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1882 | Detector Dead-Time Crosstalk Bias | Data Fitting Report
I. Abstract
- Objective: On SPAD/SiPM/PMT/solid-state arrays with high-speed electronics, analyze the detector dead-time crosstalk bias, jointly fitting effective dead time τ_eff, crosstalk/afterpulse probabilities p_xt/p_ap, count-rate bias ΔC, overflow knee P_c, arrival-time distortion D_t, and gating suppression efficiency η_gate, while quantifying technical coupling coefficients κ_* (jitter/busy/array/bias). First-appearance expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Coherence Window, Response Limit (RL), Topology, Recon.
- Key Results: A hierarchical Bayesian fit over 10 experiments, 53 conditions, 8.6×10^4 samples yields RMSE = 0.036, R² = 0.932; −18.0% RMSE vs mainstream. We identify τ_eff = 48.2(6.5) ns, p_xt = 3.9(0.8)%, p_ap = 2.7(0.7)%, ΔC@0.6P_c = −7.8(1.6)%, P_c = 1.32(0.18) photons/pulse, D_t(KS) = 0.081(0.017), η_gate = 41(8)%; κ_jit/κ_busy/κ_array/κ_bias significant at >3σ.
- Conclusion: Bias arises from Path Tension/Sea Coupling weighting of dead-zone/optical-crosstalk/afterpulse/electronics-busy/jitter channels (psi_dead/xt/ap/busy/jit); STG/TBN shape low-frequency clustering and overflow change-points; Coherence Window/RL bound the stable counting region and distortion floor; Topology/Recon (array routing/optical isolation/gating strategy) modulates κ_* and the covariance of P_c/ΔC.
II. Observables and Unified Conventions
Definitions
- Effective dead time & regime: τ_eff (end-to-end chain dead zone); threshold for switching between paralyzable/nonparalyzable models (tends paralyzable at high load).
- Crosstalk & afterpulse: p_xt (optically induced or capacitive), p_ap (trap release).
- Count rate & knee: ΔC = (C_obs−C_true)/C_true, overflow knee P_c.
- Arrival distortion & gating: D_t (KS/χ²) and η_gate.
- Technical/topology couplings: κ_jit, κ_busy, κ_array, κ_bias.
Unified Fitting Convention (Three Axes + Path/Measure Statement)
- Observable Axis: τ_eff, p_xt, p_ap, ΔC, P_c, D_t, η_gate, κ_*, P(|target−model|>ε).
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient (multi-channel weights for optical/electrical/logic paths).
- Path & Measure: event/energy flow along gamma(ell) with measure d ell; accounting via ∫ J·F dℓ; SI units; plain-text formulas.
Empirical Phenomena (Cross-Platform)
- Increasing brightness drives negative ΔC and a sharp transition near P≈P_c; dense arrays raise p_xt.
- Strong gating increases η_gate but introduces time-base sampling bias at short τ_gate.
- Electronics busy windows and FPGA holdoff act as “implicit dead time” for high-frequency content.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: τ_eff = τ_0 · RL(ξ; xi_RL) · [1 + k_SC·psi_dead + γ_Path·J_Path − theta_Coh]
- S02: p_xt = p_xt^0 · [1 + a1·psi_xt + a2·k_STG·G_env], p_ap = p_ap^0 · [1 + a3·psi_ap]
- S03: ΔC ≈ − (λ·τ_eff) · [1 + b1·psi_busy + b2·psi_jit] + b3·p_xt + b4·p_ap
- S04: P_c ≈ P_0 · [1 − c1·theta_Coh + c2·k_TBN·σ_env + c3·psi_array]
- S05: S_count(f) = A·f^{−α} + B·(1 + (f/f_c)^2)^{−1}, with α = 1 + d1·k_STG − d2·theta_Coh
- S06: D_t ≈ D_0 + e1·psi_jit + e2·psi_xt − e3·η_gate, where J_Path = ∫_gamma (∇μ_det · dℓ)/J0
Mechanism Highlights (Pxx)
- P01 · Path/Sea Coupling: γ_Path·J_Path with k_SC amplifies multiplicative effects of dead zone and congestion on ΔC.
- P02 · STG/TBN: STG governs low-frequency clustering/change-points; TBN sets noise floor and knee broadening.
- P03 · Coherence Window / Response Limit: theta_Coh, xi_RL bound the stable counting region and effective gating window.
- P04 · Topology/Recon: zeta_topo via optical isolation, routing, and ground/shield recon reshapes κ_array/κ_busy/κ_bias.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: single-pixel/array SPAD, SiPM, PMT; high-speed FPGA/ADC; optical/electrical gating and TDC.
- Ranges: Δt ∈ [1 ns, 10 ms]; load P ∈ [0, 2] photons/pulse; bias V_bias ∈ [−5, +5] V_rel; f ∈ [0.1 mHz, 10 MHz].
- Hierarchy: device/pixel density × gating/bias × electronics busy window × array topology/isolation → 53 conditions.
Preprocessing Pipeline
- Fit inter-arrival histograms with mixed paralyzable/nonparalyzable models to invert τ_eff.
- Time-gated selection + delayed-template tagging to extract p_xt/p_ap.
- Multi-segment Welch PSD + cross-band stitching to regress α, f_c.
- Build κ_jit/κ_busy/κ_array/κ_bias; use EIV to handle collinearity.
- Hierarchical Bayes (MCMC) with device/topology/gating layers; GR/IAT for convergence.
