27-EFT.WP.Packets.Light v1.0 | Chapter 2 — Physical Baselines (Wave–Particle / Phase–Group / Dispersion)

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Chapter 2 — Physical Baselines (Wave–Particle / Phase–Group / Dispersion)

One-sentence goal: Unify the wave–particle duality, phase/group velocities, and dispersion/attenuation conventions of optical packets in fiber and free-space media, and provide minimal equations and implementation anchors that feed two-form arrival-time computation, compensation, and release.

I. Scope & Objects

1. Inputs
   • Path & media: segmented path gamma(ell) (fiber / fso / device), medium functions n_eff(f, x), attenuation alpha(f, x), dispersion spectrum beta(omega) and second-order dispersion beta2, PMD statistics D_pmd.
   • Source & signal: transmit spectrum X(f) (unit(X) = "√W" or normalized a.u.), pulse envelope A(z, t).
   • Reference conditions: RefCond (temperature / pressure / humidity; stress / bend radius; weather / turbulence Cn2), compute timebase tau_mono, and publication time ts.
2. Outputs
   • Per-segment propagation quantities: phase velocity v_phi, group velocity v_g, arrival-time component dT_g, link transfer function H(f), residuals of dispersion / PMD / loss with uncertainties u / U.
   • Inputs to the two forms: integrals of n_eff and n_g, and the physical decomposition of the two-form gap delta_form.
3. Boundary
   This chapter focuses on first-order linear optics and engineering approximations: under weak nonlinearity / weak coupling, SPM/XPM are reported only as minimal augmentations; quantum noise enters the power-spectral model via ASE / N0 only.

II. Terms & Variables

• Frequency & wavenumber: f, omega = 2*pi*f, k0 = omega / c_ref, beta(omega) (propagation constant).
• Indices & velocities: phase index n_phi(f), group index n_g(f) = n_phi - f * ( d n_phi / d f ), phase velocity v_phi = c_ref / n_phi, group velocity v_g = c_ref / n_g.
• Arrival time & dispersion: group-delay density dT_g = ( n_g / c_ref ) d ell, total group delay T_g = ( ∫_{gamma} ( n_g / c_ref ) d ell ); second-order dispersion beta2 = d^2 beta / d omega^2; dispersion parameter D = - ( 2*pi*c_ref / lambda^2 ) * beta2.
• Attenuation & gain: power attenuation alpha(f) (unit = "[1/m]"), device gain G, noise factor n_sp.
• PMD: first-order differential group delay tau_DGD, statistical parameter D_pmd (unit = "ps/√km").
• Nonlinearity (optional): nonlinear coefficient gamma_nl, effective area A_eff, self-phase modulation term i*gamma_nl*|A|^2 A.
• Dimensional examples: unit(T_g) = "[T]", unit(beta2) = "[s^2/m]", unit(alpha) = "[1/m]".

III. Postulates P602-*

• P602-1 (Physical consistency of two forms): Arrival-time publication must present both the constant-factored and general forms in parallel and record the discrepancy delta_form; per-segment splits of dispersion / PMD / loss must be traceable to the path inventory and RefCond.
• P602-2 (Explicit measures): Any integral must declare the path and measure ( ∫_{gamma(ell)} • d ell ), and segmental sums must be explicit ∑_{seg}.
• P602-3 (Dimensional conservation): Every quantity entering equations declares unit / dim and passes check_dim( y - f(x) ).
• P602-4 (Weak nonlinearity by default): Use the linear dispersion–loss model by default; if nonlinear terms are enabled, record gamma_nl / A_eff and power limits, and include them in the uncertainty budget.
• P602-5 (Traceable environment): Effects of temperature / stress / bending, and meteorology / turbulence enter n_eff / alpha / PMD via corr_env(x; RefCond).

IV. Minimal Equations S602-*

1. Two-form arrival time (per-segment capable)
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫_{gamma} n_eff(f,x) d ell ).
   • General: T_arr = ( ∫_{gamma} ( n_eff(f,x) / c_ref ) d ell ).
   • Two-form gap: delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
2. Group delay and pulse broadening
   • Group delay: T_g = ( ∫_{gamma} ( n_g / c_ref ) d ell ).
   • RMS broadening due to second-order dispersion (Gaussian-pulse approximation):
     sigma_out^2 = sigma_in^2 + ( beta2 * L_eff )^2 * ( ∆omega^2 ) / 2 (with L_eff the effective length).
3. Power and OSNR approximations
   • Segment loss: P_out = P_in * exp( - ( ∫_{seg} alpha(f) d ell ) ) * G_seg.
   • Link OSNR (linear): OSNR_lin ≈ P_sig / ( ∑ n_sp * h * f_0 * (G - 1) * RBW ) (sum over devices).
4. First-order PMD model
   DGD approximation: tau_DGD ≈ D_pmd * sqrt( L_gamma_km ); pulse-power penalty aggregates with dispersion broadening.
5. FSO scintillation (reference)
   Log-amplitude variance: sigma_X^2 ≈ 0.563 * k0^(7/6) * ( ∫ Cn2(z) z^(5/6) dz ); incorporate into equivalent attenuation alpha_fso.
6. Nonlinear extension (optional)
   Generalized NLSE (envelope):
   ∂A/∂z + ( alpha/2 ) A + i ( beta2/2 ) ∂^2 A/∂t^2 = i * gamma_nl * |A|^2 A
   (for weak-nonlinearity segments, use only to estimate first-order corrections to extra delay and spectral broadening).

