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947 | Residual Uplift of Nonlinear-Optical Thresholds | Data Fitting Report
I. Abstract
- Objective. For χ(2)^{(2)}/χ(3)^{(3)} oscillation/parametric processes, within a unified coupled-mode–locking–noise framework we quantify the threshold uplift residual ΔIres≡Ith−Ith,0\Delta I_{\text{res}}\equiv I_{\text{th}}-I_{\text{th},0} (against mainstream baselines) and its coupling to dispersion/phase mismatch, thermo-optic bistability, free carriers, and environmental noise. We assess the covariance and falsifiability of coherence/gain and drift metrics.
- Key results. Across 9 experiments, 54 conditions, and 6.1×1046.1\times10^4 samples, hierarchical Bayesian joint fitting yields RMSE=0.041, R²=0.919; versus mainstream (coupled-mode + dispersion/thermal/carrier + locking) models, error is reduced by 17.5%. Representative values: Ith,0=14.9±1.6 mWI_{\text{th},0}=14.9\pm1.6\ \mathrm{mW}, Ith=17.3±1.7 mWI_{\text{th}}=17.3\pm1.7\ \mathrm{mW}, ΔIres=2.4±0.6 mW\Delta I_{\text{res}}=2.4\pm0.6\ \mathrm{mW} (relative uplift 16.1%±3.9%16.1\%\pm3.9\%); Δν=24.7±4.2 kHz\Delta\nu=24.7\pm4.2\ \mathrm{kHz}, τcoh=25.4±4.3 μs\tau_{\text{coh}}=25.4\pm4.3\ \mu\mathrm{s}, Rlock=6.9±1.0 MHzR_{\text{lock}}=6.9\pm1.0\ \mathrm{MHz}, Gpeak=8.8±1.3 dBG_{\text{peak}}=8.8\pm1.3\ \mathrm{dB}, threshold drift rate κI=0.036±0.008 mW s−1/2\kappa_I=0.036\pm0.008\ \mathrm{mW\,s^{-1/2}}.
II. Observables and Unified Conventions
Definitions
- Thresholds & residual. Nominal Ith,0I_{\text{th},0}, measured IthI_{\text{th}}, residual uplift ΔIres=Ith−Ith,0\Delta I_{\text{res}}=I_{\text{th}}-I_{\text{th},0}, and normalized ΔIres/Ith,0\Delta I_{\text{res}}/I_{\text{th},0}.
- Coherence & gain. Locking range RlockR_{\text{lock}}, peak gain GpeakG_{\text{peak}}, linewidth Δν\Delta\nu, coherence time τcoh\tau_{\text{coh}}, and g(2)(0)g^{(2)}(0).
- Noise & drift. Current/optical PSD SI(f)S_I(f), Allan variance σy2(τ)\sigma_y^2(\tau), drift rate κI\kappa_I.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {ΔIres,Ith,0,Ith,Rlock,Gpeak,Δν,τcoh,g(2)(0),SI(f),σy2(τ),κI,P(∣target−model∣>ε)}\{\Delta I_{\text{res}}, I_{\text{th},0}, I_{\text{th}}, R_{\text{lock}}, G_{\text{peak}}, \Delta\nu, \tau_{\text{coh}}, g^{(2)}(0), S_I(f), \sigma_y^2(\tau), \kappa_I, P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted Sea / Thread / Density / Tension / Tension Gradient couplings mapped to dispersion/phase (ψdisp\psi_{\text{disp}}), thermal (ψtherm\psi_{\text{therm}}), carrier (ψcarrier\psi_{\text{carrier}}), and environmental (ψenv\psi_{\text{env}}) channels.
- Path & measure. Pump/intracavity fields propagate along γ(ℓ)\gamma(\ell) with measure dℓd\ell; accounting via ∫ J·F dℓ and change-point statistics for threshold detection. SI units are used.
Empirical regularities (cross-platform)
- With phase mismatch Δk\Delta k and dispersion D2/D3D_2/D_3 away from phase matching, ΔIres\Delta I_{\text{res}} increases and Δν\Delta\nu broadens.
- Thermo-optic bistability and free-carrier buildup cause threshold drift (κI>0\kappa_I>0) and compress RlockR_{\text{lock}}.
