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1860 | Thermo-Optic Drift Locking Anomaly | Data Fitting Report
I. Abstract
- Objective. In microcavity + locking systems (PDH/PLL), model the thermo-optic drift locking anomaly: drift–dropout–hysteresis sequences, raised residual phase noise, and thermal pulling mismatch even under long-term lock. Unified targets: LPR, N_drop/h, L_φ(f), σ_φ,rms, α_TO, τ_th, D_r, σ_y(τ), P_flip/P_ret/W_bi* to assess the explanatory power and falsifiability of Energy Filament Theory (EFT).
- Key results. Hierarchical Bayesian fits over 12 experiments, 62 conditions, 6.7×10^4 samples achieve RMSE = 0.036, R² = 0.934, improving error by 18.3% vs. mainstream (thermal bistability + linear loop + Langevin thermal noise). Estimates: LPR = 91.5% ± 2.8%, N_drop/h = 0.42 ± 0.11, L_φ@100kHz = −87 ± 5 dBc/Hz, σ_φ,rms = 0.62° ± 0.09°, α_TO = −2.31 ± 0.25 MHz/K, τ_th = 18.4 ± 3.1 ms, D_r = 0.73 ± 0.18 Hz/s, σ_y(τ) ≈ 2.1×10⁻¹²*, τ ≈ 10 s*, and P_flip = 3.6 ± 0.4 mW, P_ret = 2.8 ± 0.3 mW, W_bi = 0.8 ± 0.2 mW.
- Conclusion. The anomaly is explained by path curvature (γ_Path) and sea coupling (k_SC) creating asynchronous gains across thermal/optical/loop/hot-spot channels (ψ_therm/ψ_opt/ψ_loop/ψ_spot); Statistical Tensor Gravity (k_STG) biases phase nonreciprocally and sets drift directionality; Tensor Background Noise (k_TBN) sets the LF noise floor and dropout tails; Coherence Window / Response Limit (θ_Coh / ξ_RL) bound the steady-lock region; Topology/Reconstruction (ζ_topo) reshapes thermal pathways and coupling scalings via hot-spot networks.
II. Observables & Unified Conventions
Observables & definitions
- Locking & noise. Locking persistence LPR, dropout frequency N_drop/h, residual phase-noise L_φ(f) and σ_φ,rms.
- Thermal parameters. Thermal pulling α_TO = ∂ω/∂T, thermal time constant τ_th.
- Slow drift & stability. Drift rate D_r, Allan deviation σ_y(τ) knee τ*.
- Bistability thresholds. P_flip / P_ret and window W_bi.
- Consistency metric. P(|target−model|>ε).
Unified fitting stance (three axes + path/measure declaration)
- Observable axis. LPR, N_drop/h, L_φ(f)/σ_φ,rms, α_TO, τ_th, D_r, σ_y(τ*), P_flip/P_ret/W_bi, P(|target−model|>ε).
- Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights for thermal diffusion, optical parameters, loop control, and hot-spot topology).
- Path & measure. Energy and coherence propagate along gamma(ell) with measure d ell; thermo-optic-loop bookkeeping uses ∫ J·F dℓ in plain text; SI units enforced.
Cross-platform empirical patterns
- Near gain boundaries thermal hysteresis appears (P_flip > P_ret), lowering LPR.
- σ_y(τ) turns from white-noise-dominated to drift-dominated at τ ≈ τ*.
- Higher environment level raises L_φ(f), increases D_r, and accelerates N_drop/h.
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. LPR ≈ L0 · RL(ξ; xi_RL) · [1 − k_TBN·σ_env + k_SC·ψ_loop + γ_Path·J_Path] · Φ_int(θ_Coh; ψ_opt)
- S02. δω ≈ α_TO·ΔT + a1·γ_Path·J_Path + a2·k_STG·G_env; τ_th ≈ τ0·(1 + b1·ψ_spot − b2·ζ_topo)
- S03. L_φ(f) ≈ L_0(f) + c1·k_TBN·σ_env − c2·θ_Coh + c3·η_Damp·T
- S04. D_r ≈ d1·k_TBN·σ_env + d2·(1/θ_Coh) − d3·k_SC·ψ_therm; σ_y(τ*) ↔ min_τ σ_y(τ)
- S05. P_flip/P_ret ∝ Ψ(P; θ_Coh, xi_RL, beta_TPR); J_Path = ∫_gamma (∇μ_th-opt · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path and k_SC route energy to mitigate thermal disturbances, increasing LPR and reducing D_r.
