SEP 9 — Broadband beam measurements and propagation statistics#


SEP	9
Title	Broadband beam measurements and propagation statistics
Author	John Travers
Drafting agent	GPT-6 Astra; GPT-6 (2026-10-22 coordination with SEP 31; 2026-10-23 split to SEP 34)
Status	Draft
Review status	Not reviewed
Reviewers	—
Type	Feature
Created	2026-09-27
Resolution	—
Requires	—
Supersedes	—
Superseded-by	—
Implementation	—
Manual pages	—

Part I — Proposal#

1. Abstract#

Add beam measurements to modal propagation, including both pulse-integrated fluence widths and time-resolved widths, with a frequency-dependent reconstruction and an explicit measurement domain. Complete the propagation-statistics catalogue with per-mode pulse durations, filtered total peak power, general spectral filters, species pressures and separately named nonlinear-resolution diagnostics. Preserve existing statistics, defaults and propagation trajectories. The reference is the reconstructed physical field and its declared flux convention, with analytic tests, independent spatial and temporal refinement, and parity on NumPy, Numba CPU and Kokkos CPU. A width is a measurement of a specified distribution, not an effective area converted into a radius or a universal measure of beam quality.

2. Motivation and scope#

A multimode fibre can change its transverse light distribution without changing its total energy or even its modal energies: relative modal phases alone change the interference pattern. A mode’s profile can also change across a broadband spectrum, and different parts of a pulse can occupy different transverse regions. The useful questions are therefore distinct:

How wide is the energy distribution integrated over the complete pulse?
How wide is the power distribution at each resolved time within it?
How do those answers change when a spectral band is selected?
Is the field centred, elliptical, hollow or split into several lobes?
Is an apparent change physical, or a consequence of too few modes, too few integration nodes, insufficient temporal sampling or a clipped measurement domain?

Both fluence and time-resolved widths are required deliverables. Spectral selection and time resolution are independent: a band-limited width is not a time slice. The location is the guide cross-section at the stated propagation position. These are not measurements after outcoupling, imaging optics or free-space propagation.

The proposal covers ordinary modal routes with an actual transverse profile and the existing general modal route with its frequency-dependent E/H reference. The latter is qualified only on combinations the propagation route itself admits; this SEP does not add general-modal envelope propagation. A dispersion-only GNLSE declaration has no spatial shape. It can supply a declared effective-area intensity, but cannot supply a beam width. Scalar and vector profiles, taper motion, material regions, modal coherence and saved-field amplitude conventions are in scope.

Existing radial, Cartesian, line and hybrid planar statistics remain supported; their common flux measure follows SEP 34’s medium-flux correction. Adding further time-resolved spatial measurements is outside this SEP. SEP 31 also owns transported observation-plane fields; the measurements here describe fields inside the propagation route. No new detector material model, polarisation analyser, finite-exposure response, noise ensemble, beam-quality propagation ratio \(M^2\), mode solver, response model or adaptive numerical controller is proposed. A width along a fibre is not necessarily a free-space beam waist. No new dependency is required.

3. Context#

3.1 Capability audit#

The following inventory separates absent measurements from existing equivalents and deliberate choices. It is an inventory of capabilities, not a request to reproduce another implementation’s function names or reductions. The current definitions are in statistics, power and intensity, src/solprop/stats.py, src/solprop/spatial.py and the backend statistics methods.

Measurement or facility	Present behaviour	Disposition in this SEP
Spectral centroid	`omega0`, per mode and polarisation, with power/metric weighting	Keep; no missing centroid statistic
Energy	`energy`; `physical_energy` on a general modal basis	Keep the route-specific meanings
Energy in wavelength/frequency intervals	`energy_window`, half-open angular-frequency intervals	Keep exact tiling; add a wavelength constructor for explicit filters
Energy through an arbitrary spectral window	Hard intervals only in run statistics	Add a labelled complex amplitude-transfer specification and `energy_filtered`
Peak power per mode and of the simultaneous sum	`peak_power`, `peak_power_total`	Keep; sum traces before taking the total peak
Filtered peak power	`peak_power_window` per mode for hard intervals	Add `peak_power_total_window` and general filtered counterparts
Inner and outer temporal FWHM of the total pulse	`fwhm_t`, `fwhm_t_outer`	Keep their scalar shapes
Inner and outer temporal FWHM per mode	No catalogue entry; the total widths do not replace these	Add `fwhm_t_modes`, `fwhm_t_outer_modes`
Effective-area average peak intensity	Can be computed from peak power and an area, but has no distinct statistic	Add explicit `effective_peak_intensity` for one mode with a declared area
On-axis modal peak intensity	`peak_intensity`, using the current analytic Poynting closure	Keep; do not silently change it to a transverse maximum
Transverse maximum modal intensity	Absent; an informal roadmap item	Add `peak_intensity_transverse` on the declared diagnostic domain
Fibre beam widths	`fwhm_r` and `beam_radius` exist for spatial geometries, not modal reconstruction	Add modal fluence and time-resolved width families
Electron density and ionisation rate	Backend statistics already exist, with additional ordered species statistics	Keep their population and sampling conventions
Gas density and total pressure	Profile statistics; ordered `species_density` also exists	Keep
Pressure of each constituent	No ordered species-pressure statistic	Add `species_pressure` with an explicit EOS interpretation
Core radius and ZDW	Existing opt-in profile and modal statistics	Keep; multiple-ZDW root selection remains a separate known issue
Position and step length	Saved-plane positions, periodic-row positions and indexed step history	Already present; no duplicate `z`/`dz` statistics
Relative transverse integration error	`quadrature_error`, a cached embedded-rule alarm	Keep; it does not measure modal truncation
Absolute and band-selected integration residuals	No catalogue entries	Add fresh, per-member nonlinear quadrature diagnostics with stated norms
Nonlinear source reconstruction in a retained modal space	Not implemented	Add an opt-in spatial residual; do not equate it to quadrature error
Number of transverse evaluation points	Fixed rule extents are available from the problem	Record rule/probe counts as diagnostic metadata, not a fictitious adaptive error
Collector, custom closures, device traits, analytic transform plans	`StatsSpec`, one catalogue, backend seam and periodic channel	Infrastructure already exists; arbitrary executable callbacks are not new observables

The ionisation comparison needs particular care. Ordinary multimode plasma is sampled at the nonlinear nodes; the single-channel density/rate pair refers to the densest final-population node. The mixture path additionally exposes each species’ largest rate over nodes and time. These are deliberately different from measuring only on the guide axis. They are not missing statistics to be replaced. Likewise, the established peak intensity is not \(P/A_{\rm eff}\).