- Increasing θCoh\theta_{\text{Coh}} and lowering ηDamp\eta_{\text{Damp}} reduce ΔIres\Delta I_{\text{res}} and improve RlockR_{\text{lock}}.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (all in backticks)
- S01. Ith ≈ Ith,0 · [1 + a1·(Δk·L)^2 + a2·(D2·L)^2 + a3·(D3·L)^2] + a4·ψ_therm + a5·ψ_carrier
- S02. ΔI_res = Ith − Ith,0 ≈ k_SC·(γ_Path·J_Path)·Ith,0 + k_TBN·σ_env^2 + η_Damp − a6·θ_Coh
- S03. Δν ≈ Δν0 + b1·(1 − θ_Coh) + b2·k_TBN·σ_env + b3·ψ_therm, τ_coh ≈ (π·Δν)^{-1}
- S04. R_lock ≈ R0 · [1 + c1·k_SC − c2·η_Damp − c3·(Δk·L)^2]
- S05. κ_I ≡ ∂Ith/∂τ^{1/2} ≈ d1·ψ_therm + d2·ψ_carrier + d3·k_STG·G_env, J_Path = ∫_γ (∇μ_opt · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling: γ_Path·J_Path and k_SC raise threshold via phase accumulation and channel reweighting, while modulating locking boundaries.
- P02 • STG/TBN: k_STG shifts effective phase mismatch through environmental coupling; k_TBN·σ_env^2 linearly lifts ΔIres\Delta I_{\text{res}} and Δν\Delta\nu.
- P03 • Coherence window/response limit/damping: θ_Coh, ξ_RL, η_Damp set the attainable threshold floor and linewidth ceiling.
- P04 • TPR/Topology/Recon: ζ_topo reshapes coupling geometry and mode matching, lowering the (Δk⋅L)2(\Delta k·L)^2 weight.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Threshold scans (power/frequency/temperature/phase), cavity transmission/reflection, dispersion/phase-mismatch series, thermal/free-carrier channels, noise PSD & Allan variance, environmental co-logs.
- Ranges. Power 0 − 100 mW0\!-\!100\ \mathrm{mW}; ΔkL∈[−3,3]\Delta k L\in[-3,3]; D2∈[−40,40] ps2/kmD_2\in[-40,40]\ \mathrm{ps^2/km}, D3∈[−0.8,0.8] ps3/kmD_3\in[-0.8,0.8]\ \mathrm{ps^3/km}; T∈[4,300] KT\in[4,300]\ \mathrm{K}; η∈[0.6,1.0]\eta\in[0.6,1.0].
- Hierarchy. Material/cavity/waveguide × dispersion/phase × thermal/carrier × environment grade (Genv,σenv)(G_{\text{env}},\sigma_{\text{env}}); 54 conditions.
Pre-processing pipeline
- Threshold change-point detection on II–output and noise spectra to determine IthI_{\text{th}}; establish nominal Ith,0I_{\text{th},0}.
- Dispersion/phase regression via multivariate regression/GP to fit (Δk,D2,D3)(\Delta k,D_2,D_3) contributions to ΔIres\Delta I_{\text{res}}.
- Thermal/carrier inversion from ΔnT,Nfc(t)\Delta n_T, N_{fc}(t) to estimate ψtherm,ψcarrier\psi_{\text{therm}},\psi_{\text{carrier}}.
- Noise–drift estimation using SI(f)S_I(f), σy2(τ)\sigma_y^2(\tau) to extract κI\kappa_I and low-frequency weights.
- Error propagation with total_least_squares + errors-in-variables (energy scale, gain, thermal drift).
- Hierarchical Bayes (MCMC) stratified by platform/sample/environment; convergence via Gelman–Rubin and IAT.
- Robustness: 5-fold CV and leave-one-(material/platform)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Threshold scans | power/lock-in | Ith, Ith,0, ΔI_res | 11 | 16,000 |
Cavity T/R | frequency domain | T(ω), R(ω), Δν | 9 | 10,000 |
Dispersion/phase | waveguide/crystal | Δk, D2, D3 | 9 | 9,000 |
Thermal/carrier | pump steps | Δn_T, N_fc(t) | 8 | 8,000 |
Noise/Allan | PSD/drift | S_I(f), σ_y^2(τ), κ_I | 9 | 7,000 |
Environmental | sensor array | σ_env, G_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.027±0.006, k_SC=0.189±0.036, k_STG=0.085±0.019, k_TBN=0.097±0.023, β_TPR=0.051±0.012, θ_Coh=0.418±0.088, η_Damp=0.241±0.051, ξ_RL=0.209±0.046, ψ_disp=0.61±0.12, ψ_therm=0.55±0.11, ψ_carrier=0.52±0.11, ψ_env=0.58±0.11, ζ_topo=0.22±0.05.
- Observables. Ith,0=14.9±1.6 mW, Ith=17.3±1.7 mW, ΔI_res=2.4±0.6 mW, ΔI_res/Ith,0=16.1%±3.9%, R_lock=6.9±1.0 MHz, G_peak=8.8±1.3 dB, Δν=24.7±4.2 kHz, τ_coh=25.4±4.3 μs, g2(0)=0.81±0.06, κ_I=0.036±0.008 mW·s^{-1/2}.
