A surface plasmon polariton (SPP) is a light wave bound to a metal–dielectric interface. Its wavenumber follows the standard dispersion relation, with the metal's permittivity given by the free-electron (Drude) model:
k_spp = k₀ √( εm·εd / (εm + εd) )
εm(ω) = 1 − ωp² / (ω² + iωγ)
k₀ = 2π/λ₀, ω = 2πc/λ₀
On a taper, geometry adds a second effect: as the local radius r shrinks toward the metal's skin depth δ = λ₀/(2π√|Re εm|), the effective mode index rises steeply (adiabatic nanofocusing, Stockman 2004):
n(r) ≈ Re(n_spp) · (1 + δ/r)
λ_spp(r) = λ₀ / n(r)
Because the mode slows down and squeezes into a smaller cross-section at the same time, energy-flux conservation along the taper forces the near-field intensity to pile up at the apex:
|E(r)|² ∝ n(r)/r (relative to the base)
L_spp = 1 / (2k₀·Im n_spp) — Ohmic loss length
- Metal — switches the Drude parameters ωp, γ (gold vs. silver; silver has lower loss, so it focuses further before decaying).
- λ₀, r₀, θ — set the driving wavelength and the cone's geometry, which together fix how compressed the wave gets before it hits the ~2 nm apex floor (below that, nonlocal/quantum effects the model doesn't capture take over).
- Ohmic damping γ× — scales the Drude scattering rate γ (defect density, grain boundaries, temperature all raise it in a real sample); higher damping shortens the propagation length and caps how bright the apex hot spot can get.
- Send Wave Pulse — launches a visible wavefront so you can watch the ripple spacing (local λ_spp) shrink and the glow (field enhancement) brighten as it nears the tip. Drag anywhere on the taper to probe the exact local radius, index and wavelength.
The lower-left panel plots the mode index n(z) and local wavelength λ_spp(z) along the taper; the lower-right panel is an Argand (complex-plane) view of the Drude permittivity εm(ω) driving all of it. This is the working principle behind apertureless near-field microscopy (s-SNOM), tip-enhanced Raman spectroscopy (TERS) and plasmonic photothermal probes.