V. Metrology Pipeline M60-2 (Ready → Modeling → Estimation → Checks → Persistence)

1. Ready: collect segment and device inventories (hash / id / validity), freeze RefCond, unify frequency grid and units (Hz, m, s, dB).
2. Modeling: build n_eff(f, x) / n_phi / n_g, alpha(f), beta(omega) / beta2, D_pmd, Cn2(z); for free-space segments, estimate equivalent attenuation and phase perturbation.
3. Estimation: integrate per segment to obtain T_g and the two forms of T_arr; compute link-level OSNR by cumulative product/sum; estimate broadening and PMD penalties.
4. Checks:
   • check_dim(T_arr) = "[T]"; delta_form ≤ tol_Tarr;
   • residual dispersion / PMD bounds; OSNR above threshold; nonlinearity within limits;
   • record uncertainties u / U and their provenance (model / measurement / environment).
5. Persistence: write manifest.packet.phys.* = { gamma.hash, RefCond, T_parts, T_arr_forms, delta_form, alpha / D / beta2 / PMD, OSNR, u / U, contracts.*, signature }.

VI. Contracts & Assertions C60-2x (Suggested Thresholds)

• C60-201 (Two-form gap): delta_form_p95 ≤ tol_Tarr (recommend tol_Tarr = 1e−3 * T_arr).
• C60-202 (Residual dispersion): | D_res | ≤ D_max (per modulation / symbol rate).
• C60-203 (PMD bound): tau_DGD_p95 ≤ tau_DGD_max (modulation-specific).
• C60-204 (OSNR floor): OSNR_dB ≥ OSNR_min_dB; for coherent reception, include Q ≥ Q_min.
• C60-205 (Nonlinearity gate): the combination of P_tx and gamma_nl must not drive SPM drift beyond threshold; on violation, lower power / enable segmental compensation.
• C60-206 (Unit compliance): all published fields pass check_dim with correct linear/log mappings.

VII. Implementation Bindings I60-2* (interfaces, I/O, invariants)

• I60-21 build_medium_models(segments, env) -> { n_eff(f,x), alpha(f), beta(omega), beta2, D_pmd }
• I60-22 segment_integrals(gamma_seg, n_eff, c_ref) -> { T_form1, T_form2, delta_form }
• I60-23 compute_group_delay(n_g, gamma) -> T_g
• I60-24 estimate_dispersion_broadening(beta2, L, pulse) -> sigma_out
• I60-25 estimate_pmd_penalty(D_pmd, L) -> tau_DGD
• I60-26 link_osnr(devices, rbw, ref) -> OSNR_lin_dB
• I60-27 fso_scintillation(Cn2, geom) -> { sigma_X2, alpha_fso }
• I60-28 nonlinear_check(P_tx, A_eff, gamma_nl, link) -> report
• I60-29 emit_phys_manifest(results, policy) -> manifest.packet.phys
  Invariants: non_decreasing(ell); check_dim(*) passes; two-form and segment checks are traceable; RefCond aligns with device versions.

VIII. Cross-References

• Two-form arrival time & path correction: Chapter 8 and EFT.WP.Metrology.PathCorrection v1.0, Chapters 10/11.
• Framing & timebase: Chapter 3 and EFT.WP.Metrology.TimeBase v1.0, …Sync v1.0.
• Propagation impairments & compensation: Chapter 6 (CD / PMD / nonlinearity compensation).
• OSNR / Q / EVM: Chapter 11 (metrology & panels).
• FSO environment: see …PathCorrection and field meteorology inventories.

IX. Quality & Risk Control

• SLI / SLO: delta_form_p99, cd_residual, pmd_drift, osnr_drift, latency_p99.
• Fallback strategies: dispersion / PMD drift → write back online compensation parameters; OSNR reduction → raise gain / reroute / lower code rate; nonlinearity over-limit → reduce power / segmental amplification; FSO scintillation → switch to backup link.
• Audit: segment / device hashes, environment records, two-form gap and residual-history, panel replay and dual-environment recomputation deltas.

Summary

• This chapter establishes the engineering baselines for phase / group velocities, dispersion, attenuation, and PMD in optical-packet propagation, yielding the minimal equations and checks required for two-form arrival time and link metrology.
• Deliverables are published under manifest.packet.phys.*, providing the unified physical anchor for framing, compensation, metrology, and arrival-time harmonization in Chapters 3–8 / 11.