A spaser (Surface Plasmon Amplification by Stimulated Emission of Radiation) is the plasmonic analogue of a laser: instead of amplifying a photon field in an optical cavity, it amplifies a localized surface-plasmon mode confined to a metal nanoparticle, using a surrounding shell of gain molecules (dye or quantum dots) pumped into population inversion. This is the 2D counterpart of our 3D spaser visualizer — same rate equations, drawn as a flat cross-section plus two live analysis plots instead of an orbit-able 3D scene.
This simulator integrates the normalized two-level gain / single-mode plasmon rate equations every frame, and additionally sweeps the pump axis to steady state on every parameter change to draw the threshold curve on the right:
dn2/dt = P(1 - n2) - n2 - G·n2·N_ph / 50
dN_ph/dt = G·n2·N_ph - κ·N_ph + β·n2
G = g · (molecule density / 100) (effective gain-coupling)
n2 = fraction of gain molecules in the excited state
N_ph = normalized plasmon occupation number
P = pump rate, κ = plasmon (ohmic + radiative) loss rate
β·n2 = small spontaneous-emission seed feeding the mode
Below threshold, spontaneous emission dominates and N_ph stays near zero. Once the round-trip gain G·n2 exceeds the loss κ, stimulated emission runs away: excited molecules are triggered to decay in phase with the existing plasmon field, feeding it coherently until gain saturation (the -G·n2·N_ph term depleting n2) clamps the field at a steady, amplified value — exactly the relaxation-oscillation-then-clamp behaviour real lasers and spasers exhibit. The threshold pump rate follows from linearising around N_ph → 0: P_th = κ / (G − κ) when G > κ (no finite threshold exists if the coupling can never overcome the loss). The left-side plot below re-derives that same threshold numerically: at every pump value it integrates the ODE pair forward from a cold start until N_ph settles, then plots the settled value — no closed-form shortcut, just the same integrator run 81 times.
- Pump rate — how fast molecules are re-excited; raising it drives more of the shell into inversion. Also sets where the live marker sits on the threshold-sweep plot.
- Plasmon loss κ — ohmic damping in the metal core plus radiative leakage; a lossier / smaller nanoparticle needs more pump to lase.
- Coupling g — how strongly each molecule's dipole couples to the near-field plasmon mode (depends on molecule–metal spacing and orientation).
- Molecule density — more gain molecules in the shell means more total stimulated-emission events per plasmon cycle, lowering the pump needed to cross threshold.
- Orange sparks travel from an excited molecule toward the nanoparticle core — individual stimulated-emission events, whose rate the core's glow (and the right-hand time trace) tracks directly.