Letter Cite This: ACS Photonics XXXX, XXX, XXX−XXX
Three-Dimensional Multipole Rotation in Spherical Silver Nanoparticles Observed by Cathodoluminescence Zac Thollar,† Carl Wadell,† Taeko Matsukata,† Naoki Yamamoto,† and Takumi Sannomiya*,†,‡ †
Department of Materials Science and Engineering, School of Materials and Chemical Technologies, Tokyo Institute of Technology, 4259 Nagatsuta, Midoriku, Yokohama, 226-8503 Japan ‡ JST, PRESTO, 4259 Nagatsuta, Midoriku, Yokohama, 226-8503 Japan S Supporting Information *
ABSTRACT: A spherical metallic nanoparticle is the simplest and most frequently used example of plasmonic nanostructures. In such a highly symmetric structure the plasmon modes consist of degenerate multipoles, which cannot be separately observed by only utilizing energy resolved means. We here demonstrate nanoscale optical field mappings of degenerate multipole modes in spherical silver nanoparticles using an angle- and polarization-resolved cathodoluminescence technique combined with scanning transmission electron microscopy. By properly selecting the detection angle and polarization, the observed optical field maps of spherical silver nanoparticles exhibit dipole or quadrupole features, depending on the energy. The angle-dependent multipole patterns visualize their three-dimensional rotation, which can be explained as superposition of the degenerate modes due to the spherical symmetry. KEYWORDS: cathodoluminescence, spherical nanoparticle, multipole, surface plasmon, scanning transmission electron microscopy ptical field enhancement of a plasmonic nanoparticle antenna has been of interest for various applications, such as photocatalysis, biosensoing, and light energy conversion.1,2 Of the different particle types, spherical nanoparticles are most widely used since their production based on colloid synthesis is simple, reliable and controllable, suited for both industrial application and research targets.3−5 The spherical shape, which has the highest geometrical symmetry, is also advantageous as an isotropic antenna that works equally in all directions. According to Mie theory, the electromagnetic field of a spherical particle can be expanded in the angular directions in polar coordinates as a sum of multipoles.6 Typically the most fundamental dipole mode is utilized because it can easily couple to a plane wave due to symmetry matching. Higher order modes, although more difficult to excite by plane waves, have higher Q-factors and a more confined electromagnetic field on the surface compared to the dipole mode.7 Higher order modes may also be more efficiently excited by a plane wave when coupled to a dipole.8 It has also been proposed that directionality from a single nanoanntena can be achieved by combining dipole and quadrupole modes.9 In all cases, it is important to precisely control the radiation properties and field enhancement of each mode in order to realize an efficient nanoantenna. Therefore, direct comparison of radiation properties and corresponding field distribution of multipole modes is necessary. To spatially resolve the optical fields at nanoscales, various techniques are available such as scanning nearfield optical microscopy (SNOM), photoelectron emission microscopy
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© XXXX American Chemical Society
(PEEM), and electron microscopy based spectroscopy.10,11 Among these measurement techniques, electron microscopy based methods offer the highest spatial resolution with simultaneous spectroscopy, which can be categorized into two different spectroscopy methods, namely electron energy loss spectroscopy (EELS) and cathodoluminescence (CL).12,13 Electron microscopy based techniques are advantageous also because high resolution images showing structural information can be simultaneously obtained. To resolve mutipole modes in highly symmetric structures with degenerate modes, CL is most suited because of its light polarization ability and the possibility to carry out angle-resolved measurements.14−17 In this study, we experimentally measure the optical field distribution and radiation direction of selected multipole modes using angle-resolved CL equipped to a scanning transmission electron microscopy (STEM) instrument. We focus on spherical nanoparticles to investigate rotation of degenerate modes. This study also gives fundamental understanding of how the modes are excited and detected when the degenerate modes rotate.
