This is the same Förster resonance energy transfer (FRET) physics as the 3D cascade, redrawn as two native 2D diagrams instead of a flattened 3D scene: an efficiency-vs-distance curve (top) and an energy-level ladder (bottom) — the standard Jablonski-style way spectroscopists actually plot this process.
E(r) = 1 / (1 + (r / R₀)⁶)
k_FRET(r) = (1 / τ_D) · (R₀ / r)⁶
R₀ = Förster radius: the r at which E = 50%
The top plot draws E(r) for the current R₀ and marks your chosen spacing r directly on the curve — you can see the inverse-sixth-power "cliff" instead of inferring it from sphere brightness. The bottom ladder places each quantum dot on its own energy rung (QD1 highest, QD4 lowest); every pumped exciton runs the exact same stochastic race used in the 3D version — transfer (rate k_FRET) vs. donor decay (rate 1/τ_D) — but a dropped exciton is drawn falling straight down off its rung (radiative loss) rather than vanishing, and a surviving hop steps down one rung toward lower energy, exactly like a real Jablonski diagram.
- Dot spacing r — the edge-to-edge distance between adjacent quantum dots; increasing it suppresses transfer as 1/r⁶ (watch the marker slide down the curve).
- Förster radius R₀ — set by the spectral-overlap integral J and dipole orientation factor κ²; larger R₀ shifts the whole curve outward.
- Donor lifetime τ_D — the intrinsic excited-state lifetime; a longer-lived donor gives FRET more time to win the race before radiative decay.
- Excite donor — injects one exciton at the QD1 rung and lets it random-walk down the ladder (or fall off early) using the real per-hop probabilities above.
Real-world relevance: exactly this cascade geometry is used in QD-based light-harvesting antennas, FRET biosensors, and multi-step energy-transfer LEDs, where a directional relay concentrates excitation onto a single low-bandgap emitter.