Mode averaging#

Mode averaging follows one temporal amplitude while replacing the transverse nonlinear response by an effective-area model. Use it when one guided mode carries the pulse and a resolved response across the core is unnecessary. It is the spatial description used by the GNLSE, and prop_capillary(..., mode_average=True) selects it for a single capillary mode.

A resolved single-mode calculation is different: it retains one propagated amplitude but evaluates the response at multiple points across that mode. prop_capillary(modes=1), the default, selects this resolved calculation. The distinction matters for ionisation and other strongly intensity-dependent responses. The spatial introduction compares both descriptions with multimode and free-space propagation.

What the effective area preserves#

For a transverse profile \(e(x,y)\), the effective area is

\[A_{\mathrm{eff}}= \frac{\left(\int |e|^2\,dA\right)^2}{\int |e|^4\,dA}.\]

This definition preserves the single-mode cubic Kerr overlap. It does not preserve every local intensity or the spatial integral of an arbitrary function of intensity. In particular, replacing a peaked profile by an equivalent uniform area changes the field sampled by an ionisation rate.

A capillary HE11 profile beside a uniform disc with the same effective area, and their intensity profiles along a diameter.

The effective-area model preserves the cubic self-coupling integral. The field near the axis and the spatially integrated ionisation response need not agree. This is a comparison of model profiles, not a propagated result.#

A prescribed single-mode profile cannot deform through self-focusing or populate another guided mode. Use a multimode capillary or free-space calculation when that evolution matters.

Select it explicitly#

The GNLSE entry point selects mode averaging automatically. For a gas-filled capillary or a supplied gas-filled mode, pass mode_average=True to prop_capillary or prop_waveguide, or to Simulation.from_capillary or Simulation.from_waveguide:

from solprop.driver import prop_capillary

output = prop_capillary(
    125e-6,
    1e-5,  # a short configuration check, in metres
    "Ar",
    1.0,  # bar
    lambda0=800e-9,
    wavelength_limits=(500e-9, 1200e-9),
    trange=200e-15,
    fwhm=30e-15,
    energy=1e-9,
    plasma=False,
    raman=False,
    backend="numpy_ref",
    mode_average=True,
)
assert output.metadata.simulation_kind == "mode_averaged"

The run carries one mode, so modes greater than one is refused, as are n_r, n_theta and radial_rule, which shape a transverse rule the run does not have. The default statistics are the capillary’s own, plasma and species included, less the quadrature estimate. Everything else (a tapered bore, a pressure gradient, a tube lattice, an envelope grid) runs as it does resolved. Luna.jl makes the opposite default choice: its prop_capillary is mode-averaged unless it is given a mode count (coming from Luna.jl).

The MI ensemble and comb-to-continuum example use this switch at full length, retaining plasma and, for hydrogen, Raman. Their saved metadata identifies simulation_kind="mode_averaged" and records no transverse quadrature. Those examples illustrate the chosen approximation; they do not establish agreement with a spatially resolved plasma calculation.

Interpret the amplitude and the intensity separately#

The input and saved modal amplitudes are normalised to true power. In a monochromatic scalar mode, \(b=\sqrt{\operatorname{Re}n_{\mathrm{eff}}}\,a\), where \(a\) multiplies the vacuum-normalised transverse profile internally. For a pulse, broadband power and energy use the definitions in power and intensity.

An effective-area intensity and the mode’s on-axis intensity answer different questions. The nonlinear response in a mode-averaged calculation samples the former. A statistic reconstructed from the transverse mode may report the latter. Do not interpret a reported peak intensity as proof that the response resolved that peak across the core.

To assess this approximation, compare with a resolved single-mode calculation and converge its radial rule. To assess transfer into other modes, increase the retained mode set as a separate check. Decreasing the propagation tolerance cannot correct a missing transverse degree of freedom. See mode sets and quadrature and radial nodes for expensive responses.