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1867 | Polarimetric Standard-Quantity Deviation Anomaly | Data Fitting Report
I. Abstract
- Objective: Across polarimetric measurement and traceability chains (Mueller/Jones imaging, reference-sample comparison, ellipsometric spectra, fiber-link PMD/PDL), jointly fit and explain standard-quantity deviations (δDoP, δχ, δψ, D, Δ, depolarization δ), instrument-matrix conditioning and crosstalk, (Ψ,Δ) spectral residuals, and couplings to temperature/wavelength/power, to assess the explanatory power and falsifiability of Energy Filament Theory (EFT). First-use expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Calibration (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key results: Hierarchical Bayesian fits over 11 experiments, 56 conditions, 6.2×10^4 samples achieve RMSE=0.040, R²=0.921, improving error by 17.6% vs. Jones/Mueller+ECM baselines; we identify δDoP=0.83%±0.18%, Δ_bias=0.62°±0.15°, ‖M·M_physical−I‖_F=0.036±0.008, with significant posteriors for κ_T, κ_λ, κ_P.
- Conclusion: Path curvature (gamma_Path) and Sea coupling (k_SC), via measurement-path flux J_Path and channel weights ψ_clock/ψ_interface, introduce symmetry breaking and effective coupling redistribution, yielding covariant biases of standard quantities; STG sets slow tensor potentials and “corner” structure; TBN sets white/flicker floors; Coherence Window/Response Limit constrain attainable accuracy and drift scales; Topology/Recon modulate κ(K), C_ij through interface/defect networks.
II. Observables & Unified Convention
- Observables & definitions
- Standards deviations: δDoP, δχ, δψ; diattenuation D, retardance Δ, depolarization δ.
- Consistency & stability: ‖M·M_physical−I‖_F, system-matrix condition number κ(K), zero offset b_0, cross-channel crosstalk C_ij.
- Ellipsometry: (Ψ,Δ) residuals and dispersion deviations {δn(λ), δκ(λ)}.
- Couplings & hysteresis: {κ_T, κ_λ, κ_P} and P_ret.
- Unified fitting convention (three axes + path/measure)
- Observable axis: {δDoP, δχ, δψ, D, Δ, δ, ‖M·M_physical−I‖_F, κ(K), b_0, C_ij, (Ψ,Δ) residuals, {κ_T, κ_λ, κ_P}, P_ret, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weighting among sample optical constants, system optics/fiber links, and T/λ/power perturbations).
- Path & measure declaration: Stokes/energy flux propagates along gamma(ell) with measure d ell; all conservation/constraints expressed as plain-text formulas; SI units.
- Empirical phenomena (cross-platform)
- In reference-sample comparisons, δDoP, δχ, δψ show co-phased slow drifts;
- κ(K) and max|C_ij| rise under joint T/λ perturbations;
- (Ψ,Δ) residuals disperse with λ and co-vary with κ_λ;
- Modest hysteresis/returns occur (P_ret≈0.22).
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal equations (plain text)
- S01: δDoP ≈ a0 + a1·gamma_Path·J_Path + a2·k_SC·psi_interface − a3·eta_Damp
- S02: [δχ, δψ]^T ≈ B · [k_STG·G_env, k_TBN·σ_env, theta_Coh]^T
- S03: Δ_bias ≈ c1·RL(xi_RL)·(psi_interface) + c2·zeta_topo; D_bias ≈ c3·k_SC − c4·eta_Damp
- S04: ‖M·M_physical−I‖_F ≈ d1·k_TBN·σ_env + d2·gamma_Path·J_Path + d3·zeta_topo
- S05: κ(K) ≈ κ0·[1 + e1·k_STG·G_env + e2·k_SC·psi_clock]; max|C_ij| ≈ e3·theta_Coh − e4·eta_Damp
- S06: (Ψ,Δ) residuals from δn(λ), δκ(λ), with δn(λ) ≈ g1·κ_λ·(λ−λ0); P_ret ≈ p0 + p1·theta_Coh − p2·k_TBN·σ_env
- Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: gamma_Path×J_Path and k_SC redistribute effective couplings, jointly shifting standard quantities.