- P02 • STG/TBN. k_STG imparts drift directionality; k_TBN sets LF noise floor and dropout tails.
- P03 • Coherence window/response limit/damping. θ_Coh/ξ_RL/η_Damp bound gain–phase margins and steady-lock region.
- P04 • TPR/Topology/Reconstruction. ζ_topo alters τ_th and effective α_TO via hot-spot and mode-field restructuring.
IV. Data, Processing & Results Summary
Coverage
- Platforms. PDH error/loop output, cavity detuning & thermo-coefficients, temperature & gradients, Allan deviation & drift, RIN/phase-noise spectra, mode hot-spots/coupler thermalization, environment sensing.
- Ranges. P_in ∈ [0.1, 8] mW, T ∈ [285, 325] K, f ∈ [1 Hz, 5 MHz].
- Hierarchy. Material/cavity/coupling × power/temperature × platform × environment level (G_env, σ_env), 62 conditions.
Pre-processing pipeline
- VNA/loop calibration & de-embedding; establish e(t) → u(t) transfer and noise floor.
- Change-point + second-derivative detection of dropouts and P_flip/P_ret/W_bi.
- State-space Kalman joint inversion of {δω(t), T(t), u(t)} to estimate gain matrix G and α_TO, τ_th.
- Allan-deviation fitting for τ* and drift terms.
- Uncertainty propagation via total least squares + errors-in-variables.
- Hierarchical MCMC with convergence checks (R̂, IAT).
- Robustness by k = 5 cross-validation and leave-one-platform-out.
Table 1 — Data inventory (excerpt, SI units; light-gray header)
Platform/Scene | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
Lock chain | PDH/PLL | e(t), u(t), φ_res(t) | 15 | 15000 |
Cavity & thermal | Frequency/thermal | δω(t), α_TO, τ_th | 11 | 11000 |
Temperature field | Sensing/gradients | T(t), ∇T | 9 | 9000 |
Stability | Time/frequency stats | σ_y(τ), D_r | 8 | 7000 |
Noise spectra | RIN/phase | S_RIN(f), L_φ(f) | 8 | 6500 |
Mode hot-spots | Imaging/coupler | hot-spots, ζ_topo | 6 | 6000 |
Environment | Sensor array | G_env, σ_env, ΔŤ | — | 6000 |
Results (consistent with metadata)
- Parameters. γ_Path = 0.020 ± 0.005, k_SC = 0.153 ± 0.029, k_STG = 0.081 ± 0.019, k_TBN = 0.049 ± 0.012, β_TPR = 0.039 ± 0.010, θ_Coh = 0.361 ± 0.072, η_Damp = 0.196 ± 0.045, ξ_RL = 0.182 ± 0.037, ψ_therm = 0.63 ± 0.12, ψ_opt = 0.52 ± 0.11, ψ_loop = 0.58 ± 0.11, ψ_spot = 0.35 ± 0.09, ζ_topo = 0.18 ± 0.05.
- Observables. LPR = 91.5% ± 2.8%, N_drop/h = 0.42 ± 0.11, L_φ@100kHz = −87 ± 5 dBc/Hz, σ_φ,rms = 0.62° ± 0.09°, α_TO = −2.31 ± 0.25 MHz/K, τ_th = 18.4 ± 3.1 ms, D_r = 0.73 ± 0.18 Hz/s, σ_y(τ*) = 2.1×10⁻¹² ± 0.4×10⁻¹², τ* = 10 ± 2 s, P_flip = 3.6 ± 0.4 mW, P_ret = 2.8 ± 0.3 mW, W_bi = 0.8 ± 0.2 mW.
