rigid cluster is formed by $N$ particles arranged in a fixed spatial configuration; each particle may be nonspherical and oriented arbitrarily with respect to the common “cluster” reference frame $(O, x, y, z)$ . The orientation and position $r_{i}$ of each particle ( $i = 1 \dots N$ ) is arbitrary but fixed in this reference frame; in practice such particles would be held in place by a template [28, 29, 30], our calculations assume that the template has no impact on the optical properties of the structure. Incident light takes the form of a plane wave with wave vector $k$ from an arbitrary direction $(φ, θ)$ in the cluster reference frame, and we consider both possible states of circular polarization (left, $L$ , or right $R$ ). For each direction of incidence and either polarization we can calculate the local degree of chirality $C$ in the vicinity of the particles, and map its spatial distribution (schematically represented by purple and green gradients). The quantity $〈 C 〉$ derived in this work represents the average value for the chosen polarization when the cluster as a whole is randomly oriented with respect to incident light, or equivalently, when $k (φ, θ)$ spans the full solid angle in the cluster reference frame.
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Figure 2
(a), (b) 2D color maps of the normalized orientation-averaged local degree of optical chirality $〈 \bar{C} 〉$ around a single Si sphere with radius $R = 50 nm$ excited at $λ = 480 nm$ (a), and a Ag sphere with $R = 30 nm$ at $λ = 468 nm$ (b). Note that the patterns are spherically symmetric. (c)–(e) Corresponding plot of the radial dependence of $〈 \bar{C} 〉, 〈 | E |^{2} 〉, 〈 | B |^{2} 〉$ with origin at the surface of the sphere. Note that $〈 | E |^{2} 〉$ is displayed on a log scale to better compare the relative decay profiles, and that we scale the magnetic field by the speed of light $c = 299 792 458 m / s$ to retain the same order of magnitude as the electric field. (f) Radial dependence of the phase angle between $E$ and $B$ at the location $P$ along the $x$ direction as shown in the inset, for three particular incidence directions, $k_{x}$ (solid line) and $k_{y}$ or $k_{z}$ (dashed line, both directions are equivalent).
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Figure 3
(a) Map of the normalized local degree of optical chirality $〈 \bar{C} 〉$ for a dimer of silver spheres with radius $R = 30 nm$ and gap $5 nm$ , calculated on a 2D plane passing through the sphere centres. The wavelength of excitation is $λ = 413 nm$ , corresponding to the peak extinction. (b) Same configuration as in (a) for a dimer of 50-nm-radius Si spheres with $5 nm$ gap, illuminated at $λ = 490 nm$ (peak extinction). (c), (d) Calculated $〈 \bar{C} 〉$ spectra at four different locations around each dimer, as depicted in the inset. The point M is in the middle of the gap and points A, B, C are $0.1 nm$ away from the nanoparticle's surface. (e), (f) Surface-averaged of $〈 \bar{C} 〉$ across the particles' surface for each dimer. The inset in (f) traces the maximum surface-averaged $〈 \bar{C} 〉$ for a Si dimer with varying sphere radius (20–200 nm) and constant gap ( $5 nm$ ).
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Figure 4
Predicted spectrum of the local degree of optical chirality at the point $P$ half-way between the last two particles, as depicted in the inset. The structure consists of a helix of five prolate gold spheroids, with helix axis $z$ , radius $100 nm$ , pitch $700 nm$ , angular-step $δ = π / 4$ . The spheroids are oriented along the helix, and have semi-axes $a = b = 30 nm$ and $c = 50 nm$ . The structure is illuminated with a right circularly polarized plane wave along the $x$ direction (blue dashed line), $y$ direction (green dashed line), or $z$ direction (purple dashed line). The average over these three incident directions (red dashed line) is compared to the rigorous orientation-averaged value $〈 \bar{C} 〉$ (solid black line).
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