23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 8 — Fiber/Cable Paths (Thermal Drift / Dispersion / Connectors)

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Chapter 8 — Fiber/Cable Paths (Thermal Drift / Dispersion / Connectors)

One-Sentence Goal
Center the model on n_g(f,T) and build a unified treatment of group delay, temperature coefficients, chromatic/velocity dispersion, and connector-induced extra delay for optical fibers and cables; provide round-trip symmetry auditing and calibration conventions, and persist outputs as manifest.path.fiber.*.

I. Scope and Objects

1. Inputs
   • Medium & structure: cable_spec = { type, L_gamma, fiber/coax class, lambda or f, PMD_coef, D(lambda), S = dD/dlambda, alpha_L, dn_g/dT, splices, connectors }.
   • Environment: temp_profile(x,t) or equivalent T_eff(t); segmented temperatures and weights for rooms/shafts/conduits.
   • Signaling & directionality: single-fiber bidirectional or dual-fiber bidirectional; lambda_AB, lambda_BA or f_AB, f_BA; WDM/DWDM.
   • Measurements: RTT (round trip), OTDR/S-params, loopback / comparison links, ts.
   • Reference: RefCond (data sources, update cadence, versioning, installation topology hash).
2. Outputs
   • T_fiber(f) or T_fiber(lambda), with components: T_geom, T_TCD, T_CD, T_PMD, T_conn, and round-trip asymmetry residual epsilon_asym.
   • Quality fields: delta_form, u(T_fiber), tags, and policy cards.
3. Applicability and Boundaries
   • Group-delay modeling for SMF, PMF, dispersion-compensating fiber (DCF), coaxial cable, and twisted pair in the propagation domain.
   • Digital switching, buffering, and protocol-stack processing delays belong to the Instrument volume and enter here only as interface variables.

II. Terms and Variables

• n_phi(f,T) phase index; n_g(f,T) group index; unit = "1".
• T_fiber(f) = ( ∫_{gamma} n_g(f,T) d ell ) / c_ref, unit = "s", dim = "[T]".
• L_gamma = ( ∫_{gamma} 1 d ell ), unit = "m".
• alpha_L = ( 1 / L ) * dL/dT (linear thermal expansion), unit = "1/K".
• dn_g/dT (thermo-optic contribution to group index), unit = "1/K".
• D(lambda) (chromatic-dispersion parameter), unit = "ps/(nm•km)"; S = dD/dlambda, unit = "ps/(nm^2•km)".
• beta2 (group-velocity dispersion), unit = "s^2/m"; PMD_coef (first-order PMD), unit = "ps/√km"; DGD (differential group delay), unit = "ps".
• v_p(f,T) (phase velocity, cable), VF = v_p/c_ref (velocity factor).
• tau_conn (connector/device group delay), tau_ripple(f) = - d arg(H_conn)/d omega.
• epsilon_asym = ( T_AB - T_BA ) / 2 (round-trip asymmetry residual).

III. Axioms P808-*

• P808-1 (Piecewise Path Integral) — Express propagation delay by segmentwise integration:
  T_med = ( ∫_{segments} ( n_g(f,T(x,t)) / c_ref ) d ell ) + ∑ tau_conn, with segment-level RefCond recorded.
• P808-2 (Group-Speed Convention) — Use n_g, not n_phi, for arrival-time questions in fibers/cables.
• P808-3 (Additive Thermal Drift) — Under small drifts and weak nonlinearity, the first-order temperature-induced delay change is additive, with dT/dT determined jointly by dn_g/dT and alpha_L.
• P808-4 (Separable Dispersion) — First-order dispersion (D/beta2) dominates the relative delay across carriers/wavelengths; enable higher orders by policy with explicit tags.
• P808-5 (Round-Trip Symmetry Audit) — Any one-way inference from RTT must explicitly audit epsilon_asym; never assume zero when lambda_AB ≠ lambda_BA or when devices differ by direction.
• P808-6 (Connector Equivalence) — Connectors, splices, filters, and active modules are treated here as LTI elements with tau_conn + tau_ripple(f), parameterized by calibration or S-params.
• P808-7 (Two-Form Consistency) — Compute T_form1 = ( 1 / c_ref ) * ( ∫ n_g d ell ) and T_form2 = ( ∫ ( n_g / c_ref ) d ell ) in parallel and persist delta_form.