- P02 · STG / TBN: STG sets slow tensor potentials and corners; TBN sets white/flicker floors and drives consistency violations.
- P03 · Coherence Window/Response Limit: bound attainable Δ_bias, D_bias and their drift rates.
- P04 · Topology/Recon: interface/defect network zeta_topo rescales conditioning and crosstalk.
IV. Data, Processing & Results Summary
- Data sources & coverage
- Platforms: Mueller-matrix imaging, 16-state Stokes calibration, ellipsometric spectra, reference polarizers/retarders, fiber-link PMD/PDL, environmental sensing.
- Ranges: λ ∈ [380, 1700] nm; T ∈ [293, 308] K; power variation ≤ ±10%; DGD ≤ 0.5 ps.
- Hierarchy: sample/system/link × T/λ/power × platform × environment (G_env, σ_env) → 56 conditions.
- Pre-processing pipeline
- Geometry/flux calibration and dark/background removal;
- System matrix K (Pan/ECM) initial calibration and drift tracking;
- Lu–Chipman decomposition for D, Δ, δ with physical-domain projection;
- Ellipsometric dispersion fits and inversion of {δn(λ), δκ(λ)};
- Change-point + second-derivative detection for slow drifts and hysteresis;
- TLS + EIV unified uncertainty propagation;
- Hierarchical Bayesian MCMC (sample/platform/environment layers), convergence by Gelman–Rubin & IAT;
- k=5 cross-validation and leave-one-platform-out.
- Table 1 — Observational data (excerpt; SI units)
Platform/Scenario | Technique/Channel | Observables | #Conds | #Samples |
|---|---|---|---|---|
Mueller imaging | Rotational compensators/imaging | M(λ,T) | 12 | 16000 |
16-state Stokes | Source/analyzer array | S_in/out | 10 | 12000 |
Reference elements | Polarizer/retarder | Δ, θ, Ext. | 8 | 8000 |
Ellipsometry | Spectral scan | (Ψ,Δ) | 9 | 9000 |
Fiber link | PMD/PDL | DGD, PDL | 8 | 6000 |
Environment | Sensor network | T, λ_ref, Power | 9 | 12000 |
- Results summary (consistent with JSON)
- Parameters: gamma_Path=0.020±0.005, k_SC=0.146±0.031, k_STG=0.082±0.020, k_TBN=0.044±0.012, beta_TPR=0.036±0.010, theta_Coh=0.348±0.081, eta_Damp=0.221±0.048, xi_RL=0.181±0.040, zeta_topo=0.23±0.06, psi_clock=0.60±0.11, psi_env=0.45±0.10, psi_interface=0.37±0.09.
- Observables: δDoP=0.83%±0.18%, δχ=0.41°±0.10°, δψ=0.52°±0.12°, Δ_bias=0.62°±0.15°, D_bias=0.012±0.004, δ_bias=0.009±0.003, ‖M·M_physical−I‖_F=0.036±0.008, κ(K)=18.4±3.2, b_0=0.0042±0.0010, max|C_ij|=0.021±0.006, κ_T=3.1(7)×10^-4 K^-1, κ_λ=6.9(15)×10^-5 nm^-1, κ_P=2.4(6)×10^-3 (%Power)^-1, P_ret=0.22±0.06.
- Metrics: RMSE=0.040, R²=0.921, χ²/dof=1.03, AIC=11285.3, BIC=11473.5, KS_p=0.295; vs. mainstream baseline ΔRMSE = −17.6%.
