Plasmonic nanofocusing spectral interferometry (PNSI) measures broadband, amplitude- and phase-resolved scattering spectra of single nanostructures with ~10 nm resolution by interfering surface plasmon polaritons nanofocused to the apex of a conical gold taper with SPPs back-reflected from that apex. Esmann and co-workers (2020) showed that the extracted complex coupling spectrum directly probes the projected local density of optical states (LDOS) only in the weak-coupling limit where tip–sample multiple scattering is negligible; when the tip perturbs the sample, higher-order scattering terms modify the retrieved resonance energies and linewidths. Still open is a quantitative framework that makes PNSI a reliable LDOS probe beyond this limit: a correction or inversion scheme accounting for higher-order tip–sample multiple scattering, complemented by strategies for suppressing or controlling tip-induced perturbations while retaining sufficient signal. Later work has not delivered the required inversion theory, so the problem remains open.

Background

When light is confined to nanoscale metallic structures, the rate and spectrum of spontaneous emission, energy transfer, and absorption of a nanoscale emitter are governed not only by its intrinsic properties but by the electromagnetic environment, quantified by the local density of optical states (LDOS) — the number of available photonic modes per unit volume, frequency, and direction, expressible as the imaginary part of the electromagnetic Green's dyadic. Mapping the LDOS at nanometer spatial resolution with spectral and phase sensitivity is central to understanding and controlling light–matter interactions in plasmonic and photonic nanostructures, but available probes are limited: far-field spectroscopy is diffraction-limited, electron-beam spectroscopy has its own resolution and interpretation limits, and conventional scanning near-field optical microscopy (SNOM), which raster-scans a sharp illuminated probe over a sample, historically lacked phase-stable broadband quantitation in the visible and near-infrared.

Plasmonic nanofocusing offers a distinct route. A surface plasmon polariton (SPP — a light wave bound to and propagating along a metal–dielectric interface) is launched by grating coupling onto the shaft of a conical gold taper; as the taper radius shrinks, adiabatic compression concentrates the SPP into a spatially isolated, spectrally broadband nanoscale light source at the apex, with minimal background from the far-field illumination. Esmann et al. built on this platform (first in the companion work 'Vectorial near-field coupling', Nature Nanotechnology 14, 698–704, 2019, which resolved tip-induced resonance shifts and broadenings of a single gold nanorod via near-field coupling to transverse and longitudinal taper polarizabilities) to demonstrate plasmonic nanofocusing spectral interferometry (PNSI, Nanophotonics 9, 491–508, 2020): SPPs backpropagating from the perturbed apex interfere with the launched SPPs at a small protrusion on the taper shaft, forming an inherently phase-stable spectral interferogram that is detected in the far field. From these interferograms the complex-valued local coupling spectrum β(ω) of a single nanorod's longitudinal localized surface plasmon resonance (localized electromagnetic eigenmode of a metal nanoparticle) is reconstructed — absorptive Lorentzian line shape, resonance energy, and linewidth — and spatial maps of tip-induced linewidth broadening (a Purcell-effect-like additional damping channel into taper SPPs, following the mode intensity at the rod apices; the Purcell effect is the enhancement of an emitter's decay rate by its environment) and red shifts of up to ~50 meV near the rod rim were measured.

A coupled-dipole model — approximating the taper apex and the nanorod as point dipoles with Lorentzian polarizabilities — underpins the analytic inversion: the measured interferogram is written as S(ω)=I0(ω)∣1+σ(ω)eiωτ0∣2S(\omega)=I_0(\omega)|1+\sigma(\omega)e^{\mathrm{i}\omega\tau_0}|^2, with τ0\tau_0 the SPP round-trip delay and σ(ω)\sigma(\omega) the complex reflection coefficient, and in the first-order (single-scattering) limit β(ω)\beta(\omega) reduces to the LDOS projected along the taper axis, ρe,z(ω)∝Im Gzz\rho_{e,z}(\omega)\propto\mathrm{Im}\,G_{zz}. The critical limitation, stated explicitly in the source work and long familiar from apertureless near-field microscopy (where tip–sample coupling in elastic light scattering has been modeled since at least Raschke and Lienau, Applied Physics Letters 83, 5089–5091, 2003), is that this LDOS interpretation holds only when multiple near-field scatterings between tip and sample are negligible: when the tip strongly perturbs the sample, β(ω)\beta(\omega) contains higher-order terms and the extracted spectral parameters — resonance energy and linewidth — are modified, so what is measured is the LDOS of the tip–sample hybrid, not of the sample alone. The model also neglects substrate and taper-shaft contributions to the Green's function and any nonlocal metal response. The source work left open how to remove these limitations: it neither developed an inversion beyond the perturbative truncation, nor demonstrated practical strategies (operation at larger tip–sample separations with acceptable signal-to-noise, apex engineering such as attaching a small dielectric scatterer, or less invasive probe designs) for suppressing or controlling the perturbation while keeping signal.

Problem Statement

Establish quantitative retrieval of the projected local density of optical states (LDOS) from plasmonic nanofocusing spectral interferometry (PNSI) in the regime where the tip–sample coupling is not negligible, as left open by Esmann et al. 2020. Concretely, develop and validate a signal-formation and inversion framework — going beyond the first-order perturbative truncation of the coupled-dipole treatment, i.e., incorporating higher-order (non-perturbative) tip–sample multiple scattering — that maps the measured spectral interferograms S(ω)S(\omega) (including spatial maps at fixed tip–sample distance) to the unperturbed projected LDOS and the intrinsic spectral parameters (resonance energy, linewidth) of the sample. This serves the overarching objective of turning PNSI from a weak-coupling-limited measurement of the tip–sample hybrid into a quantitative, minimally invasive probe of the sample's own projected LDOS. The framework must also quantify and control the tip-induced perturbations of the measured resonance parameters while retaining sufficient signal. Validation must demonstrate quantitative recovery of resonance energies and linewidths (and the projected LDOS) against independent ground truth — full-wave simulations and/or unperturbed reference measurements — over a coupling range extending beyond the weak-coupling limit, i.e., into regimes in which tip-induced resonance shifts and linewidth broadenings are non-negligible, at the ~10 nm spatial resolution and broadband spectral coverage characteristic of the technique.

References

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