Boosting Optical Nanocavity Coupling by Retardation Matching to Dark Modes

Plasmonic nanoantennas can focus light at nanometer length scales providing intense field enhancements. For the tightest optical confinements (0.5–5 nm) achieved in plasmonic gaps, the gap spacing, refractive index, and facet width play a dominant role in determining the optical properties making tuning through antenna shape challenging. We show here that controlling the surrounding refractive index instead allows both efficient frequency tuning and enhanced in-/output coupling through retardation matching as this allows dark modes to become optically active, improving widespread functionalities.

: Effect of surrounding refractive index on the nearfield of plasmonic gap modes. a) Percentage enhancement of each mode with increase in ℎ for a high refractive index ( =1.5) dielectric coating. b) Nearfield enhancement as a function of the refractive index of the embedding dielectric medium ( ). c) Relative nearfield enhancements in percentage vs refractive index of the embedding dielectric medium ( ). Figure S2: Reproducibility of polymer coated samples. Three additional repeats of the PMMA polymer coating were performed and darkfield scattering spectra from 63, 763, 203 particles were collected. Binning the main peak position shows a histogram with a clear reoccurring peak position between 625-650nm. The scattering intensities for the average spectra show excellent reproducibility (2.9, 2.9, 2.7, 2.9%). frequency

Supplementary note 2: Illumination geometries
To determine the illumination geometry of the objective used, the DF illumination was visualised on a white sheet of paper placed perpendicular to the image plane. The angle of DF illumination was measured to be 75° ( Figure S3). which is dimensionless as required. We use the peak values on resonance for ,E.
To obtain the mode volumes, the nearfield profile across the gap is used, E( , ). The mode volume is defined as with an extra factor of ½ for the odd (11),( 21) modes which have an additional sin(ϕ) dependence. Note that the volume so far omits out the field which penetrates slightly into the metal facet on either side of the nanocavity, as well as the much weaker field around the rest of the NP. The decay length of light into the metal follows from the resonant wavevector ∥ = 2 for facet diameter and Bessel zero , since ⊥~∥ and = ⊥ −1 = /2 ~5 nm, thus larger than . The extra metallic volume contribution is then 2. . ( ) but the field is smaller by = , which gives = ( ) 2 3 4 = 2 /4 ~ 850 nm 3 , which is twice as large as the original gap volume. As a result, the total volume is three times the original integration above. Indeed to match previous estimates for ~0.6 at ℎ=0 (from COMSOL calculations), a volume roughly double the integration volume is required.
Note that the integrated scattered flux is derived from the peak scattered flux (in a particular direction). This is different for the normally radiating modes ( 1) and the high angle modes ( 0), and is evaluated using calculated mode patterns as shown in ref [1].
The total Q-factor can be extracted from the linewiths, = /Δ , but we note that there is considerable confusion in the literature about whether an extra prefactor of 2 is included. To reproduce the known values of = 0.7 at ℎ=0, the appropriate formula is = 2 /Δ . Since = 2 / , we extract = 2 / and = − so that the radiative efficiency for each mode can be defined as = /( + ) Finally, we obtain the field enhancement inside the nanogap. In ref [2], the antenna mode coupled energy was estimated from the polarizability of the NP to give the energy coupled into the nanogap where ~2 is the spectral enhancement here, which gives with an extra coupling factor included for how well each mode couples, and which describes the effective antenna cross section. As noted above, we scale and to match the calculated , (ℎ = 0) for (10) for coating refractive index = 1.6. The extracted parameters for each mode are plotted below ( Figure S5), except for (21) as the initial uncoated intensity of this mode is too weak to be extracted. This gives an understanding for how the dielectric layer coating acts ( Figure S7) to give: -increased E with dielectric layer height from the increased antenna cross section, strongest for (20, 11) modes, -strong increase of (11) in near-field and scattering due to increases in both in/out coupling ( , ), -(20) radiating strongly, but has poor in-coupling hence only appears weakly in for low ℎ, -larger linewidth of (20) which comes as both radiative and non-radiative emission is faster, -a linewidth of (11) dominated by non-radiative components, while (20) is dominated by radiative components.
From this it is clear that the dielectric layer helps in-coupling, but particularly for higher order modes. The dielectric layer also drastically increases the outcoupling for the (11) mode, which can be intuitively considered as helping plasmons escape out around the edge of the nanocavity onto the surface of the NP. Figure S8: Near-field enhancement across the centre of the gap, for no dielectric coating.
Using this model the SERS enhancements can be predicted, since the enhancement (ignoring input/output wavelength differences) is E 4 . 3 The nanocavity centre favours 'even' modes due to the central antinode, but we integrate over all molecules overlapping the modes to get the predicted SERS emission = E 4 /( 2 ) where is the size of a molecule ( Figure S9). This model shows that, independent from wavelength tuning, as a result of the retardation matching alone the local refractive index ( ) can provide an increase in SERS of nearly 10x when fully embedded (here calculated for =1.6).