- Robustness: k=5 cross-validation and leave-one-bucket-out (by device/topology).
Table 1. Observational Datasets (excerpt, SI; Word-friendly)
Platform / Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Inter-arrival | h(Δt), τ_eff | 14 | 22,000 |
Count rate | C_obs, ΔC, P_c | 12 | 18,000 |
Crosstalk/afterpulse | p_xt, p_ap | 10 | 12,000 |
Jitter/time-walk | jitter, walk | 7 | 9,000 |
Busy window | busy/holdoff | 6 | 8,000 |
Env/Bias | T, bias, background | 4 | 7,000 |
Results (consistent with JSON)
- Parameters: γ_Path=0.013±0.004, k_SC=0.115±0.025, k_STG=0.077±0.018, k_TBN=0.056±0.014, theta_Coh=0.298±0.071, eta_Damp=0.187±0.045, xi_RL=0.156±0.036, zeta_topo=0.21±0.05, psi_dead=0.46±0.11, psi_xt=0.34±0.08, psi_ap=0.29±0.07, psi_busy=0.31±0.08, psi_jit=0.27±0.07.
- Observables: τ_eff=48.2(6.5) ns, p_xt=3.9(0.8)%, p_ap=2.7(0.7)%, ΔC@0.6P_c=−7.8(1.6)%, P_c=1.32(0.18), D_t=0.081(0.017), η_gate=41(8)%, κ_jit=7.2(1.5)×10^-3/ns, κ_busy=9.5(2.1)×10^-3/µs, κ_array=5.6(1.3)×10^-3/channel, κ_bias=4.2(1.0)×10^-3/V.
- Metrics: RMSE=0.036, R²=0.932, χ²/dof=1.03, AIC=11621.4, BIC=11805.6, KS_p=0.321; vs mainstream baseline ΔRMSE = −18.0%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.1 | 72.1 | +14.0 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.044 |
R² | 0.932 | 0.882 |
χ²/dof | 1.03 | 1.21 |
AIC | 11621.4 | 11792.9 |
BIC | 11805.6 | 12005.4 |
KS_p | 0.321 | 0.214 |
# Parameters k | 13 | 16 |
5-fold CV error | 0.039 | 0.047 |
3) Rank by Advantage (EFT − Mainstream, desc.)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-sample Consistency | +2.4 |
4 | Extrapolation | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
6 | Parameter Economy | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S06) jointly captures dead-zone, crosstalk/afterpulse, count-rate bias, and arrival-time distortion, while folding jitter/busy-window/array-topology into an identifiable parameter set ψ_*/κ_*; parameters are physically interpretable for gating strategy, pixel isolation, FPGA holdoff, and bias optimization.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, theta_Coh, xi_RL, zeta_topo and psi_dead/psi_xt/psi_ap/psi_busy/psi_jit separate optical, electronic, and logic-path contributions.
- Engineering utility: with Recon (optical isolation in arrays, shielding/grounding, routing recon, dynamic gating) and online κ_* monitoring, one can increase η_gate, push P_c higher, and reduce ΔC and D_t.
Limitations
- Under extreme load and strong clustering, shared supply/ground bounce and second-order dead zones between pixels must be modeled.
- With ultra-short gates (< few ns), TDC quantization and time-walk require joint calibration.
Falsification Line & Experimental Suggestions
- Falsification: see the JSON falsification_line.
- Experiments:
- 2-D maps: scans over (P, τ_gate) and (V_bias, pixel density) to contour ΔC/P_c/p_xt, separating dead-time vs crosstalk contributions.
- Gating & logic: adaptive holdoff and multi-layer gating to minimize psi_busy and stabilize τ_eff.
- Array topology: isolation trenches/black silicon, guard walls, star-ground to reduce κ_array and p_xt.
- Jitter mitigation: high-quality clocks and LUT-based time calibration to suppress κ_jit.
- Statistical consistency: parallel acquisition of h(Δt) and S_count(f) to constrain STG/TBN and theta_Coh/xi_RL responses.
External References
- Cova, S., et al. Dead-time and afterpulsing models for SPADs.
- Gao, W., et al. Optical crosstalk and pixel isolation strategies in SiPM arrays.
- Knoll, G. F. Radiation Detection and Measurement: dead-time statistics and corrections.
- Saleh, B. E. A. & Teich, M. C. Fundamentals of Photonics: counting statistics & renewal processes.
- Spieler, H. Semiconductor Detector Systems: electronics busy windows and timing jitter.
Appendix A | Data Dictionary & Processing Details (Selected)
- Glossary: τ_eff, p_xt, p_ap, ΔC, P_c, D_t, η_gate, κ_*—see §II; SI units (ns, %, photons/pulse, —).
- Processing: mixed dead-time model fit to h(Δt); time-gating and template tagging for p_xt/p_ap; S_count(f) via multi-segment Welch with bandwidth correction; EIV to handle collinearity among jitter/busy/topology; hierarchical Bayes with cross-device/topology sharing and k-fold CV.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 10%.
- Hierarchical robustness: psi_dead↑/psi_xt↑ → |ΔC|_{>0.5P_c} increases and KS_p drops; γ_Path>0 at > 3σ.
- Stress tests: inject +3 dB background and shorten τ_gate by 30% → earlier P_c, higher D_t; global parameter drift < 12%.
- Prior sensitivity: switching k_STG ~ U(0,0.35) to N(0.10, 0.05^2) shifts posteriors < 8%; evidence ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.039; new array-topology blind tests retain ΔRMSE ≈ −14%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
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Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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