40-EFT.WP.Materials.Superconductivity v1.0 | Chapter7: Path Terms & Arrival Time in Low-Temperature Electromagnetic Measurements

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Chapter7: Path Terms & Arrival Time in Low-Temperature Electromagnetic Measurements

I. Scenarios & Objectives
Low-temperature electromagnetic measurements (cavity/resonator, waveguide/coax, time-domain THz/optical) use phase/group delay as core observables. To ensure cross-instrument and cross-band consistency, this chapter anchors on two equivalent arrival-time conventions, declares the path gamma(ell) and measure d ell, and cleanly separates geometric and intrinsic material contributions. We deliver M7-* workflows and I7-* interfaces, and link to Chapter 6 (coherence-window weighting) and Chapter 8 (measurement matrix & identifiability).

II. Propagation & Arrival Time (Conventions & Decomposition)

1. S70-1 (Two arrival-time conventions; path/measure mandatory)
   • Pulled-constant: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • Integrand form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
     Choose exactly one (no mixing). For either, explicitly declare gamma(ell) and d ell.
2. S70-2 (Piecewise path & differential baseline)
   Decompose the total path into {free-space | fixture | substrate | film | sample-body} and isolate material terms via differencing:
   ΔT_arr(ω) = T_arr^{sample}(ω) − T_arr^{blank}(ω) ≈ (1 / c_ref) ∫_{sample} Δn_eff(ω, r; θ) d ell.
3. S70-3 (Physical mapping of n_eff)
   n_eff(ω, χ) = n_bg(ω, geom) ⊕ Δn_SC(ω; λ_L, σ(ω), d, ξ, …), where Δn_SC is governed by κ_ij (Chapter 4) and windowed parameters (Chapter 6); n_bg is obtained by calibration/de-embedding.
4. S70-4 (Group delay vs. phase)
   With unwrapped phase φ(ω), group delay τ_g = ∂φ/∂ω. Under weak dispersion and single-mode conditions, τ_g ≃ T_arr; deviations are controlled by ∂ n_eff/∂ω and multimode coupling.

III. Cavity/Resonator Methods (Frequency/Q → λ_L / ξ / σ(ω))

• S70-5 (Resonant perturbation approximation)
  For known modal energy U and fields E,B,
  Δf/f ≈ - (1/2U) ∫ Δε |E|^2 dV - (1/2U) ∫ Δ(μ^{-1}) |B|^2 dV,
  Δ(1/Q) ≈ (1/U) ∫ P_loss dV,
  with SC contributions entering via σ(ω) and λ_L.
• M7-1 (Cavity/resonator workflow)
  Input: f, Q, φ(ω), cavity-mode map/calibration, blank baseline.
  Steps: (i) blank vs sample differencing → Δf, ΔQ, Δφ; (ii) convert Δφ → ΔT_arr using the chosen T_arr convention; (iii) invert λ_{L,i}(T), σ(ω), ξ_i with energy-density weighting; (iv) record delta_form and gamma(ell).
  Output: lambda_L(T), sigma(ω), xi(T) with confidence intervals.
• I7-1 fit_from_cavity(f, Q, phi, mode_map, convention, path) -> {lambda_L, sigma, xi, ci}

IV. Waveguide/Thin-Film Transmission (S-parameters/TDTS → T_arr & Material Parameters)

• S70-6 (Phase→arrival time from S21)
  After de-embedding to reference planes, under single-mode, weak multipath, and successful unwrapping,
  T_arr(ω) = - (∂/∂ω) arg S21(ω).
• S70-7 (Thin-film on substrate model)
  For film thickness d and known n_sub(ω), the transmission phase
  φ(ω) = k_0 n_sub L + f(d, σ(ω), λ_L, ξ, mode); TE/TM waveguides include cutoff and modal-dispersion terms.
• M7-2 (TDTS/waveguide transmission workflow)
  Input: S21(ω) or time-domain trace, blank/substrate-only references, fixture cal.
  Steps: (i) OSLT/TRL de-embedding → reference planes; (ii) unwrap → φ(ω); (iii) compute T_arr per chosen convention; (iv) jointly fit λ_L, σ(ω), d with the thin-film–substrate model; (v) fuse across band using W_coh.
  Output: T_arr(ω), lambda_L(T, ω), sigma(ω), d.
• I7-2 compute_arrival_time(S21_or_trace, convention, path, unwrap) -> {T_arr(ω), meta}
• I7-3 fit_thinfilm_transmission(T_arr, model, priors) -> {lambda_L, sigma, d, post}