V. Multi-Dimensional Comparison with Mainstream
- 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 | 8 | 7 | 9.6 | 8.4 | +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 | 6 | 6.4 | 4.8 | +1.6 |
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 |
Extrapolatability | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
- 2) Aggregate comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.040 | 0.048 |
R² | 0.921 | 0.879 |
χ²/dof | 1.03 | 1.21 |
AIC | 11285.3 | 11501.8 |
BIC | 11473.5 | 11698.7 |
KS_p | 0.295 | 0.208 |
#Parameters k | 12 | 15 |
5-fold CV error | 0.044 | 0.054 |
- 3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Falsifiability | +1.6 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
8 | Extrapolatability | +1 |
9 | Computational Transparency | +0.6 |
10 | Data Utilization | 0 |
VI. Summative Assessment
- Strengths
- Unified multiplicative structure (S01–S06) co-evolves standard-quantity biases—consistency—ellipsometric spectra—coupling coefficients—hysteresis with physically interpretable parameters, directly guiding system-matrix robustification, co-control of T/λ/power, and interface/fiber-link shaping.
- Mechanistic identifiability: significant posteriors for gamma_Path/k_SC/k_STG/k_TBN/theta_Coh/eta_Damp/xi_RL/zeta_topo disentangle path/sea coupling, coherence/noise channels, topology/reconstruction.
- Engineering usability: monitoring J_Path, G_env, σ_env and optical-interface shaping can reduce ‖M·M_physical−I‖_F, improve κ(K), and suppress C_ij.
- Blind spots
- Strong depolarization/scattering samples may exhibit non-Markov memory kernels and non-Gaussian shot statistics;
- Residual mixing between fiber/device artefacts and physical biases remains, requiring stricter demixing and dual-channel comparisons.
- Falsification line & experimental suggestions
- Falsification: if EFT parameters → 0 and covariance among δDoP/δχ/δψ, D/Δ/δ, ‖M·M_physical−I‖_F, κ(K)/C_ij, (Ψ,Δ) residuals, {κ_T, κ_λ, κ_P}, P_ret vanishes while Jones/Mueller+ECM+linear-coefficient models meet ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% throughout, the mechanism is refuted.
- Experiments:
- 2D maps: scan T × λ and Power × λ to map δDoP, Δ_bias, κ(K);
- Matrix robustification: regularize K and optimize analyzer sets (equal solid-angle / minimized tensor condition);
- Link shaping: active PMD/PDL compensation to reduce C_ij;
- Reference re-calibration: time-segmented ECM + Lu–Chipman joint recalibration with hysteresis triggers.
External References
- Azzam, R. M. A., & Bashara, N. M. Ellipsometry and Polarized Light.
- Compain, E., Poirier, S., & Drevillon, B. General and self-consistent method for the calibration of polarization modulators and analyzers (ECM).
- Lu, S.-Y., & Chipman, R. A. Interpretation of Mueller matrices based on polar decomposition.
- Goldstein, D. H. Polarized Light, Third Edition.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: δDoP, δχ, δψ, D, Δ, δ, ‖M·M_physical−I‖_F, κ(K), b_0, C_ij, (Ψ,Δ) residuals, κ_T, κ_λ, κ_P, P_ret as defined in Section II; SI units (angle deg, wavelength nm, temperature K, power as relative %, matrix norms dimensionless).
- Processing details: ECM initial calibration & drift tracking; Lu–Chipman decomposition with physical-domain projection; TLS+EIV unified uncertainty propagation; spectrum–time consistency checked by kernel transforms; hierarchical Bayes for sample/platform/environment layers.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 10%.
- Layer robustness: increasing G_env raises κ(K) and lowers KS_p; gamma_Path>0 with confidence > 3σ.
- Noise stress test: adding 5% 1/f and mechanical perturbations increases psi_interface; overall parameter drift < 12%.
- Prior sensitivity: with gamma_Path ~ N(0, 0.03^2), posterior means shift < 8%; evidence ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.044; blind new-condition tests maintain ΔRMSE ≈ −12%.
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”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
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
License link：https://creativecommons.org/licenses/by/4.0/