IV. Minimal Equations S808-*

• S808-1 (Principal Group Delay, Fiber)
  T_fiber(f) = ( ∫_{gamma} n_g(f,T) d ell ) / c_ref + ∑ tau_conn(f),
  check_dim( T_fiber ) = "[T]".
• S808-2 (Temperature Coefficient and Thermal Drift)
  With T_fiber = ( n_g L ) / c_ref,
  dT_fiber/dT = ( L / c_ref ) * ( dn_g/dT + n_g * alpha_L ), unit = "s/K".
  For segments:
  dT/dT = ( ∑ L_i * ( dn_g/dT |_i + n_g |_i * alpha_L |_i ) ) / c_ref.
• S808-3 (Cable Formulation)
  T_cable(f) = L / v_p(f,T) = ( L / c_ref ) * sqrt( epsilon_eff(f,T) * mu_r ),
  dT_cable/dT = ( L / c_ref ) * ( 0.5 / sqrt( epsilon_eff * mu_r ) ) * d epsilon_eff/dT + ( L * alpha_L / v_p ).
• S808-4 (Dispersion and Inter-Wavelength Delay)
  tau_g(lambda) = ( n_g(lambda) L ) / c_ref,
  D(lambda) = ( 1 / L ) * ( d tau_g / d lambda ),
  relative delay for lambda_1, lambda_2:
  Delta_T_CD ≈ L * ∫_{lambda_1}^{lambda_2} D(lambda) d lambda ≈ L * ( D(lambda_0) * Delta_lambda + 0.5 * S * Delta_lambda^2 ).
  Relation to group-velocity dispersion:
  beta2 = - ( lambda^2 / ( 2 * pi * c_ref ) ) * D.
• S808-5 (First-Order PMD Model)
  DGD_rms = PMD_coef * sqrt( L_km ),
  equivalent delay jitter T_PMD_rms = k_p * DGD_rms (k_p ∈ [0,1] depends on polarization state and devices).
• S808-6 (Connector Group Delay and Ripple)
  tau_conn(f) = - d arg( H_conn(f) ) / d omega,
  total connector term T_conn = ∑ tau_conn(f); with only nominal data, use equivalent length L_eq via tau_conn ≈ ( n_g * L_eq ) / c_ref.
• S808-7 (Round Trip and Asymmetry)
  RTT = T_AB + T_BA + T_inst, one-way expectation
  T_1w ≈ ( RTT - T_inst ) / 2 - epsilon_asym, with
  epsilon_asym = 0.5 * ( T_AB - T_BA ) ≈ 0.5 * ( Delta_T_CD( lambda_AB, lambda_BA ) + Delta_T_conn + Delta_T_env ).
• S808-8 (Two-Form Difference)
  delta_form = | ( 1 / c_ref ) * ( ∫ n_g d ell ) - ( ∫ ( n_g / c_ref ) d ell ) |, subject to contract thresholds.

V. Metrological Workflow M80-8

1. Ready — Collect cable_spec, link topology, lambda_AB/BA or f_AB/BA, temperature sensing and historical OTDR/S-params; align to tau_mono.
2. Geometry & Segmentation — Build segment list { L_i, class_i, install_env_i } from OTDR/as-built drawings; bind alpha_L, dn_g/dT, D(lambda), S per segment.
3. Baseline — Per S808-1/3, compute T_geom = ( ∑ n_g |_i * L_i ) / c_ref (or use the cable form).
4. Thermal Drift — Project temp_profile through S808-2/3 to obtain T_TCD; if missing, use T_eff with default coefficients and tag.
5. Dispersion
   • Single link, multi-wavelength: estimate Delta_T_CD(lambda) via S808-4, write directional fields.
   • Round trip, two directions: form epsilon_asym_CD and feed one-way inference.
6. PMD — From PMD_coef and L_gamma, estimate T_PMD_rms and include as a random component of u; down-weight for PM fibers appropriately.
7. Connectors — From S-params/datasheets compute or table-look up tau_conn, tau_ripple; tag abnormal echoes or strong ripple.
8. Two Forms & Audit — Compute T_form1, T_form2, delta_form; for RTT links compute and audit epsilon_asym.
9. Persist — Emit
   manifest.path.fiber = { T_fiber, parts:{ T_geom, T_TCD, T_CD, T_PMD, T_conn }, asym:{ epsilon_asym, lambda_AB, lambda_BA }, qc:{ delta_form, u }, RefCond, cable_spec.hash, tags }.
10. Monitor — Maintain temp_coeff_drift, cd_drift, asym_drift, and dashboard metrics.