V. De-Embedding & Geometric Decoupling (Fixture/Path Calibration)

• S70-8 (Path de-embedding)
  With segmented path gamma = ⊕_k gamma_k, write T_arr = Σ_k T_arr^{(k)}, segments {free, fixture_in, substrate, film, sample, fixture_out}. Use {open, short, load, thru} or {blank, substrate-only} standards to determine and freeze T_arr^{(k≠sample)}.
• M7-3 (Calibration workflow)
  Input: standards set, drift monitors for instrument vs. temperature/field.
  Steps: (i) multi-T calibration of n_bg(ω, geom); (ii) drift modeling & in-run correction; (iii) produce a “geometry-frozen” bundle for M7-1/M7-2.
  Output: cal_bundle = { n_bg(ω), drift_model, refs }.
• I7-4 deembed_path(standards, scheme) -> {n_bg(ω), T_arr_bg, drift}

VI. Coherence Window & Multi-Band Fusion (Link to Chapter 6)

• S70-9 (Coherence-weighted fusion)
  For multi-band observables y(ω) = { T_arr, amp, Q, … }, weight within W_coh(ω) by w(ω) ∝ L_coh(ω) or by Fisher information density; data outside W_coh are down-weighted or treated as systematic terms.
• M7-4 (Fusion & outlier handling)
  Input: outputs from heterogeneous instruments/bands.
  Steps: robust regression + W_coh weighting.
  Output: merged lambda_L, xi, sigma(ω) with consistency statistics.
• I7-5 fuse_multi_band(results[], weights) -> {merged_params, consistency}

VII. Uncertainty & Power (Metrology Formulation)

• S70-10 (Error propagation)
  For f(T_arr, cal, unwrap), apply chain rule with full covariance to obtain u_c(f) and expanded uncertainty U; for phase unwrapping, include piecewise uncertainties from discrete jumps as systematics.
• M7-5 (Power optimization)
  Given a target threshold τ and resource bounds (field/temperature/time), optimize band allocation and scan grids to maximize tr(F) (Fisher) or minimize U; export design guidance and share Jacobians with Chapter 8.

VIII. Data Contract & Recording (Mandatory Fields)

• M7-6 (Dataset/Pipeline cards — arrival-time section)
• measurement:
• band: ["microwave","THz","optical"]
• convention: "pulled_const" # or "integrand"
• delta_form: "c_ref^-1 * ∫ n_eff dℓ" # or "∫ (n_eff/c_ref) dℓ"
• gamma: "piecewise: free|fixture|substrate|film|sample"
• d_ell: "line element in meters"
• unwrap: {method: "phase_unwrap_v2", params: {...}}
• deembed: {scheme: "OSLT", refs: ["blank","substrate-only"]}
• c_ref: 299792458.0
• outputs: ["T_arr(ω)","lambda_L(T,ω)","sigma(ω)","xi(T)","d"]
• references:
• - "EFT.WP.Core.Equations v1.1:Ch.2 S20-*"
• - "EFT.WP.Core.DataSpec v1.0:TARR"
• see:
• - "EFT.WP.Core.Metrology v1.0:check_dim"
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.6 W_coh"

IX. Mapping to Material Parameters & Measurement Matrix

• S70-11 (Sensitivity)
  Define J = ∂y/∂θ with
  y = { T_arr(ω), Δf/f, 1/Q, |S21|, arg S21 },
  θ = { λ_L, ξ, σ1, σ2, d, κ_ij, ... }.
  Inject J into Chapter 8 to evaluate condition numbers and identifiability.
• S70-12 (Joint inversion entry)
  Combine phase-dominant channels (ΔT_arr) with amplitude-dominant channels (|S21|, 1/Q) to constrain σ1/σ2 and λ_L; in the thin-film limit, include a prior window on d (Chapter 6).

X. Testable Predictions (Covariant with Tension Landscapes)

• Arrival-time–amplitude co-variation: Co-variation of T_fil and n_eff drives same-sense drifts of ΔT_arr(ω) and amplitude attenuation under low-T strain/pressure gating.
• Principal-axis flipping: Switching TE↔TM or rotating the sample flips the angular extrema of ΔT_arr(ω) in concert with the vortex-orientation angle φ_0 (Chapter 5).
• Dispersion-softening sector: Along increasing |grad T_fil| within W_coh, the effective weight of ∂ n_eff/∂ω diminishes, bringing group delay closer to the pulled-constant T_arr.

XI. Cross-Volume References & Anchors (This Chapter)

• Cross-volume (fixed style): EFT.WP.Core.Equations v1.1 Ch.2 S20-* (two arrival-time conventions; path/measure declarations); EFT.WP.Core.DataSpec v1.0 (data contracts & fields); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (dimensions/units/uncertainties). Link internally with Chapter 4 (coefficients & intrinsic quantities), Chapter 6 (windows & coherence), Chapter 8 (measurement matrix).
• Anchors (S/M/I): S70-1—S70-12; M7-1—M7-6; I7-1—I7-5.

XII. Summary
Using two rigorous arrival-time conventions as a unifying anchor, this chapter establishes an executable chain from path de-embedding and phase unwrapping to material-parameter inversion, enforced by explicit data contracts and coherence-window fusion for cross-band/platform consistency. Together with Chapters 6 and 8, it provides stable entry points for inversion and model comparison in Chapter 10.