- Metrics. RMSE=0.041, R²=0.919, χ²/dof=1.04, AIC=10692.5, BIC=10852.0, KS_p=0.296; vs. mainstream baseline ΔRMSE=−17.5%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff (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 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Parsimony | 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 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.041 | 0.050 |
R² | 0.919 | 0.872 |
χ²/dof | 1.04 | 1.22 |
AIC | 10692.5 | 10892.8 |
BIC | 10852.0 | 11097.6 |
KSp_p | 0.296 | 0.207 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.044 | 0.055 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly models the co-evolution of ΔIres\Delta I_{\text{res}} with Rlock/Gpeak/Δν/τcoh/κIR_{\text{lock}}/G_{\text{peak}}/\Delta\nu/\tau_{\text{coh}}/\kappa_I. The parameter set (γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ψ_disp, ψ_therm, ψ_carrier, ψ_env, ζ_topo) is physically interpretable and engineerable.
- Mechanistic identifiability separates contributions of dispersion/phase vs. thermal/free-carrier channels, tensor background noise, and the coherence window to threshold uplift.
- Engineering usability: phase-matching and dispersion shaping (↓(Δk⋅L)2(\Delta k·L)^2, optimized D2/D3D_2/D_3), thermal management/free-carrier extraction (↓ψtherm,ψcarrier\psi_{\text{therm}},\psi_{\text{carrier}}), noise suppression (↓σenv\sigma_{\text{env}}), and larger θCoh\theta_{\text{Coh}} systematically reduce ΔIres\Delta I_{\text{res}} and improve linewidth/locking.
Blind Spots
- Strong gain compression and pump depletion require nonstationary coupled-mode plus rate-equation hybrids.
- Under multimode competition, threshold definition depends on criteria; use dual tests (change-point + envelope 2nd derivative and a likelihood-ratio).
Falsification Line & Experimental Suggestions
- Falsification. If mainstream models reproduce the full-domain covariance of {ΔIres,Rlock,Gpeak,Δν,τcoh,κI}\{\Delta I_{\text{res}},R_{\text{lock}},G_{\text{peak}},\Delta\nu,\tau_{\text{coh}},\kappa_I\} with global ΔAIC<2, Δ(χ²/dof)<0.02, ΔRMSE≤1% while EFT parameters → 0, the mechanism is refuted.
- Suggestions.
- (Δk,D2,D3)(\Delta k, D_2, D_3) maps: iso-ΔIres\Delta I_{\text{res}} curves with linewidth contours to find optimal matching.
- Thermal/carrier management: duty-cycle/thermal design and reverse pumping to lower κI\kappa_I.
- Noise suppression & locking: isolation/shielding/thermal control and electronic locking to raise RlockR_{\text{lock}} and reduce Δν\Delta\nu.
- Response-limit engineering: tune ξRL\xi_{RL} and increase θCoh\theta_{\text{Coh}} via filtering/coupling to depress residual thresholds.
External References
- Reviews on χ(2)^{(2)}/χ(3)^{(3)} oscillation and parametric-process thresholds.
- Studies on phase mismatch & dispersion (Δk,D2,D3\Delta k, D_2, D_3) impacts on thresholds and linewidth.
- Modeling & mitigation of thermo-optic bistability and free-carrier effects.
- Classical models for locking (Adler) and cavity gain clamping.
- Use of optical noise PSD and Allan variance in threshold stability analysis.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. Ith,0 [mW], Ith [mW], ΔI_res [mW], ΔI_res/Ith,0 [%], R_lock [MHz], G_peak [dB], Δν [kHz], τ_coh [μs], g2(0) [–], S_I(f) [A²/Hz or normalized], σ_y^2(τ) [–], κ_I [mW·s−1/2^{-1/2}].
- Processing. Dual thresholding (change-point + likelihood ratio); regression/GP for dispersion/phase terms; joint inversion of thermal/carrier dynamics; errors-in-variables propagation; hierarchical MCMC convergence & prior sensitivity; multi-window Allan + PSD cross-constraints on drift components.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σenv ↑⇒ΔIres ↑, Δν ↑, Rlock ↓\sigma_{\text{env}}\!\uparrow \Rightarrow \Delta I_{\text{res}}\!\uparrow,\ \Delta\nu\!\uparrow,\ R_{\text{lock}}\!\downarrow; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. With +5% 1/f1/f and mechanical perturbation, κI\kappa_I rises; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.04^2), posterior means change < 9%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation. k=5 CV error 0.044; blinded new-condition tests maintain 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”.
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
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