3.2 Existing reconstruction and execution#

The ordinary Mode.field(points, z) interface has no frequency argument. modes.quadrature.mode_matrix uses a vacuum-normalised transverse profile. In particular, StepIndexMode declares a profile wavelength: its dispersive propagation constant does not make that frozen transverse profile dispersive. A diagnostic of this route must retain that approximation and name it. It must not solve new profiles at other frequencies and apply them to amplitudes propagated in a different basis.

The general modal route already carries tracked, complex, frequency-dependent E/H fields through translate.modal_tables.ModalTransformTables. processing.modal checks the guide, gauge and table identities and reconstructs a plane, including a moving guide’s nodes. Its reconstructed_flux uses the reconstructed cross-frequency E/H product. This is different from the general route’s run statistic analytic_flux (the two shared that name until the accessor was renamed), which is formed from metric-normalised amplitudes. Their energy integrals agree; their time traces need not. New beam measurements must use the reconstructed product and must not reinterpret the existing statistic.

backends.numpy_ref.backend, backends.numba_cpu.backend, backends.spatial and backends.driver.StepDriver own the reference, compiled and periodic evaluation. In solprop-kokkos, src/solprop_kokkos/backend.py packs/unpacks native statistics; src/general.cpp owns the modal statistics pass, and src/general_modal.cpp the general modal execution. The current Kokkos temporal FWHM path transfers a total power trace for host crossing interpolation; a new family of per-mode/time-resolved measurements must budget that transfer rather than assume all statistics are scalars.

The design builds on ADR-010 (quadrature), ADR-012 (statistics scheduling), ADR-017 (Fourier boundary), ADR-030 (power normalisation), the kernel contract and the capability gates in output.capability and stats.GENERAL_MODAL_ACCESSORS. It requires none of the other draft physics SEPs. Statistics observe the accepted state after boundary treatments; cached stage arrays are reusable only when they represent that same state, position, noise convention, filter and sampling grid. Beam measurements use the propagating field, including input noise when it is part of that field. They do not silently add the modified-noise drive that is present only in a nonlinear-response evaluation. A source diagnostic, in contrast, uses the run’s declared nonlinear-drive noise convention and records that scope.

4. Theory#

4.1 The measured field and the spectral filter#

Use SI and the optics convention of conventions: inverse transforms carry \(e^{-i\omega t}\). At a fixed \(z\), write the analytic fields as

\[ \widetilde{\mathbf E}(\mathbf r,\omega) =\sum_m a_m(\omega)\mathbf e_m(\mathbf r,\omega),\qquad \widetilde{\mathbf H}(\mathbf r,\omega) =\sum_m a_m(\omega)\mathbf h_m(\mathbf r,\omega). \]

The channel index here includes any declared polarisation component. The E/H profiles have units V/(m \(\sqrt{\mathrm W}\)) and A/(m \(\sqrt{\mathrm W}\)) on a unit-power reference. In an ordinary modal route the internal amplitude is instead the existing vacuum-normalised \(a\); its magnetic equivalent is described below. Complex profiles and amplitudes must be expressed in the same Fourier convention and tracked gauge. Conjugating a saved amplitude without converting its profile reference is not a physical reconstruction.

Let \(T_j(\omega)\) be a dimensionless scalar amplitude transfer, applied to both fields before reconstruction. The unfiltered request is \(T_0=1\) over the represented active spectrum. The filtered flux is

\[ S_j(\mathbf r,t)=\frac12\operatorname{Re} [\mathbf E_j(\mathbf r,t)\times\mathbf H_j(\mathbf r,t)^*] \cdot\hat{\mathbf z},\qquad [S_j]=\mathrm{W/m^2}. \]

This is the analytic Poynting flux, with the real instantaneous sum-frequency term excluded. It is not \(|E|^2\) in an arbitrarily chosen refractive index and does not include a detector’s finite response time. Only transverse components enter the axial cross product, although the full vector reference must retain its longitudinal components for consistency and nonlinear-source diagnostics.

For ordinary profiles \(\mathbf g_m=\mathbf e_m/\sqrt{N_m}\), where \(N_m=\varepsilon_0c\int|\mathbf e_m|^2dA/2\), retain the existing closure

\[ \widetilde{\mathbf E}_k=\sum_m a_{mk}\mathbf g_m,\quad \widetilde{\mathbf E}_{w,k}=\sum_m w_{mk}a_{mk}\mathbf g_m,\quad S_j=\frac{\varepsilon_0c}{2}\operatorname{Re} [\mathbf E_j\cdot\mathbf E_{w,j}^*],\quad w_{mk}=\operatorname{Re}n_{{\rm eff},m}(\omega_k,z). \]

The route’s explicit index override remains authoritative. Reconstructing a saved true-power amplitude requires \(a=b/\sqrt w\) first. Even at one frequency, local intermodal terms carry the arithmetic mean \((w_m+w_n)/2\); replacing \(a\) with \(b\) in the vacuum synthesis would give geometric-mean weights. This is an observable difference in an interference pattern. The ordinary closure is an approximation of that propagation model, not a substitute for the general route’s actual H fields.

4.2 Full-pulse fluence and exact discrete normalisation#

The fluence through the plane is

\[ F_j(\mathbf r)=\int S_j(\mathbf r,t)\,dt =\frac{1}{4\pi}\operatorname{Re}\int |T_j(\omega)|^2 [\widetilde{\mathbf E}(\mathbf r,\omega) \times\widetilde{\mathbf H}(\mathbf r,\omega)^*]_z\,d\omega, \qquad [F_j]=\mathrm{J/m^2}. \]

The continuous expression uses analytic positive-frequency spectra. In an envelope representation the same integral uses the represented detuning band, with physical frequency retained for profiles and filters. Parseval removes cross-frequency terms only after integration over the complete temporal window; coherent cross-mode terms at each frequency remain. Thus common spectral phase changes do not change fluence, but can change time-resolved widths. Relative modal spectral phase can change both.

For the raw DFT arrays, let \(N=n_t\) and let \(d_k=1\) on an envelope grid. On a field grid \(d_k=2\) for interior positive-frequency bins and \(d_k=1\) at DC and Nyquist, with the endpoint reality constraints of the analytic-signal construction. Then

\[ \eta_k=\frac{\Delta t}{N}d_k^2,\qquad F_j(\mathbf r)=\frac12\sum_k\eta_k|T_{jk}|^2 \operatorname{Re}[\mathbf E_k(\mathbf r)\times\mathbf H_k(\mathbf r)^*]_z. \]

This is the discrete definition. The ordinary closure replaces the half cross product by \(\varepsilon_0c\operatorname{Re}(\mathbf E_k\cdot\mathbf E_{w,k}^*)/2\). No Simpson weights or frequency-centroid approximation are added. The integration measure is the physical area, including \(r\,dr\,d\theta\) on polar rules. A radial line profile itself is not multiplied by \(r\) before a half-maximum search.

