Modes from tabulated data#

Measured or independently calculated effective indices can supply dispersion and loss without an analytic guide model. First decide whether a table describes an evacuated hollow guide, a filled guide, or a solid-core fibre: adding gas to an already filled table would count the material twice.

When the effective index comes from a measurement or from a separate mode solver, a table of it can stand in for an analytic model. There are three ways to use one:

  • an evacuated hollow table plus a declared gas, TabulatedHollowMode, which adds the gas’s susceptibility to the table and so follows a pressure change or gradient (below);

  • a filled hollow table, TabulatedMode, whose table is the whole index of one particular fill (below);

  • a solid-core fibre: the bare IndexTable given to prop_gnlse as its fibre= (below).

The two hollow classes keep Marcatili’s transverse field, so they run on the capillary route as a supplied modes=[...] set. A table never extrapolates, and interpolation between its rows is the main source of error; the last subsection says how to check it.

IndexTable takes loaded arrays of angular frequency and complex effective index, with an optional effective-area column. There is no file-format convention: read the rows with NumPy, h5py or a loader of your choice, then call IndexTable(omega, neff_values) or IndexTable.from_wavelength(wavelength, neff). All units are SI. Both constructors accept the rows in either order and sort them into increasing frequency. The second argument of the plain constructor is neff_values, not neff, because a field and a method cannot share one name: table.neff(omega) evaluates the interpolant.

The table copies its arrays and makes them read-only. It needs at least four distinct positive frequencies, finite indices with nonnegative imaginary part, and positive finite areas when present. It interpolates real refractivity \(\operatorname{Re}(n-1)\) with a cubic spline in angular frequency, and differentiates that spline analytically. Default loss interpolation is a cubic spline of \(\log\operatorname{Im}n\), which stays positive between knots when loss spans many decades. It requires strictly positive imaginary rows. For data with zero-loss rows, or to interpolate the imaginary index itself rather than its logarithm, use loss_interpolation="linear": a cubic spline of \(\operatorname{Im}n\). If that spline undershoots zero, both its value and its derivative are clamped to zero, with one warning log per table.

IndexTable.from_loss(wavelength, index, loss_db_per_m, ...) converts power loss using \(\alpha=\ln(10)\,L_{\rm dB}/10\) and \(\operatorname{Im}n=c\alpha/(2\omega)\). The table’s alpha is in m⁻¹; its beta, dbeta_domega, refractivity, dneff_domega and effective_area use the same interpolants. Area interpolation is cubic; a nonpositive interpolated area is refused, rather than clipped.

An evacuated hollow table and its gas fill#

For a separated longitudinal and transverse wavevector,

\[ \beta^2=n_g^2 k^2-\kappa^2,\qquad k=\omega/c. \]

An evacuated table gives \(\kappa^2/k^2=1-n_v^2\). Holding that transverse quantity fixed while adding gas susceptibility \(\chi_g=n_g^2-1\) gives

\[ n_{\rm eff}^2=n_v^2+\chi_g. \]

TabulatedHollowMode applies this rule through the full Marcatili evaluation. For \(r=n_v-1\) it supplies \(w=-r(2+r)\) and \(w'=-2(1+r)r'\); these preserve the small refractivity and analytic derivative. It evaluates the declared gas, and never samples silica. The rule is exact for Marcatili’s fill-independent waveguide term: at the table’s own knots it reproduces the same MarcatiliMode to rounding, on a mixture’s rationalised arithmetic as well as on a single gas’s. Between knots the error is the spline’s: doubling the knots from 50 to 100 to 200 reduces the index error by factors of 15.98–16.26 and the derivative error by 8.13–8.36, fourth and third order. The transverse fields, normalisation and polarisation partners are Marcatili’s.

import numpy as np
from solprop.constants import C
from solprop.modes import IndexTable, MarcatiliMode, TabulatedHollowMode

omega = 2 * np.pi * C / np.linspace(200e-9, 2e-6, 200)
vacuum = MarcatiliMode(15e-6, cladding=1.45)
table = IndexTable(omega, vacuum.neff(omega))
mode = TabulatedHollowMode(15e-6, "Ar", 10.0, table=table)

A pressure gradient varies the gas contribution while keeping the table and radius fixed; a taper is refused. model="reduced" and a supplied cladding are also refused. The usual loss policy acts after the table model: keep, remove, scale or replace its total attenuation. Hollow-table species absorption is still attributable to the declared gas.

The rule neglects changes in wall contrast and the gas-driven resonance shift: the filled table keeps its wall resonances where the evacuated guide has them. A gas lowers the wall’s index contrast, and the antiresonance condition \(\lambda_m \propto t\sqrt{n_w^2-n_g^2}\) moves every resonance by \(\Delta\lambda/\lambda \approx -\chi_g/[2(n_w^2-1)]\): about a quarter of a per cent towards shorter wavelengths (−0.24 % at 800 nm) for 10 bar of argon in a silica wall. Close to a resonance the filled guide’s index and loss therefore differ from the table’s by more than the smooth-row figures below. The rule’s error is measured on the antiresonant fibre whose finite-element data are described in accuracy against a finite-element solution, with the evacuated FEM rows as the table and the 10-bar argon rows (293 K, kingston source) as the answer. Rows within 3 % of a wall resonance are counted as near it; the split is descriptive, not a validity region.