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RESULTS AND DISCUSSION In the CL technique, the measured signal is proportional to the radiative component of the electromagnetic local density of Special Issue: Recent Developments and Applications of Plasmonics Received: October 30, 2017 Published: December 20, 2017 A
DOI: 10.1021/acsphotonics.7b01293 ACS Photonics XXXX, XXX, XXX−XXX
Letter
ACS Photonics state (EMLDOS) parallel to the electron path.18 The electromagnetic radiation from a spherical particle can be described as a sum of radiation from each electric multipole mode.19 The magnetic field HSm and electric field ESm of the electric multipole components in the far field (kr ≫ 1) can be expressed as HSm = ( −i)S + 1
eikr LYSm(θ , φ) kr
complex coupling constant CSm for each mode. The charge distributions and radiation patterns for three dipole (S = 1) and five quadrupole (S = 2) basis functions are shown in Figure 1. Detailed derivation and expression of the formula are in the Supporting Information.
(1)
ESm = Z0HSm × n
(2)
where k is the wavenumber, r is the position vector from the origin, L = −ir × ∇ is the orbital angular-momentum operator, YSm is a set of spherical harmonics, Z0 is the impedance of vacuum, and n is the unit vector in the radial direction. By introducing vector spherical harmonics 1 XSm(θ , φ) = LYSm(θ , φ) , the time-averaged Poynting S(S + 1)
vector ⟨S⟩ of the radiative field for each mode can be described as Z 1 Re[ESm(r ) × H *Sm(r )] = 0 |HSm(r )|2 n 2 2 Z0 S(S + 1) 2 = |XSm(θ , φ)| n 2(kr )2
⟨S(r )⟩ ∝
(3)
We now introduce basis functions for vector spherical harmonics as follows: XS+m =
1 (XSm + XS − m) = XS+m, θ eθ + XS+m, ϕeϕ 2
(4)
XS−m =
1 (XSm − XS − m) = XS−m, θ eθ + XS−m, ϕeϕ 2i
(5)
XS+m
XS−m
Both and become purely real or imaginary vectors * since XS − m = XSm for m = odd and XS − m = − XS*m for m = even. θ and φ vector components are indicated with superscripts. eθ and eφ are the unit vectors in the θ and φ directions, respectively. Since X is proportional to the magnetic component, the corresponding electric field component E can be obtained by taking vector products of X and n. Considering that the radiation intensity I is proportional to the amplitude of the time-averaged pointing vector ⟨S⟩ for each degenerate mode, the total angle-dependent radiation intensity I with different polarization components for given S can be expressed as
Figure 1. Angle-dependent amplitude plots of ES±m for dipoles (S = 1) and quadrupoles (S = 2) with different polarizations (nonpolarized, θpolarized or φ-polarized) of the radiated light. For the corresponding charge distribution plots, the color scale is applied to a sphere.
To perform the mode selective observation, we used angular resolved STEM-CL. The used STEM setup consists of a parabolic mirror inserted at the sample position to collimate the light radiated from the sample. The light is transferred to a spectrometer through an optical lens system as schematically illustrated in Figure 2.14 By placing a pinhole mask in the beam path, it is possible to select the light emission angle and perform angle-resolved measurement. We note that p-polarization corresponds to θ-polarization in eqs 6−8, and spolarization to φ-polarization, when the pinhole mask position is close to the Y or Z axis.14,20 Spherical silver particles were deposited by thermal evaporation in argon atmosphere on ultrathin carbon films with a SiO2 layer (thickness ∼ 100 nm) to achieve mechanical and electrical stability (Figure 2b). The coordinate definitions on the sample and in the instrument are indicated in Figure 2a,c. The experimental details are found in Method section. We first aim to selectively observe in-plane multipoles with their charge distributions parallel to the substrate (X−Y plane), corresponding to |m| = S modes in eqs 6−8 and in Figure 1. Such multipole modes have no radiation in the Z direction except for the dipole mode (S = 1). Therefore, higher order inplane modes cannot couple to normal-incident plane waves
ISnon‐pol(θ , ϕ) 2
∝ CS0ES0(θ , ϕ) +
∑
{CS+mES+m(θ , ϕ) + CS−mES−m(θ , ϕ)}
(6)
0