Integrating the reconstructed fluence over the full represented domain must recover the route’s energy to the independently qualified spatial/table error. For a general basis that energy is \(\sum_k\eta_k a_k^\dagger W_k a_k\), not a sum of coordinate powers. This is the central normalisation test. A core-only aperture is a different measurement and reports its captured energy; it must not be renormalised and presented as the whole beam. Evanescent cladding tails matter especially to second moments, so energy convergence alone does not certify a width.

4.3 When a coherency-matrix reduction is exact#

At each frequency the local flux is a Hermitian quadratic form \(a_k^\dagger Q_k(\mathbf r)a_k\). If \(Q_k(\mathbf r)=\sum_{\ell=1}^L q_{\ell k}Q_\ell(\mathbf r)\) with a certified separation, spectral accumulation can be performed first:

\[ C_{j\ell}=\sum_k\eta_k|T_{jk}|^2q_{\ell k}a_ka_k^\dagger, \qquad F_j(\mathbf r)=\sum_\ell\operatorname{tr}[Q_\ell(\mathbf r)C_{j\ell}]. \]

For the ordinary fixed-profile closure an exact special case is

\[ C_{j,mn}=\sum_k\eta_k|T_{jk}|^2 \frac{w_{mk}+w_{nk}}2\,a_{mk}^*a_{nk},\qquad F_j=\frac{\varepsilon_0c}{2}\sum_{mn} C_{j,mn}\,\mathbf g_m^*\cdot\mathbf g_n. \]

The second expression deliberately uses the conjugate orientation to the first matrix convention; its indices state the contraction explicitly. Tests must use complex profiles and nontrivial phases, since real profiles can conceal a misplaced conjugation. All quantities are evaluated at the measurement’s \(z\).

A single unweighted matrix is valid only when the spatial kernel is independent of frequency up to a known scalar. Frequency-dependent profile shapes, mode-index weights or detector response cannot be discarded. In particular, evaluating every profile at the pulse’s centroid frequency is not broadband reconstruction. The general implementation synthesises at each frequency; a low-rank approximation needs its own certified error and is not required by this SEP.

4.4 Time-resolved measurements#

For a requested time, use \(D_j(\mathbf r)=S_j(\mathbf r,t)\) in the same spatial measurement operator used for \(D_j=F_j\). In continuous notation the temporal product includes the kernel

\[ S_j(\mathbf r,t)=\frac{1}{2(2\pi)^2}\operatorname{Re} \iint T_j(\omega)T_j(\omega')^* [\widetilde{\mathbf E}(\mathbf r,\omega) \times\widetilde{\mathbf H}(\mathbf r,\omega')^*]_z e^{-i(\omega-\omega')t}\,d\omega\,d\omega'. \]

Do not replace this by independent spectral intensities or a same-frequency metric square root. Implement it by synthesising E/H spectra and inverse-transforming them, not by materialising the double-frequency kernel. A frequency-independent profile allows temporal modal synthesis first; a dispersive profile generally does not. On envelope grids the removed carrier cancels in the Hermitian product.

Times are in seconds in the run’s co-moving frame, with its existing time origin. An integer oversampling factor selects a zero-padded temporal grid, preserving physical amplitudes and the treatment of field-grid endpoints. Requested samples are an explicit subset of that grid; no periodic wrapping of an out-of-window time is allowed. Omitting the subset means all times on that grid when a temporal statistic is requested; a fluence-only request allocates no temporal output. Zero padding refines an already represented field and cannot recover unrepresented optical bandwidth. Sampling must resolve the difference frequencies of the flux, not merely a convenient set of pulse times. Temporal convergence is qualified independently of propagation error and transverse sampling.

In general,

\[ \operatorname{width}\!\left(\int S_j\,dt\right) \ne\int \operatorname{width}(S_j)\,dt, \]

and neither side is a width measured at the instant of peak total power. All these operations are nonlinear once centring, normalisation or a threshold is introduced. The delivered time series must therefore include the time axis and the power through the measurement aperture so that a width during a negligible tail is identifiable.

4.5 Centroids, second moments and half-maximum widths#

For an admissible distribution \(D\) over the declared aperture \(A\), set

\[ U_D=\int_A D\,dA,\quad \boldsymbol\mu_D=\frac{\int_A\mathbf rD\,dA}{U_D},\quad \Sigma_D=\frac{\int_A(\mathbf r-\boldsymbol\mu_D) (\mathbf r-\boldsymbol\mu_D)^T D\,dA}{U_D}. \]

\(U_D\) is energy for fluence and power for a time slice. Centroids are in m and the covariance is in m². With eigenvalues \(\lambda_1\ge\lambda_2\), report second-moment diameters \(d_i=4\sqrt{\lambda_i}\), and the optional equivalent radial moment \(r_2=\sqrt{2\operatorname{tr}\Sigma_D}\). This is the usual second-moment beam-width convention [1, 2], applied here to the explicitly specified energy or power distribution; no claim of a complete ISO measurement procedure is made. Keep the covariance itself so a circular beam does not need an arbitrary orientation.

For the axis-aligned Gaussian \(D=D_0\exp[-2(x^2/w_x^2+y^2/w_y^2)]\), direct Gaussian integration gives \(\Sigma_{xx}=w_x^2/4\), \(\Sigma_{yy}=w_y^2/4\), and \(d_i=2w_i\). On a circular Gaussian \(r_2=w\) and the radial full width at half maximum is \(w\sqrt{2\ln2}\). Radius, diameter and FWHM must never share an unlabeled output.

For a general modal beam define fwhm_r from the azimuthal average about the fluence centroid, or the slice’s own centroid for fwhm_r_time:

\[ \overline D(r)=\frac1{2\pi}\int_0^{2\pi} D(\boldsymbol\mu_D+r(\cos\theta,\sin\theta))\,d\theta. \]

Use half the maximum of this averaged profile over the certified sampled domain. Starting at \(r=0\), take the first outward bracketed crossing and return twice its radius. No search goes beyond the aperture’s largest complete centred circle. If the central value is below the threshold, as in a sufficiently hollow beam, the central FWHM is undefined. This is a radialised width, explicitly labelled as such; it is not a line cut at one azimuth and is not a principal-axis diameter. Centroid and covariance expose the information this scalar removes. The existing spatial Cartesian width uses a related radial average; its implementation and defaults are not changed here.

