Nanometer-Scale Spatial and Spectral Mapping of Exciton Polaritons in Structured Plasmonic Cavities

Exciton polaritons (EPs) are ubiquitous light-matter excitations under intense investigation as test beds of fundamental physics and as components for all-optical computing. Owing to their unique attributes and facile experimental tunability, EPs potentially enable strong nonlinearities, condensation, and superfluidity at room temperature. However, the diffraction limit of light and the momentum content of fast electron probes preclude the characterization of EPs in nanoscale structured cavities exhibiting energy-momentum dispersion. Here we present fully relativistic analytical theory and companion numerical simulations showing that these limitations can be overcome to measure EPs in periodic nanophotonic cavities on their natural energy, momentum, and length scales via lattice electron energy gain spectroscopy. With the combined high momentum resolution of light and nanoscale spatial resolution of focused electron beams, lattice electron energy gain spectroscopy can expose deeply subwavelength EP features using currently available monochromated, aberration-corrected scanning transmission electron microscopes.

Figure 1

Optical and electron excitation of 1D arrays. (a) Scheme of 1D array excited by a plane wave and/or a focused electron beam with trajectory parallel or perpendicular to the lattice periodicity direction. (b) Per-NP optical extinction spectra for an infinite 1D array of 90 nm diameter Ag NPs with periodicity $a=415\mathrm{nm}$ as a function of $K_{||}$. The solid line denotes the energy of the isolated NP dipolar LSP, while the dashed lines mark the energy-momentum dispersion of the $\pm 1$ photonic diffraction orders. (c) ${\tilde{\mathrm{\Gamma }}}_{\mathrm{EEL}}^N(\omega )$ for a $0.7c$ electron moving along the perpendicular trajectory versus the number of NPs $N$ in finite array realizations. The normalized optical extinctions of the single NP (dashed gray line) and array (dashed black line; $N=501$) are displayed for reference. (d) ${\tilde{\mathrm{\Gamma }}}_{\mathrm{EEL}}(\omega )$ for an electron moving along the parallel trajectory with $N=501$ versus velocity. (e) Line out spectra from (d) at the dashed vertical lines. In panels (c)–(e) the electron trajectory passes 5 nm outside of the NP surface.

Figure 2

Lattice electron energy gain spectroscopy. (a) Analytic LEEG probability spectra for the same Ag NP array as a function of $K_{||}$. (b) Analytical (solid) and numerical (dashed) evaluation of ${\mathcal{P}}_{\mathrm{EEG}}$ spectra for a single particle and 1D array. Inset shows a spatial map of ${\mathcal{P}}_{\mathrm{EEG}}$ at the LPP energy around one particle. Here, $v=0.7c$ and $I_0={10}^9\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

Position-dependent LEEG probability and point absorption spectra for the same 1D array embedded inside an excitonic medium. (a) Spatial distribution of the optically excited electric field within the $xy$ plane in the absence of the emitter medium. The dashed gray line indicates the array axis. (b) Scheme showing the 1D array inside the emitter medium slab. (c) Normalized ${\mathcal{P}}_{\mathrm{EEG}}$ (dashed) and ${\sigma }_{\mathrm{abs}}$ (solid) spectra at points P1 and P2 in panel (a) for the 1D array inside excitonic media of varying $f$. The vertical dotted lines at 2.95 eV mark the resonance energy of the bulk excitonic transition $\hslash {\omega }_0$.

Figure 4

Spatially and spectrally resolving variations in LPP EPs using LEEG spectroscopy. (a) Normalized relative energy gain and absorption spectra at P1 and P2 from Fig. 3 for the array-slab system ($f=0.0445$, $K_{||}=0$). Insets show the in-plane electric response field for the combined array-slab system at the UP and LP energies marked by vertical gray lines. The color map corresponds to the out-of-plane component of the response field. (b) ${\mathcal{P}}_{\mathrm{EEG}}$ spectrum images at the UP and LP energies. (c) Point absorption spectrum images at the UP and LP energies.