Single-Photon Detector: Dead Time & Dark Counts — 2D

Incident Light

Photon rate λ 2.0 MHz

Detection efficiency η 30%

Detector

Dead time τ_d 50 ns

Dark count rate 50 kHz

Afterpulse prob. p_ap 3%

Dead-Time Model

Live Readout

Click rate

0 kHz

Total clicks

True photon clicks

Dark + afterpulse

Live busy fraction

Signal-to-noise

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How it works

Same detector physics as the 3D version, run as an honest event-driven simulation rather than a formula lookup. Incident photons and dark counts are two independent Poisson processes; every arrival is a fresh exponential-interarrival draw from its own rate:

P(k events in Δt) = (rΔt)^k e^(−rΔt) / k!,  Δt = −ln(1−U)/r

Each photon triggers a click with probability η. Every registered click — photon, dark, or afterpulse — opens a dead-time window τ_d. Two real gating rules are selectable:

Non-paralyzable — an arrival during dead time is dropped and does not extend the blind window. Registered rate saturates: R = ρ/(1+ρτ_d), ρ = ηλ+dark rate.
Paralyzable — an arrival during dead time is dropped but re-arms the window from that moment, so a busy detector under high flux can be paralyzed almost indefinitely: R = ρ·e^−ρτ_d, which rolls back down to zero at high ρ instead of saturating.

The bottom panel doesn't just plot those formulas — it re-runs the same event-driven generator at ~24 fixed values of λ (dashed line shows the closed-form prediction for comparison, solid dots are the actual simulated count). The top timeline shows every click as it happens (blue = true photon, amber = dark count, red = afterpulse), with a red band marking the live dead-time window.

Photon click Dark count Afterpulse