Use linear interpolation between ordered bracketing samples and refine the diagnostic radius/angle samples independently of the nonlinear quadrature. A resolved ring can have finite second moments and undefined central FWHM. Multiple crossings, tangent crossings and shallow slopes need explicit resolution flags; a root finder alone does not establish that the correct lobe was resolved.

Analytic flux, including local fluence, is not guaranteed positive everywhere for an arbitrary coherent modal superposition. The existing convention retains its sign. A width interprets \(D/U_D\) as a positive spatial distribution, so an appreciable negative region, nonpositive total, unresolved domain or a dark time slice cannot silently become a beam width. Return NaN for the affected width and a reason code; retain the signed aperture energy/power. Nonfinite samples raise instead. Do not clip the flux or replace it by \(|E|^2\). An operation-count bound distinguishes round-off-sized negative values from physical signed flux; sums still use the signed samples. Only eigenvalues consistent with zero within the propagated round-off bound may be rounded to zero for a square root. A user-visible relative signal floor can mark weak temporal slices; its value is stored and does not alter the field or energy.

4.6 Remaining scalar statistics#

Per-mode temporal widths reuse the current ordinary-route trace \(P_m(t)=\sum_p\operatorname{Re}[a_{mp}(t)a_{w,mp}(t)^*]\) and its existing inner/outer crossing rules. They sum polarisations within a spatial mode. No basis-independent per-mode energy or duration is promised for a general nonorthogonal basis; these names are refused there, preserving the existing capability distinction.

Apply a filter before forming either member of the flux product. For ordinary modal filtered peaks,

\[ P_{j,m}^{\rm peak}=\max_t P_{j,m}(t),\qquad P_{j,\rm total}^{\rm peak}=\max_t\sum_m P_{j,m}(t). \]

Taking the largest mode peak or adding peaks attained at different times is wrong. An amplitude transfer \(T\) weights spectral energy by \(|T|^2\), not \(T\); phase affects filtered temporal peaks even though it does not affect filtered energy. Hard half-open energy_windows retain their exact-tiling semantics. Arbitrary overlapping filters have no such sum rule. Wavelength intervals map monotonically to reversed frequency endpoints, \(\omega_{\rm lo}=2\pi c/\lambda_{\rm hi}\) and \(\omega_{\rm hi}=2\pi c/\lambda_{\rm lo}\). No wavelength-density Jacobian belongs in a transfer function multiplying a spectrum already sampled in frequency.

effective_peak_intensity is explicitly \(\max_t P(t)/A_{\rm eff}(z)\), W/m², only for a single mode with a finite positive declared effective area. It is a nonlinear effective-area diagnostic, not the spatial maximum. An area alone gives neither the field profile nor its width; for a broadband frozen-profile model the area retains that model’s reference-wavelength approximation.

peak_intensity_transverse is \(\max_{q,t}S(q,t)\) on the recorded diagnostic nodes and temporal grid, with the position/time of the sampled maximum available in metadata or companion outputs. It is a sampled maximum, whose convergence must be tested. Its temporal search covers the complete reconstructed grid, independently of a subset selected for width output. It does not rename or change modal peak_intensity.

species_pressure reports an ordered vector in Pa, following the SI rule for new quantities. The existing pressure output in bar remains unchanged. For a shared fill, report the partial-pressure paths already declared by that mixture’s EOS construction. For independently supplied number-density paths, evaluate each constituent’s own EOS at its local density and temperature and label the result partial_eos. Do not assume room temperature or claim that the sum is a thermodynamic mixture pressure outside the mixture model’s domain. The ideal-gas limit is \(p_s=\rho_s k_BT\); this SEP introduces no new mixing law or constants.

4.7 Quadrature resolution and modal truncation are different#

The existing quadrature_error compares the fine and embedded coarse projections of a stage’s source and retains a cached alarm. It is deliberately not an on-axis source reconstruction, not an error of the accepted field and not a controller input. Preserve that meaning, including undefined cases with no embedded rule.

New plane diagnostics evaluate the accepted state with the same physical response, noise treatment and temporal/spectral windows as the run, using diagnostic buffers. Let \(N_f\) and \(N_c\) be fine/coarse nonlinear drives in the lab-frame internal amplitude derivative, after physical response factors and spectral masking, with the linear term and numerical absorber excluded. For filter \(T_j\), define

\[ \|v\|_{j,W}^2=\sum_k\eta_k|T_{jk}|^2v_k^\dagger W_kv_k,\quad R_{j,\rm quad}=\|N_f-N_c\|_{j,W},\quad e_{j,\rm quad}=R_{j,\rm quad}/\|N_f\|_{j,W}. \]

Use the ordinary diagonal power weight or the general route’s flux metric as appropriate. \(R_{j,\rm quad}\) has units \(\sqrt{\mathrm J}/\mathrm m\), since the drive is an amplitude derivative. This is a norm, not an RMS of arbitrarily scaled DFT coefficients. A vanishing source has zero absolute residual and undefined relative residual; missing coarse quadrature is separately unavailable. A general guide currently has no embedded coarse rule: it must refuse these requests by name until one is supplied and independently qualified. Do not fabricate a zero error. Band-selected residuals reveal poorly resolved weak emission hidden by a large pump.

For modal truncation use a separately defined source-space projection. The first implementation covers instantaneous and delayed polarisation responses, ordinary and general modal. At every retained frequency, sample the physical nonlinear polarisation \(p_k\) (C/m²) on a diagnostic rule \(V\) of positive area weights. If \(B_k\) contains the electric profile columns in the propagated span, the weighted least squares reconstruction is

\[ \widehat p_k=B_k(B_k^\dagger VB_k)^{-1}B_k^\dagger Vp_k,\qquad e_{j,\rm span}^2= \frac{\sum_k\eta_k|T_{jk}|^2\|p_k-\widehat p_k\|_V^2} {\sum_k\eta_k|T_{jk}|^2\|p_k\|_V^2}. \]

Compute the projection with a rank-revealing weighted QR/SVD or an equivalently conditioned factorisation, not an explicit inverse. Use a separate validation rule to qualify sampling; projecting and testing only at interpolation nodes can conceal missing spatial structure. The result measures representability of the polarisation in the retained electric-profile span under a stated norm. It is invariant under a nonsingular change of coordinates within that span, but it is not an emitted-energy fraction, not a UPPE reciprocal-source projection, and not an error bound on propagation. A pointwise on-axis residual would miss modes vanishing there and is not the recommended diagnostic.