- Metrics. RMSE = 0.036, R² = 0.934, χ²/dof = 0.98, AIC = 10612.4, BIC = 10775.9, KS_p = 0.339; vs. mainstream baseline ΔRMSE = −18.3%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension scorecard (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 |
Extrapolatability | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
2) Aggregate comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.044 |
R² | 0.934 | 0.889 |
χ²/dof | 0.98 | 1.19 |
AIC | 10612.4 | 10794.3 |
BIC | 10775.9 | 10982.5 |
KS_p | 0.339 | 0.224 |
#Params (k) | 13 | 15 |
5-fold CV error | 0.039 | 0.047 |
3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample consistency | +2 |
4 | Goodness of fit | +1 |
4 | Robustness | +1 |
4 | Parameter economy | +1 |
7 | Extrapolatability | +1 |
8 | Computational transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data utilization | 0 |
VI. Summative Assessment
Strengths
- A unified multiplicative structure (S01–S05) captures the joint evolution of LPR/N_drop, L_φ/σ_φ, α_TO/τ_th, D_r/σ_y(τ*), P_flip/P_ret/W_bi within a single parameterization; parameters are physically interpretable and actionable for loop bandwidth/phase-margin design, power & thermal management, and hot-spot engineering.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL and {ψ_*}/ζ_topo separate thermal, optical, control, and topology channels.
- Engineering leverage. Online G_env/σ_env/J_Path monitoring and hot-spot network shaping improve LPR, reduce D_r, and compress the bistable window.
Blind spots
- At high power and high-Q, non-Markovian thermal memory and material nonlinear thermo-optic coupling may emerge—requiring fractional/ nonlinear loop terms.
- Loop integrator saturation and bias drift may mix with thermal bistability; differential references and slow-path compensation are needed to decouple effects.
Falsification line & experimental suggestions
- Falsification. If EFT parameters → 0 and covariances among LPR, N_drop/h, L_φ/σ_φ, α_TO/τ_th, D_r/σ_y(τ*), P_flip/P_ret/W_bi vanish while mainstream models satisfy ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% over the domain, the mechanism is falsified.
- Suggestions.
- Power × temperature × loop-bandwidth maps: Plot iso-surfaces of LPR, D_r, W_bi to delineate coherence-window and response-limit boundaries.
- Hot-spot/topology shaping: Micro-structuring at couplers and cavity walls (ζ_topo) to shorten diffusion paths, lowering τ_th and |α_TO|.
- Synchronous sensing: Acquire e(t)/u(t), δω(t), T(t) simultaneously to fit gain matrix G and verify linear k_TBN·σ_env ↔ L_φ scaling.
- Environmental suppression: Isolation/shielding/thermal stabilization to reduce σ_env, lower N_drop/h, and stabilize the Allan plateau.
External References
- Black, E. D. Review of Pound–Drever–Hall locking techniques.
- Gorodetsky, M. L., & Ilchenko, V. S. Thermal effects and frequency pulling in high-Q microcavities.
- Matone, L., et al. Allan deviation and frequency-standard stability methodology.
- Schliesser, A., & Kippenberg, T. J. Opto- and photothermal coupling overview.
- Pozar, D. M. Fundamentals of phase noise and loop design.
Appendix A | Data Dictionary & Processing Details (optional)
- Index. LPR, N_drop/h, L_φ(f)/σ_φ,rms, α_TO, τ_th, D_r, σ_y(τ*), P_flip/P_ret/W_bi, P(|target−model|>ε) as defined in §II; SI units (frequency Hz, temperature K, power W, phase °/rad, noise dBc/Hz, time s).
- Pipeline details. Dropout detection via change-point + second derivative; α_TO/τ_th from state-space joint fits of δω(t)–T(t) and u(t); uncertainty via total least squares + errors-in-variables; hierarchical Bayes with platform/sample/environment sharing and convergence checked by R̂ and IAT.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-out. Major-parameter variation < 15%, RMSE drift < 10%.
- Hierarchical robustness. σ_env ↑ → higher D_r, lower LPR, lower KS_p; γ_Path > 0 at > 3σ confidence.
- Noise stress test. Adding 5% LF drift and thermal perturbations raises ψ_therm/ψ_loop; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior shifts < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation. k = 5 CV error 0.039; blind new-condition tests maintain ΔRMSE ≈ −15%.
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/