pubs.acs.org/journal/apchd5 Article Plasmonic Nanocavity Modes: From Near-Field to Far-Field Radiation Nuttawut Kongsuwan, Angela Demetriadou, Matthew Horton, Rohit Chikkaraddy, Jeremy J. Baumberg,* and Ortwin Hess* Cite This: https://dx.doi.org/10.1021/acsphotonics.9b01445 Read Online ACCESS Metrics & More Article Recommendations *sı Supporting Information ABSTRACT: In the past decade, advances in nanotechnology have led to the development of plasmonic nanocavities that facilitate light−matter strong coupling in ambient conditions. The most robust example is the nanoparticle-on-mirror (NPoM) structure whose geometry is controlled with subnanometer precision. The excited plasmons in such nanocavities are extremely sensitive to the exact morphology of the nanocavity, giving rise to unexpected optical behaviors. So far, most theoretical and experimental studies on such nanocavities have been based solely on their scattering and absorption properties. However, these methods do not provide a complete optical description of the nanocavities. Here, the NPoM is treated as an open nonconservative system supporting a set of photonic quasinormal modes (QNMs). By investigating the morphology-dependent optical properties of nanocavities, we propose a simple yet comprehensive nomenclature based on spherical harmonics and report spectrally overlapping bright and dark nanogap eigenmodes. The near-field and far-field optical properties of NPoMs are explored and reveal intricate multimodal interactions. KEYWORDS: plasmonics, nanophotonics, nanocavities, quasinormal mode, near-to-far-field transformation etallic nanostructures have the ability to confine light resonances from far-field spectral peaks. Although significant M below the diffraction limit via the collective excitation of information can be obtained from the far-field spectra, they do conduction electrons, called localized surface plasmons. not reveal complete optical descriptions of the nanocavities Through recent advances in nanofabrication techniques, gaps and commonly hide information about their dark modes. For of just 1−2 nm between nanostructures have been achieved.1,2 example, an incident field from the far-field does not couple to At such extreme nanogaps, the plasmonic modes of two all available photonic modes of the system. Resonances that are nanostructures hybridize to allow an unprecedented light spectrally close also interfere with each other and often merge Downloaded via UNIV OF CAMBRIDGE on February 3, 2020 at 16:45:04 (UTC). confinement,3,4 making coupled nanostructures an ideal into single broadened peaks in far-field spectra. As a result, platform for field-enhanced spectroscopy,5,6 photocatalysis,7 analyses of the far-field scattered spectra can yield incon- 8 See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. and nano-optoelectronics. One such nanostructure is the sistencies between near-field and far-field plasmonic re- nanoparticle-on-mirror (NPoM) geometry, where a nano- sponses.28 particle is separated from an underlying metal film by a The precise morphological details of the NPoM nanogaps molecular monolayer.9,10 This geometry (which resembles the also dramatically modify their optical response.17,29 Once the prototypical dimer, but is more reliable and robust to fabricate) nanoparticles are placed on substrates, they lie on their facets, has attracted considerable interest since it enables light−matter which can have varying facet widths. Previous studies of the strong-coupling of a single molecule at room temperature,11 gap morphology of NPoMs described their gap plasmonic and it has many potential applications, including biosensing12 resonances with two sets of modes: longitudinal antenna and quantum information.13,14 modes, excited for all facet widths, and transverse waveguide A wide range of theoretical and experimental studies have modes, excited at large facet widths.18,19 Although this been conducted to investigate the optical properties of NPoM description sheds light on the asymptotic behavior of the nanocavities.12,15,16 Several studies examine resonances of 17−21 NPoMs and their influence on optical emission of single Received: October 5, 2019 22−24 molecules in the nanogaps. However, only a few studies Published: January 9, 2020 to date analyze the inherent resonant states of the NPoM − nanocavities,25 27 and most other studies have so far described their optical responses via a scattering method, inferring their © XXXX American Chemical Society https://dx.doi.org/10.1021/acsphotonics.9b01445 A ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics pubs.acs.org/journal/apchd5 Article Figure 1. (a) Schematic for a gold nanoparticle-on-mirror (NPoM) structure with the diameter 2R = 80 nm, nanogap d = 1 nm, nanogap index ngap fi = 1.45, background index nb = 1, and facet width w. (b, c, d, e) Normalized quasinormal-mode (QNM) electric elds Ez,Sm in the vertical xz-plane of NPoMs with (b, c) w = 0 and (d, e) w = 20 nm for (b, d) the (10) mode and (c, e) the (11) mode. The gray dashed lines indicate the facet edges. NPoM plasmons, it provides an incomplete picture of NPoM materials, its QNMs can be found by solving the time- resonances at intermediate facet widths. harmonic and source-free Maxwell’s equations30 In this Article, we identify and map the inherent resonant states of the NPoM geometry by employing the quasinormal 0 −∇×iμ −1 Hr̃ () Hr̃ () mode (QNM) method.30 We propose a simple yet 0 i i = ωĩ comprehensive nomenclature based on spherical harmonics −1 Er̃ () Er̃ () iεω(;r ĩ )∇× 0 i i (1) to describe the resonances of NPoMs with varying i yi y i y morphologies. A collection of spectrally overlapping bright j ε ω̃ zj z ̃j z̃ wherej (r; i) is the permittivityzj andz Hi jand Ezi are the and dark nanogap QNMs are reported, including some j zj z j z magneticj and electric fieldszj of a QNMz j that satisfyz the photonic modes that have not been reported elsewhere. k {k { k { These results imply that a quantum emitter placed inside a Sommerfeld radiation condition for outgoing waves. NPoM nanogap couples to a collection of QNMs and For dispersive materials like metals, eq 1 is, in general, a experiences a complex multimodal interaction. By calculating nonlinear eigenvalue equation. However, eq 1 can be the far-field emission of each QNM using a near-to-far-field converted into a linear equation if the material permittivities transformation (NFFT) method, we predict the total emission are described by an N-pole Lorentz−Drude permittivity profile of a dipole emitter placed inside a NPoM nanogap from the QNM collective responses. The resulting emission profiles 2 N ωpk, display rich and intricate behaviors, governed by the NPoM ε()ωε=− 1 ∑ ∞ ωω2 −+2 γω morphology. k=1 0,k i k (2) Eigenmodes of Plasmonic Nanocavities. A general i y open system is nonconservative as its energy leaks to its where ε∞ is thej asymptotic permittivity atz infinite frequency, j z surrounding environment, and therefore, its time-evolution is while ω , ωj , and γ are the plasmaz frequency, resonant p,k 0,kk k { non-Hermitian. Consequently, its resonances can no longer be frequency, and decay rate, respectively, of the kth Lorentz− described by normal modes but instead are characterized by − 31 Drude pole. For each Lorentz Drude pole in the summation, QNMs with complex eigenfrequencies. The QNM analysis is fi 27 a standard methodology to study open and dissipative systems, two auxiliary elds can be introduced of which the approximate descriptions are often provided by a 2 εω∞ pk, few QNMs. This approach has spanned a wide range of Pr̃ ()=− Er̃ () applications, including gravitational waves from black holes32 ik, 2 2 i ωωik̃ −+0, i γωk ĩ (3) and electromagnetic waves from nanoresonators.30 In nanophotonics, significant progress has been made in the ̃ ω ̃ past decades toward solving QNMs for general dispersive Jik, ()rPr=−i iik̃ , () (4) materials. Efficient QNM solvers have been developed using a variety of techniques, including the time-domain approach,33,34 ̃ ̃ − where Pi,k and Ji,k are auxiliary polarization and current, the pole-search approach,35 37 and the auxiliary-field eigenval- th th − 27 respectively, of the i QNM and the k Lorentz Drude pole. ue approach. For resonators with arbitrary shapes and In the case of a single pole permittivity (N = 1), the auxiliary materials, analytic solutions are not generally available, and fields can be inserted into eq 1 to obtain a linear eigenvalue several numerical methods have been developed that surround the resonators by perfectly matched layers (PMLs) to equation fi 27,38 approximately simulate in nite domains. ̂ Here, we represent the resonances as QNMs with complex /ψωĩ = ĩ ψĩ (5) ω̃ ω − κ ω frequencies i = i i i, where the real part, i, is the spectral position and the imaginary part, κ , is the line width, that is, the i ψ̃ = (()()()())HrErPrJr̃ ̃ ̃ ̃ T dissipation rate. For a general optical system with nonmagnetic i iiii (6) B https://dx.doi.org/10.1021/acsphotonics.9b01445 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics pubs.acs.org/journal/apchd5 Article fi fi fi − Figure 2. Normalized QNM electric elds Ez,Sm in the horizontal xy-plane at the nanogap center for the rst ve QNMs of NPoMs with (a e) w = − 0 nm and (f j) w = 20 nm. The real eigenfrequencies ωSm in eV are shown at the bottom of all panels. 000−∇×iμ −1 description, based on mode symmetries, which is valid for all 0 facet widths. −11− Figure 1b,c shows the real z-component of QNM fields ̂ iiεε∞ ∇×00 − ∞ / = ̃ i y Ezm,SS=[Re Ee m· ẑ ] for the two lowest-eigenfrequency QNMs, j00 0i z j z denoted as (10) and (11), of a spherical NPoM (w = 0 nm). At j 2 2 z fi fi j0 iiiωεp ∞ −− ω0 γ z the nanogap, these QNMs exhibit large eld con nement far j z (7) j z below the diffraction limit, which is the main characteristic of j z The treatmentj of gold permittivity is given inzsection S1.1 in gap plasmons.