Plasma/current responses require a distinct source-space convention and are explicitly deferred; a polarisation-only residual must refuse a request to label it as the total nonlinear source in such a run. Converting current to polarisation by dividing by frequency is not introduced implicitly. Adding it requires a defined treatment of DC, response factors and the measured norm. Quadrature diagnostics can still use the run’s complete assembled nonlinear drive, including plasma. Source-evaluation node counts, projection-rule counts and validation-rule counts are recorded separately; none is evidence of convergence on its own.

5. Proposed interface#

The names below are proposed, not available APIs. Extend the existing frozen StatsSpec rather than add a second callback/collector system:

filters: an ordered tuple of frozen SpectralFilter records, with a unique label and finite complex amplitude-transfer samples on the run’s coarse frequency axis. Preserve samples and their grid identity. A wavelength-band constructor takes positive finite endpoints in m and an explicit transition width in rad/s; zero means a hard half-open band, positive means the documented Planck taper. No arbitrary callable crosses the backend seam. Transfers need not be passive; values above unity are recorded as gain rather than called an energy fraction. On field grids a transfer must preserve real DC/Nyquist values where those bins are nonzero; inactive bins stay zero. Reject an incompatible endpoint transfer.
beam: an optional frozen BeamMeasurement describing the physical aperture, integration and profile-sampling rules, full-domain or finite-aperture intent, refinement/error settings, optional temporal sample indices, integer temporal oversampling factor \(\ge1\), and relative signal floor in \([0,1)\). The default floor is zero apart from the derived numerical floor. The modal reference comes from the assembled route; the record cannot substitute an unrelated profile.
nonlinear_diagnostics: an optional frozen specification of the fresh plane diagnostics, independent validation rule and response scope.

Use the first filter slot for the explicit unfiltered transfer. Shapes always keep the filter axis, including a one-filter request; do not squeeze singleton axes in the stored statistics. Here n_filters and n_stat_times are new documented axes; batch and saved-plane axes are added by the existing infrastructure.

Proposed statistic	Shape per plane	Units / meaning
`fwhm_t_modes`, `fwhm_t_outer_modes`	`(n_modes,)`	s; ordinary modal routes
`peak_power_total_window`	`(n_windows,)`	W; existing hard windows
`energy_filtered`	`(n_filters, n_modes, n_pol)`	J; ordinary modal routes
`physical_energy_filtered`	`(n_filters,)`	J; general modal metric
`peak_power_filtered`	`(n_filters, n_modes)`	W; ordinary modal trace
`peak_power_total_filtered`	`(n_filters,)`	W; ordinary modal trace
`effective_peak_intensity`	`()`	W/m²; one declared effective area
`species_pressure`	`(n_species,)`	Pa; ordered, with EOS interpretation
`beam_energy`	`(n_filters,)`	J through the declared aperture
`beam_centroid`	`(n_filters, 2)`	m; transverse Cartesian coordinates
`beam_covariance`	`(n_filters, 2, 2)`	m²; central second moments
`beam_diameters`	`(n_filters, 2)`	m; descending principal \(4\sigma\) diameters
`beam_fwhm_r`	`(n_filters,)`	m; central radialised fluence FWHM
`beam_power_time`	`(n_filters, n_stat_times)`	W; signed aperture power
`beam_centroid_time`, `beam_diameters_time`	`(n_filters, n_stat_times, 2)`	m; each slice centred independently
`beam_covariance_time`	`(n_filters, n_stat_times, 2, 2)`	m²
`beam_fwhm_r_time`	`(n_filters, n_stat_times)`	m; central radialised slice FWHM
`peak_intensity_transverse`	`(n_filters,)`	W/m²; sampled spatial/temporal maximum
`nonlinear_quadrature_residual`	`(n_filters,)`	\(\sqrt{\mathrm J}/\mathrm m\); fresh accepted-state norm
`nonlinear_quadrature_error`	`(n_filters,)`	dimensionless; fresh relative norm
`polarisation_span_error`	`(n_filters,)`	dimensionless; retained-span residual

Each beam width/moment family stores validity codes with matching filter/time axes; FWHM has a distinct validity code because it can fail when moments remain valid. The vocabulary includes valid, empty, below_signal_floor, signed_flux, no_central_lobe, unbracketed, unresolved and degenerate_orientation where orientation is requested. Nonfinite input or missing required physics is an error, not a successful measurement carrying an undefined-width code. The time axis is stored once as stats_axes/time, and the filter order/labels once in metadata.

beam_fwhm_r is the explicit filtered family; §4’s modal meaning of fwhm_r is exposed as an unfiltered scalar convenience using that same evaluator. An ordinary modal beam_radius request similarly returns the existing two-element layout (second-moment radius and central \(1/e^2\) radius), from the same radial profile. These two legacy names remain plane-only. All existing spatial shapes and meanings remain unchanged. No two separate implementations of a width are introduced.

Example sketch, with the proposed convenience constructors:

StatsSpec(
    names=("energy", "fwhm_t_modes", "peak_power_total_window",
           "beam_diameters", "beam_fwhm_r", "beam_diameters_time",
           "beam_fwhm_r_time", "beam_power_time"),
    energy_windows=frequency_intervals,
    filters=(SpectralFilter.all(grid),
             SpectralFilter.wavelength_band(grid, (700e-9, 900e-9),
                                            label="pump", transition=2e13)),
    beam=BeamMeasurement(
        aperture=measurement_aperture,
        quadrature=measurement_rule,
        profile_sampling=radial_and_angular_samples,
        time_indices=selected_indices,
        time_oversampling=2,
    ),
)

Run-time statistics and offline processing share the same measurement definition. Offline entry points require the saved amplitude/power convention and the route’s validated reconstruction reference, using the existing guide/table identity checks. They can return nodal fluence/flux for inspection; the run normally stores only reductions. A general guide cannot pass an ordinary modes= tuple and bypass these checks. Unsupported routes, filters, missing profiles, unresolved modal cutoffs, incorrect gauges or out-of-domain samples fail before propagation.

Cheap scalar additions can join the existing periodic subset after qualification. The beam families, time series, transverse maxima and fresh nonlinear diagnostics are opt-in at save planes in this proposal. They are not silently enabled at every accepted step. Users can request closer save planes; a separately benchmarked between-plane admission can be proposed later. This preserves the current cost contract and avoids treating a potentially large time series as a cheap scalar.