  • Real index. On the nine smooth rows away from resonance (690–946 nm) the rule predicts the filled index to 2.24544e-8, against 6.3e-7 from simply adding the gas index to the evacuated one. Near resonance the largest residual is 2.82871e-5.

  • Loss. The ratio of FEM to table loss is 0.99261–0.99843 on the nine smooth rows, but 0.85867–0.99843 over all fourteen rows away from resonance: it falls to 0.86 at the short-wavelength end. The rule’s transfer of loss is therefore trusted on the smooth rows only.

Those numbers describe one fibre.

A second diagnostic fills a table built from the Vincetti model itself (a 15 µm core, 300 nm walls, eight tubes of 6 µm outer radius, cladding index 1.45, 400 knots over 400–1200 nm; nominal first wall resonance at 630 nm) and compares it with the Vincetti model evaluated at the filled pressure. At 10 and 30 bar the largest real-index differences reach 9.19e-6 and 2.69e-5; near-resonance loss ratios span 0.527–1.673 and 0.161–2.558. Part of that is interpolation rather than the fill rule: even the evacuated table, where the rule does nothing, has \(\beta_2\) interpolation residuals up to 4564 fs²/m, because a 400-knot cubic does not resolve every fitted narrow resonance, and the largest of them lie outside the nominal resonance window, at the fitted tube-mode cutoffs (inside it the maximum is 2888 fs²/m). A wall-contrast refinement of the rule is not implemented.

A filled hollow table, and choosing area#

TabulatedMode takes the table as the whole complex index. It adds no gas susceptibility and returns no species absorption contribution: that loss is already in the table. Marcatili fields still supply the transverse profile. Declare gas or a mixture for its nonlinear responses, with constant material paths. A table is one fill: a 10-bar argon table paired with a 20-bar nonlinear material is a modelling error the code cannot detect. Varying material paths are refused because this table cannot describe a changing fill.

from solprop.modes import TabulatedMode

# filled_table contains the measured or computed index of this same 10-bar fill.
# mode = TabulatedMode(15e-6, "Ar", 10.0, temperature=293.0,
#                      table=filled_table, reference_wavelength=800e-9)

When an area column exists, both hollow classes require reference_wavelength, inside the table. Propagation uses the table area at that fixed reference; mode.effective_area(omega) exposes its spectral variation. With no area column, omit reference_wavelength and the inherited Marcatili area is used. Normalisation and transverse fields remain Marcatili’s even when the table supplies propagation area. On the finite-element fibre above, the evacuated and filled areas differ by at most 1.02e-4 relative on the smooth rows, so an area column is insensitive to the fill to that level. The radius still defines the field shape, not the loaded dispersion.

Solid-core input and checking the interpolant#

Pass an IndexTable as fibre= to prop_gnlse or Simulation.from_gnlse: prop_gnlse(None, length, fibre=table, n2=..., ...). The table area at lambda0 resolves \(\gamma=n_2\omega_0/(cA_{\rm eff})\); a table without area requires gamma and uses the GNLSE convention 1 m². A supplied Mode is wrapped by from_mode(): its dispersion, analytic first derivative, loss and area are retained, but its fill is ignored as a nonlinear material because gamma or n2 defines that response. A mode whose geometry moves along z is carried as a pair of declared paths – dispersion and area together – rather than frozen at the entrance; see the GNLSE page. The record nests the original mode under the GNLSE mode kind. Use the capillary route when the fill should drive nonlinear responses. See running a simulation and the executed examples.

Extrapolation is always refused. A propagation’s active band includes the spectral collars outside wavelength_limits, and dispersion probes such as beta2 evaluate neighbouring frequencies. Inspect grid.omega[grid.active_mask] before running; provide additional support for any derivative probes. The refusal gives both requested and table ranges in rad/s and nm. Narrow the grid or obtain wider data rather than extending an unverified spline.

Knot agreement checks data conversion, not resolution between rows. Leave one interior row out, rebuild, and compare the prediction at that row; do not leave an endpoint out and then extrapolate. On the evacuated finite-element rows above, the largest leave-one-out real-index error away from resonance is 9.15112e-6, much larger than the smooth fill-comparison residual. Compare beta2 with unequal-step second differences of the raw propagation constants as well. On the filled rows the two differ by up to 7.2699 % over the interior of the nine first-band rows above 580 nm, whose 29.5–33.4 nm spacing averages curvature that the derivative samples locally. This is a sparse-data diagnostic, not a physical error bar. A linear spline of the evacuated rows’ imaginary index crosses zero in 20 intervals between 200 and 456 nm; log interpolation avoids that particular defect but cannot recover missing spectral detail. Refine data around resonances and ambiguous zero-dispersion wavelengths before using them for predictions.