Avalanche Photodiode Gain — 2D Cascade & Excess-Noise Verification

This 2D companion to the 3D quantum-dot avalanche photodiode simulator strips the phenomenon down to its own independent Monte Carlo engine: a photo-injected carrier drifts across the multiplication region in exact-probability sub-steps, spawning an impact-ionization cascade whose every completed single-photon event lands one point in a live histogram. Reverse bias and the electron/hole ionization ratio k = β/α still drive the exact McIntyre local-field gain formula M = (1−k)/[exp(−(1−k)αL) − k] and its excess noise factor F(M) = kM + (1−k)(2 − 1/M), but here those analytic predictions are checked directly against the histogram's own running mean and second moment — watch the measured ⟨M⟩ and empirical F converge onto the analytic curves as more photons are injected.