6. Implementation strategies#

6.1 Recommended layers and data contract#

Keep specifications, named reference reductions and validity semantics in stats.py and the spatial measurement domain in spatial.py, with modal physics owned by modes/. Translation compiles a separate diagnostic reconstruction record: physical coordinates/area weights, aperture identity, E/H maps or certified fixed profile factors, temporal sampling, filters and their validity domain. It must not reuse a one-node effective-area transform as if it were a spatial field. General modal propagation tables remain the authority for channel identity and gauge. If diagnostic nodes differ, translation solves/evaluates profiles there and aligns them to that authority; a fresh provider solve without alignment is invalid. For a moving guide both profile and physical node coordinates belong to the plane.

Backends evaluate only validated arrays, in node chunks, and return host reductions. No new backend dependence on a mode solver or EOS is introduced. Driver/profile statistics own constituent pressures. Processing uses the same definition and identity checks as the run, with offline arrays available for independent inspection. Capability metadata must publish these new names only where the reference and backend actually support them; update all admission doors, including general-modal files, scans and batches.

Adding the diagnostic record and statistics parameters changes the seam. Bump SCHEMA_VERSION from the then-current value in the first implementation phase that carries them, update the literal contract tests and the Kokkos consumed-field tuples, and record it in the changelog. Do not preselect a version number in this Draft. Any later phase changing protected driver hooks or stage bundles repeats that rule. The new fixed axes, filter data, validity codes and reference/aperture identities require an output-schema revision with unsupported files refused and resume hashes accounting for every measurement-defining field.

6.2 Alternatives and costs#

Strategy	Cost and limitation	Decision
Reconstruct after saving the full field	Easy to inspect; misses runs that retain only statistics and transfers/stores the full state	Provide offline parity, but not as a hidden device fallback
One modal coherency matrix	About \(O(n_\omega M^2+Q M^2)\) per filter; exact for the certified separable kernel	Use for ordinary fluence when cheaper than direct synthesis
Frequency-resolved node synthesis	\(O(Q M n_\omega)\) per filter, plus reductions; retains arbitrary tracked profiles	NumPy oracle and required general-modal path
Time reconstruction in node chunks	Add \(O(Q n_t'\log n_t')\) per filter, where \(n_t'\) is the padded grid	Required for time-resolved widths and transverse peak intensity
Precomputed moment matrices	Can avoid node/time fields for some integrated moments; frequency-pair matrices cost \(O(M^2n_\omega^2)\)	Optional separable optimization only; never materialise the general pair kernel
Approximate spectral low rank	May reduce synthesis cost; introduces a new approximation near cutoffs and cancellations	Defer pending an independent error certificate

For fixed profile maps, persistent storage can be \(O(QM)\) plus a coherency matrix. Frequency-resolved maps can cost \(O(QMn_\omega)\); do not copy a dense reference a second time merely for statistics. Budget streamed/table-evaluated maps and a node chunk’s E/H buffers explicitly. Temporal reconstruction scratch scales with \(Q_{\rm chunk}n_t'\), not the full cross-section times the full time window. Filters may be processed sequentially with shared maps. Reductions of moments over nodes can accumulate one time vector per moment; radial FWHM needs a separately planned profile pass or bounded profile buffers. Recomputing that pass is an honest time/memory tradeoff, not an unlisted allocation.

Returned time-series storage grows as \(O(n_z n_{\rm filters}n_{\rm stat\_times})\) times the selected moments. Native crossing reductions are preferable when large per-mode or temporal traces would otherwise cross the device seam. Small reduced radial profiles may be transferred for the one reference interpolation algorithm if their size is admitted in the plan. Record actual transfer volumes. Neither the full state nor full nodal E/H fields are copied to implement an advertised native run statistic.

The fresh source diagnostics can cost another nonlinear evaluation at a plane, and the span test adds source sampling and a conditioned projection. Keep them opt-in and separate from the cached alarm. Partial reductions within a rounding bound fixed before measurement, no kernel allocations, no fast-math, no hidden response-state mutation, and complete memory-ledger entries apply to all three backends. No cost percentage is promised before measurements on the required short windows.

7. Verification and validation#

Use analytic fields and independently evaluated spatial integrals as the first reference. Backend agreement alone is insufficient. Each test states its error budget before comparing results; no tolerance is selected from a measured failure.

Parseval and flux normalisation. Directly integrate reconstructed time-domain E/H flux and compare to the same-frequency fluence sum on both grids, then integrate spatially against the route’s energy. Exercise isolated bins, legal DC/Nyquist endpoints, complex profiles and index weights differing by mode and frequency. Use deterministic accumulation/FFT round-off bounds plus separately measured modal-table and transverse-integration errors.
Gaussian widths. Compare centroid, covariance, \(4\sigma\) diameters and radial FWHM with §4.5 for circular and rotated elliptical Gaussians. Translation of a beam moves its centroid without changing its covariance eigenvalues. Scale the amplitude over a wide finite range without changing widths. Linear crossing interpolation is second order for a smooth nonzero-slope crossing: test three spatial refinement levels with the expected order, allowing the independently derived sampling error.
A single capillary mode and a taper. Independently integrate the squared Bessel profile with adaptive SciPy quadrature; bracket the central crossing independently. With a self-similar taper, lengths scale with core radius and covariance with its square. Compare at the actual plane, not the entrance.
Coherent multimode beating. Two orthogonal modes with a fixed spectrum and a changing relative phase have constant modal energies and a changing spatial pattern. Compare direct synthesis with the coherency contraction, including complex polarisation partners and a mode that vanishes on axis. Verify that discarding cross terms fails the constructed case.
Frequency-dependent profiles. A synthetic two-band E/H family with distinct Gaussian radii has a broadband covariance equal to the energy-weighted combination of its band covariances when their centroids coincide. With distinct centroids include the covariance of band centroids. This gives an analytic counterexample to a centroid-frequency profile. Also qualify a real general modal provider near, but within, its declared confinement domain, with widened cladding aperture and refined frequency tables.
Time dependence and spectral phase. Construct two modes with different delayed envelopes and a known spatial cross term. Verify widths and centroids at several times by direct summation; full fluence is not an average of widths. A common spectral phase changes temporal slices but leaves full fluence fixed. A pure delay translates the time axis. Refine temporal sampling at three levels; compare exact Fourier evaluation at selected times with zero-padded synthesis.
Signed flux and undefined widths. Test destructive interference with negative local flux, a dark slice, a user signal floor, an annulus, a double lobe, an unresolved crossing and a truncated aperture. Assert specific reason codes, finite raw signed totals where defined, NaNs only in the affected reductions, and exceptions for nonfinite inputs. Equal eigenvalues cannot create unstable principal-axis labels. No clipping may change the energy identity.
Gauge and convention invariance. Apply frequency-dependent channel phases and compensated amplitudes, and a declared unitary mixing of a general basis. Fluence and time-resolved reconstruction agree with the original to their contraction/table bounds. Exercise saved optics spectra, internal spectra, b-to-a conversion, an explicitly rebuilt reference and wrong-identity refusals.
Scalar completeness. Use temporally separated mode peaks to distinguish a simultaneous total peak from a sum of maxima. Check per-mode inner/outer widths, empty modes, unit/zero/complex filters, hard-window tiling, wavelength endpoints, and a phase-only filter. Check the effective-area statistic against its declared definition without equating it to the true spatial peak. Species pressures use their actual temperature paths and agree with \(\rho_s k_BT\) in the ideal limit.
Two independent resolution diagnostics. In an orthonormal analytic spatial basis, construct a source with a known component outside the retained span: the quadrature residual can converge to zero while the span residual remains. Adding the missing basis function removes the latter. Separately under-resolve the quadrature while keeping the modal space fixed. Test a weak selected band, rank deficiency, zero-source denominators, plasma refusal for the polarisation-only span test, and general-guide missing-coarse-rule refusal. Use conditioned least-squares/summation bounds; compare on an independent rule.
Execution and storage. Statistics only read the state: they add no operation to the march and do not change its schedule or stage kernels, a structural contract. Within one backend, a replayed run with statistics keeps the same steps (exact) and its propagated arrays within a rounding bound fixed before measurement of the run without them; an adaptive run keeps its arrays within the case’s accuracy envelope against a tighter or stored reference, with its accepted/rejected counts in a loose band. Cover split absorbers, rejection, replay and modified noise. Per-member diagnostics must not broadcast the largest batch error to every member. Verify partial chunks, thread-count agreement within a rounding bound fixed before measurement, periodic/plane equality for admitted cheap names (exact where the periodic row reuses the data the plane stores, otherwise within a rounding bound), memory/HDF5 round trips, resume hash mismatches, schema compatibility and scans retaining statistics without fields. Existing between-plane row-count defects are repaired with regression tests before qualifying new periodic names.