VI. Contracts and Assertions (C80-81x)

• C80-811 Completeness — cable_spec.{ type, L_gamma, lambda or f } are mandatory; reject and fall back if missing.
• C80-812 Two-Form Difference — delta_form ≤ tol_Tarr (suggest ≤ 0.02 ns).
• C80-813 Temperature Freshness — age(temp_profile) ≤ Delta_t (suggest ≤ 5 min); otherwise use T_eff and inflate u.
• C80-814 Dispersion Guard — |Delta_T_CD| ≤ cd_guard; above the guard, apply explicit correction and inflate uncertainty.
• C80-815 Round-Trip Symmetry — If |lambda_AB - lambda_BA| ≥ 1 nm or direction-dependent devices differ, compute and persist epsilon_asym; do not zero it.
• C80-816 PMD Bound — T_PMD_rms ≤ pmd_guard; if exceeded, tag pmd_high.
• C80-817 Connector Quality — return_loss ≥ RL_min, tau_ripple_p2p ≤ ripple_max; otherwise tag conn_bad.
• C80-818 Dimensional Consistency — check_dim( T_fiber ) = "[T]", check_dim( D ) = "[T][L]^-1[Lambda]^-1".
• C80-819 Direction Consistency — One-way model lambda/f must match device configuration; otherwise tag dir_mismatch.

VII. Implementation Bindings I80-*

• I80-81 model_fiber(temp, f_or_lambda, cable_spec) -> { T_fiber, parts:{T_geom,T_TCD,T_CD,T_PMD,T_conn}, qc:{delta_form,u}, tags, meta:{RefCond,cable_spec} }
  Invariants: T_fiber ≥ 0, delta_form ≤ tol_Tarr, segment list monotone.
• I80-82 estimate_TCD(cable_spec, temp_profile) -> { T_TCD, coeff, u }
• I80-83 compute_CD(lambda_AB, lambda_BA, D, S, L_gamma) -> { Delta_T_CD_AB, Delta_T_CD_BA }
• I80-84 estimate_PMD(PMD_coef, L_gamma, policy) -> { T_PMD_rms, u }
• I80-85 connector_delay(Sparams_or_catalog) -> { tau_conn(f), tau_ripple, RL }
• I80-86 calibrate_roundtrip(RTT, inst_delays, asym_model) -> { T_1w, epsilon_asym }
• I80-87 emit_path_manifest_fiber(payload, policy) -> manifest.path.fiber

VIII. Cross-References

• Two-form arrival-time and numerical integration: this volume Chapter 10 and EFT.WP.Methods.Cleaning v1.0.
• Round-trip symmetry and time-base alignment: EFT.WP.Metrology.TimeBase v1.0, EFT.WP.Metrology.Sync v1.0.
• Intra-instrument delays and differential biases: EFT.WP.Metrology.Instrument v1.0.
• Multipath echoes and connector-reflection ISI: this volume Chapter 7.

IX. Quality and Risk Control

• SLO Targets — p95( |error(T_fiber)| ) ≤ 0.2 ns, p99 ≤ 0.4 ns (controlled indoor with qualified calibration); relax one tier for outdoor conduits.
• Drift Monitoring — temp_coeff_drift and cd_drift remain stable; asym_drift near zero. Anomalies trigger inspection and re-calibration.
• Fallback — If temperature profile or cable_spec is incomplete, fall back to { default TCD coeffs + nominal D + connector library values }, and widen u and guardbands.
• Audit & Traceability — Persist cable_spec.hash, params/hash, RefCond, asym.*, contracts.*, and delta_form; manage version changes per Appendix F.

Summary
This chapter delivers a unified, computable framework for group delay, thermal drift, dispersion, PMD, and connector terms on fiber/cable paths, together with explicit round-trip symmetry auditing and calibration.
Minimal key set:
manifest.path.fiber = { T_fiber, parts:{ T_geom, T_TCD, T_CD, T_PMD, T_conn }, asym:{ epsilon_asym, lambda_AB, lambda_BA }, qc:{ delta_form, u }, RefCond, cable_spec.hash, tags }.
In concert with Chapters 5, 6, 7, 10, 11, and 12, wired-link medium terms and directional errors can be compressed to system SLOs while maintaining two-form consistency and audit-ready traceability.