Interfering Plasmons in Coupled Nanoresonators to Boost Light Localization and SERS

Plasmonic self-assembled nanocavities are ideal platforms for extreme light localization as they deliver mode volumes of <50 nm3. Here we show that high-order plasmonic modes within additional micrometer-scale resonators surrounding each nanocavity can boost light localization to intensity enhancements >105. Plasmon interference in these hybrid microresonator nanocavities produces surface-enhanced Raman scattering (SERS) signals many-fold larger than in the bare plasmonic constructs. These now allow remote access to molecules inside the ultrathin gaps, avoiding direct irradiation and thus preventing molecular damage. Combining subnanometer gaps with micrometer-scale resonators places a high computational demand on simulations, so a generalized boundary element method (BEM) solver is developed which requires 100-fold less computational resources to characterize these systems. Our results on extreme near-field enhancement open new potential for single-molecule photonic circuits, mid-infrared detectors, and remote spectroscopy.


S2 Full-wave simulations of different excitation sources
Full-wave simulations (Lumerical) comprising plane-wave and Gaussian beam sources (Fig. S2) are used to model the experimental conditions. In NPoMs (black, Fig. 2a), extreme light confinement exceeds / 0 > 200 with two dominant resonances (10) and (20) for a nanoparticle facet size of 20 nm [1,2]. To better understand the influence of the μ-resonator on light localization, we perform simulations where we consider a bare disk of 6 μm diameter and record the near-field at the center and 5 nm above the surface (red, Fig. S2a). Full analysis of the disk modes is in section S4. We also simulate the near-field of NPoR constructs (dark red). In all cases the nanoparticle is stationary at the disk center with a dielectric spacer of 1.3 nm. The superposition of NPoM and disk modes delivers near-field as high as / 0 ∼ 800 for a wavelength of 750 nm. Thus, it is clear that the high-order modes of the disk drive light localization in the nanocavity. Here we consider no facet for the nanoparticle in the NPoR construct for simplicity. Figure S2. Light confinement of NPoR for plane-wave and Gaussian excitation: a) Simulated near-field of NPoR (dark red), NPoM (black), and bare disk (red) in the gap for plane-wave excitation at 52°. b) Field enhancement for 0.8 NA Gaussian beam at normal incidence. Inset field maps correspond to cross-section along the gap. c) Excitation with Gaussian beam at an angle of 52°. d) Field map cross-sections of c) show modes of NPoR and NPoM plasmonic constructs in the gap. In all cases, the dielectric gap is 1.3 nm thick, disk diameter is 6 μm and scale bar is 10 nm.
As discussed in the main text, experimental findings show slightly reduced SERS contrast compared to theoretical predictions (Fig. S2a). One factor is that plane-excitation, used in our theoretical models, does not correspond to experimental excitation conditions. Consequently, here we examine excitation with high NA Gaussian beams and nanoparticle facets of 20 nm. In all simulations, the excitation point and nanoparticle position is at the center of the μ-resonator. For normal incidence, field enhancement is / 0 ~180 for both plasmonic constructs at a pump wavelength of 633 nm (Fig. S2b). Near-field maps shows that a Gaussian source at normal incidence cannot excite (10) and (20) modes but (21) and (11) instead (see inset, Fig. S2b) [3]. The reduced light localization can be explained if we consider an ideal Gaussian beam passing through a linear polarizer and then focusing through a high NA objective. In this case, the phase across the focal region has opposite sign (Gouy phase), meaning that opposite beam points on the back focal plane of the objective correspond to opposite phase resulting in cancellation of each other when interfering at the focal point ( =0, while and survive). One way to overcome this issue experimentally is to use a radial polarizer before the objective lens. Fortunately, real nanoparticle facets are not perfectly symmetric (containing triangle shape facets and small apexes) thus polarization along the z-axis ( ) survives, giving the SERS signals. This case is not quite our experimental findings, as it gives similar near-fields for NPoR and NPoM constructs (experimental SERS of NPoRs is threefold higher compared to NPoMs). Lastly, we investigate a combination of the two previous cases: oblique excitation (52°) with a Gaussian beam (Fig. S2c). Here light localization is higher in NPoRs compared to NPoMs comprising the same (10) and (20) resonances (Fig. S2d). This example is able to reproduce the modes expected and shows a near-field of NPoRs >3x times higher compared to NPoMs at the pump wavelength (633 nm).

S3 Description of the Computational Method
Our simulations are obtained using an in-house solver based on the Boundary Element Method (BEM) algorithm [4,5]. The BEM algorithm uses the homogeneous space Green function and the planar multilayer Green function, so that NPoRs (modelled as a faceted nanoparticle on top of a disk) and NPoMs (modelled as a faceted nanoparticle on top of an infinitely large mirror) can be appropriately described.
To demonstrate the validity of the solver, we compare total scattering intensities from a NPoR plasmonic system for the BEM solver and a full-wave solver (Lumerical) as the nanoparticle is placed at different radial positions on disk (Fig. S3). Throughout our theoretical discussions, we assume a time harmonic dependence − . The details of the algorithm and the implementation of our in-house solver can be found in [4,5]. For any given system (e.g. NPoR or NPoM constructs) and at a specific wavelength (or frequency), the main equation that we solve is In (1), stands for the system matrix which describes the complete electromagnetic behavior of the system; is the excitation vector which describes how an incident field couples to the system; and is the equivalent source vector which is the source of the scattered waves (i.e., the secondary waves). Here we use the eigenvalue problem of (1), . n n n     Z I I (2) where is the th eigenmode of the system and the corresponding eigenvalue. Clearly, both are independent of excitation. To evaluate how well a mode is coupled with an incident field, we introduce the so-called coupling efficiency, The coupling efficiency is defined as the projection (i.e., the dot product) of the excitation vector onto the th eigenmode. The integration is done with respect to the boundary of the system, .
BEM simulations record the scattering intensity as the diameter of the disk increases from 1 to 6 μm. As expected from their relative areas, the scattering intensity of the NPoR becomes over two orders of magnitude larger than the NPoM (black). In the following, we solve eigenvalue problem in (2) (4) In (4), , , and stand for the matrices describing the self-coupling of the particle, the mutual coupling between particle and disk, the coupling between disk and particle, and the self-coupling of the disk, respectively. and are the excitation vectors describing the incident field at the position of the nanoparticle and the disk. and are the equivalent sources induced at the nanoparticle and the disk. Instead of directly solving the eigenvalue corresponding to the entire system in (4), we focus on the nanoparticle and transform (4) into . p   Z I V (5) In (5), we define 11 , .
pp pd dd dp p pd dd d The eigenvalue problem for (5) reads The coupling efficiency is written as ,. We solve (7) and (8) for the NPoR construct. The following data provide a complementary view to Fig. 2b. . Blue (yellow) stands for negative (positive) charge. The coupling efficiency is calculated for a polarized plane-wave excitation at optimal incident angle of 52°.

S4. Resonances in the Disk
To model the disk modes for a finite thickness , we use an approach similar to [6] by assuming that the resonances are due to the reflection of SPPs at the boundary of the disk. We consider a disk (with permittivity 1 ) immersed in a homogeneous environment of permittivity 2 and that the center of the disk is the origin of the employed coordinate system. Inside the disk, the component of the electric field ( ) takes the following form,