Qualify NumPy first, then Numba CPU and Kokkos CPU. On identical sampled data derive parity tolerances from accumulation length, FFT errors, conditioning and the local crossing slope; do not ask for relative agreement of two undefined or vanishing widths. For a crossing, propagate a profile error as \(|\delta r|\lesssim(|\delta D|+|\delta L|)/|D'(r)|\); a nearly tangent crossing is unresolved rather than arbitrarily accurate. Record convergence in aperture, transverse rule, profile samples, frequency tables and time sampling separately.

Benchmark the default statistics, each new family, and combined requests on roughly ten accepted steps or a fibre short enough to finish in seconds. Include ordinary few-mode and frequency-dependent general-modal cases, time-resolved requests with several output sizes, batch members, chunk sizes, persistent/scratch bytes and host transfer bytes. Record the window, warm-up and measured distributions in benchmarks; do not extrapolate to a full-length propagation. GPU qualification remains the separately exercised device lane, never inferred from CPU compilation.

8. Documentation#

Extend statistics with the complete capability table, equations, units, axis shapes, cadence, validity codes, apertures and two worked examples: fluence widths and time-resolved widths of a coherent two-mode field. Extend power and intensity and general modal propagation with the distinction between the actual E/H product, ordinary frozen-profile flux and metric-envelope surrogates. Document saved-reference identity requirements in processing and filter/time-axis persistence in output.

Add a short tutorial showing unchanged modal energies with changing beam width, then a delayed two-mode pulse whose temporal slices differ from its fluence width. Keep numerical examples short; a full worked example is optional and, if added, must be executed and its generated output committed under the gallery rules. Every public constructor and reduction gets a validated docstring with units, refusals and approximation limits. Update architecture, conventions, seam and backend documentation when the implementation changes them, and the changelog in each phase. At Final, the manual links back to the archived SEP.

9. Compatibility and impact#

The new measurements do not change an existing default statistic, units, reduction, temporal trace or output shape. SEP 34 separately changes the shared medium-flux measure. All new measurements here are opt-in. In particular, total temporal FWHM stays scalar, pressure stays in bar, general-basis coordinate statistics remain restricted, peak_intensity stays on-axis on modal routes, and quadrature_error remains the cached stage alarm. Existing spatial fwhm_r is not evidence that a modal request can run without profiles. New ordinary modal widths describe the frozen-profile model when that is what propagated; a frequency-dependent diagnostic does not retroactively improve that model.

The output and seam revisions are deliberate breaking changes, with clear unsupported-file and incompatible-resume refusals; no old-file migration is required. Filter transfer samples, temporal selection, signal floors, aperture, profile approximation, guide/gauge identities, source scope and diagnostic norms all enter provenance. A backend missing any requested family refuses it by name and names a backend that supports it; no full field host fallback is hidden inside a native statistics call.

When implemented, differences from Luna must describe the coherent frequency-resolved fluence and time-resolved definitions, centroid-based radial average instead of an assumed on-axis maximum on one cut, true flux weighting, per-mode versus total widths, explicit filters, spatial residual instead of an on-axis reconstruction surrogate, and preserved plane/periodic scheduling. The broader inventory does not imply numerical identity with a differently normalised or differently sampled statistic.

The roadmap’s transverse-maximum intensity item is fully covered by this SEP. Multiple-ZDW selection, SEP 34’s shared spatial flux, and the remaining ordinary spatial accessor issues remain separate. The between-plane memory-output row-count defect is a prerequisite fix for qualification, not a statistics redesign.

10. Alternatives considered#

One width from effective area. Effective area encodes a nonlinear overlap and does not fix a profile, its centroid or its intermodal interference. Refuse this shortcut even when it happens to reproduce a Gaussian calibration.
Only a radial cut through the guide axis. This assumes the maximum, symmetry and centre that a multimode field can lose. Prefer covariance plus an explicitly radialised central width, with an undefined result for a hollow central profile.
Only pulse-integrated statistics. These conceal spatial changes within a pulse. Both fluence and time-resolved measurements are required here.
Only \(|E|^2\) or \(|b(t)|^2\). These are useful quantities with different physical meanings. They are not the adopted broadband flux and cannot silently replace it when its sign makes a conventional width undefined.
Incoherent modal intensities. This removes phase-dependent spatial beating. An incoherent ensemble requires an explicitly defined ensemble, not discarding cross terms of one deterministic propagated field.
A single frequency-independent matrix everywhere. Useful when the kernel separates, wrong when the profiles vary. Its domain is an optimization contract.
Rename the quadrature alarm as reconstruction error. Integration accuracy and modal completeness are independent; two distinct diagnostics are required.
Always compute every statistic or accept arbitrary closures. This defeats bounded device execution, meaningful cadence and serialisable provenance. Keep the catalogue and explicit capability refusals.

11. Phases#

NumPy scalar catalogue and measurement contract. Implement the per-mode durations, hard-window total peak, filters, effective-area diagnostic and species pressures with tests and manual coverage. Introduce and validate the immutable measurement records, capability vocabulary, output metadata and seam revision. Backends explicitly refuse new unsupported families. Keep the default path the same code path, so default results are unchanged (an exact no-op contract); resolve API/norm questions before accepting the proposal.
NumPy beam measurements, including temporal slices. Deliver the chunked reference reconstruction, ordinary and general modal fluence, centroids, covariance/diameters, radialised widths, temporal series, signed-flux policy, transverse peak, offline parity and the short tutorial. Qualify E/H identities, modal phase/gauge, taper motion, aperture and independent sampling convergence. Both fluence and time-resolved measurements are required to close this phase.
NumPy source diagnostics. Deliver fresh filtered quadrature norms where an embedded pair exists, and the polarisation-span residual with an independent validation rule. Document current-source and missing-rule refusals. Prove separation of quadrature error from modal truncation and read-only evaluation.
Numba CPU parity. Implement admitted requests with bounded preallocated buffers, deterministic shared bodies and reference comparisons. Qualify batch, thread count, replay and periodic scalar additions; record the short-window cost and memory ledger. Do not hide a reference-backend fallback.
Kokkos CPU parity and completion. Update consumed contract fields, native reconstruction/reductions, packed values/validity and Python adapter in solprop-kokkos. Qualify the same definitions without full state observation, including general-modal magnetic reconstruction and temporal output volume. Run both repositories’ applicable gates, publish measurements, reconcile open questions with issues/roadmap and archive only after all required CPU parity.

Each phase is independently mergeable, with the full gate, slow and coverage lanes for implementation, the appropriate documentation, and its Part II report. Until a family is implemented on a backend, its refusal remains tested. GPU work follows the standing separate qualification policy. No phase changes the propagation model to make a diagnostic easier to compute.

12. Open questions#

Q1 — acceptance, maintainers: confirm the proposed centroid-centred radial average for modal fwhm_r, and the policy that signed local flux can make widths undefined while preserving raw signed energy/power. The draft recommends these rules; they are new measurement conventions, not assumptions already approved for implementation. Both fluence and time-resolved families are in scope.
Q2 — acceptance, maintainers: approve the fresh drive norm and the separate electric-profile polarisation-span diagnostic, including its limited physical interpretation and explicit deferral of current-source span measurements.
Q3 — phase 1: settle exact public constructor names, status encoding and persistence of the time/filter axes. Retain the shapes, semantics and resume requirements above. Choose a measured default diagnostic-rule recipe only after qualifying the ordinary and general-modal domains; do not inherit nonlinear nodes as an unexamined width-resolution default.
Q4 — phases 2 and 4: choose the bounded strategy for centroid-centred radial resampling, moving-guide diagnostic tables and general-modal H synthesis. Performance may choose recomputation or additional maps, but cannot freeze the frequency dependence or substitute a new gauge. Record interpolation and aperture error estimates, including shallow-crossing failure criteria.
Q5 — phase 3 / future roadmap: a missing embedded quadrature on general guides remains a named refusal. A future current-source span diagnostic needs a separate norm/response convention; neither extension is required for Final here.
Q6 — phase 5: confirm periodic scalar qualification after the existing between-plane row-count issue is fixed. Large beam/time-series families remain plane-only; a later periodic extension needs its own measured cost admission.

13. References#

A. E. Siegman, “How to (Maybe) Measure Laser Beam Quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, OSA Trends in Optics and Photonics 17, MQ1 (1998). doi:10.1364/DLAI.1998.MQ1.
ISO, ISO 11146-1:2021: Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams. Official standard record.
F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” Journal of the Optical Society of America B 25, 1645–1654 (2008). doi:10.1364/JOSAB.25.001645. Context for broadband multimode/polarisation modelling, not a source of a beam-width formula or an implementation to reproduce.
solprop, conventions, statistics, power and intensity, general modal propagation, kernel contract, and the modules named in §3.

Source verification at drafting. The publisher records confirm the titles, authors, dates and DOIs of [1] and [3]; their accessible abstracts establish scope, but their full text was not available through the consulted public endpoints. ISO’s official record confirms [2]’s edition and restricted beam-class scope; the full normative measurement procedure was not inspected. No material coefficient or empirical beam-width constant is transcribed. The flux, Parseval, Gaussian width and projection formulas above are derived here from the package’s declared conventions; [1–3] provide context rather than unverified equation authority. An independent synthetic complex-profile calculation on both DFT layouts checked the fixed-profile contraction, temporal/spectral Parseval equality, integrated modal energy and invariance of fluence under common spectral phase; the largest relative discrepancy was \(2.9\times10^{-16}\) for 64 time samples and three modes. This checks the algebra, not provider accuracy or a propagation implementation. Implementation must check any additional external formula it adopts against its primary source and report that check. This draft does not certify ISO compliance.

Part II — Implementation record#

No implementation has started.

Part III — History#

2026-09-27: Drafted after a capability and reconstruction audit. Scope includes both full-pulse fluence and time-resolved beam widths, explicit spectral filters, and separately defined quadrature and modal-span diagnostics.
2026-10-17: Restated for SEP 25. §7 item 11’s “propagated arrays bitwise unchanged” by statistics becomes a structural rule (statistics only read the state and change no march operation) plus a rounding bound fixed before measurement for replayed runs and the case’s accuracy envelope for adaptive runs, with step counts exact on replay and in a loose band otherwise; its thread-count invariance and §6.2’s deterministic partial reductions become agreement within a rounding bound. Phase 1’s “defaults bitwise unchanged” (§11) stays exact as a no-op contract: the default takes the same code path. Periodic/plane equality stays exact only where the row reuses the data the plane stores; HDF5 round trips and resume hashes stay exact.
2026-10-22: Assigned shared uniform-medium spatial flux and transported observation planes to SEP 31. This proposal retains its in-route modal measurement scope; schema changes require no old-file compatibility layer.
2026-10-23: Reassigned the shared medium-flux prerequisite to SEP 34 following SEP 31’s review and split. SEP 31 retains field exchange and coupling; the shared flux proposal includes admitted radial adiabatic transport and complex local flux.