# A frequency-dependent electronic response

The [Kerr response](nonlinear.md#the-kerr-response) multiplies the cube of the field by
one number per gas: its electronic third-order susceptibility frozen at the run's
carrier. That number is not a property of the gas alone. A gas's hyperpolarisability
rises towards the ultraviolet — argon's Kerr coefficient at 267 nm is twice its value at
800 nm, xenon's at 400 nm nearly twice its value at 2 µm — and for a quartet of different
frequencies it depends on all four. A run whose spectrum spans an octave or more, or
which mixes widely separated colours, therefore uses a coefficient that is right for
some of its processes and wrong for others.

This page describes the two **dispersive electronic policies** of
[SEP 6](../../dev/seps/archive/sep-0006-dispersive-nonlinearity.md), which evaluate optmat's whole
fit of a gas's hyperpolarisability at the absolute frequencies of every quartet the
cubic couples, in one broadband polarisation. It is for readers who want to know which
polarisation such a run computes, what each of the two evaluators approximates and what
it costs, what happens where the field leaves the gas's data, and how to read what a run
recorded. The [tutorial](../../auto_tutorials/capillaries/plot_11_dispersive_kerr) runs
both of SEP 6's demonstration cases under all three policies.

| `MaterialSelection.electronic` | Response | What it evaluates |
|---|---|---|
| `"recommended"`, `"carrier-kerr"` (the default) | {class}`~solprop.nonlinear.responses.KerrResponse` | the Kerr coefficient at the carrier, one number per gas and run ([materials](material-selection.md#the-electronic-kerr-coefficient-at-the-carrier)) |
| `"gas-polynomial"` | {class}`~solprop.nonlinear.responses.DispersiveKerrResponse`, evaluator `"polynomial"` | optmat's gas dispersion polynomial, exactly, at every quartet |
| `"symmetric-kerr"` | {class}`~solprop.nonlinear.responses.DispersiveKerrResponse`, evaluator `"symmetric"` | a symmetric factorisation of the polynomial's Kerr diagonal (a mixture's: of the mixture's own) |

Both dispersive policies are opt-in; the carrier policy stays the default. They run on
`numpy_ref`, `numba_cpu` and the native backend's CPU build on every route: the
capillary and radial routes, Cartesian and line free space, the planar route and a
general modal guide's gas regions. The native backend's device builds are qualified for them on one
AMD Instinct generation (HIP) and one NVIDIA Hopper device (CUDA). A run is never moved to another backend or another response without being told.

## One susceptibility, all mixing processes

In the optics Fourier convention ([conventions](../../dev/conventions.md), §2) the
third-order polarisation of a real scalar field is one double integral over the input
frequencies,

$$
\widetilde P^{(3)}(\omega_o)=\frac{\varepsilon_0}{(2\pi)^2}
\iint\chi^{(3)}(-\omega_o;\omega_1,\omega_2,\omega_3)\,
\widetilde E(\omega_1)\widetilde E(\omega_2)\widetilde E(\omega_3)
\,\mathrm d\omega_1\,\mathrm d\omega_2,
\qquad \omega_3=\omega_o-\omega_1-\omega_2 .
$$

Self- and cross-phase modulation, four-wave mixing and third-harmonic generation are
not separate terms: they are slices of the same integral, $(\omega,\omega,-\omega)\to
\omega$, $(\omega_r,\omega_p,-\omega_p)\to\omega_r$,
$(\omega_p,\omega_p,-\omega_i)\to2\omega_p-\omega_i$ and
$(\omega,\omega,\omega)\to3\omega$. With a constant $\chi^{(3)}$ the integral collapses
to $\varepsilon_0\chi^{(3)}E(t)^3$, which is why the constant cubic needs no
bookkeeping of processes. A dispersive susceptibility must keep that property: one
polarisation, every quartet weighted by its own four frequencies, no process named
during propagation.

For a gas of number density $N$ with Taylor-convention second hyperpolarisability
$\gamma_T$, $\chi^{(3)} = N\gamma_T/(6\varepsilon_0)$, and solprop's per-particle
coefficient is $\gamma^{(3)} = \gamma_T/(6\varepsilon_0)$ in m⁵/V²
([materials](material-selection.md#the-electronic-kerr-coefficient-at-the-carrier)). optmat fits
$\gamma_T$ over a quartet of signed vacuum wavenumbers $\nu_j=\omega_j/2\pi c$, the
output counted negative so that the four sum to zero, with the dispersion expansion of
Bishop and De Kee, *J. Chem. Phys.* **104**, 9876 (1996),
[doi:10.1063/1.471752](https://doi.org/10.1063/1.471752) (their Eqs. 22 and 67, written there
additively in atomic units), put relative to the static value in Shelton's form:

$$
\gamma_T=\gamma_0\,[1+A S_2+B S_2^2+B' S_4],\qquad
S_2=\textstyle\sum_j\nu_j^2,\quad S_4=\sum_j\nu_j^4 .
$$

Its Kerr diagonal, $(\nu,\nu,-\nu)\to\nu$, is
$\gamma_K(\nu)/\gamma_0 = 1+4A\nu^2+(16B+4B')\nu^4$; the carrier policy uses
$\gamma_K$ at the carrier for every quartet. optmat's records carry $A$ in units of
$10^{-10}$ cm² and $B$, $B'$ in $10^{-20}$ cm⁴, which solprop takes to m² and m⁴
({class}`~solprop.physdata.nonlinear_data.DispersiveKerrModel`).

### Why a coefficient of the output frequency is wrong

The tempting shortcut multiplies the constant cubic's spectrum by the Kerr coefficient
of each *output* frequency, $\varepsilon_0\chi_K(\omega)\mathcal F[E^3]$. It is right
for monochromatic self-phase modulation and wrong for everything else. It gives every
quartet that lands on a frequency the degenerate coefficient of that frequency,
whatever its inputs were: the argon quartet $400+400-800\to266.7$ nm gets argon's Kerr
value at 266.7 nm, 50 % above the polynomial's value for that quartet, where the
carrier's value at 400 nm is only 4 % below it (table below). It also breaks the
exchange symmetry of a lossless quartet — the coefficient changes when a different
member is called the output — so the response does net work on the field, which a
transparent medium cannot. solprop does not offer it.

## The two evaluators

Both evaluate the cubic as a sum of products of spectrally filtered fields, transformed
on the run's oversampled fine grid; neither builds a table over frequency triplets. The
filters are real, even functions of **absolute** frequency, so the same response is
computed on a field grid and on an envelope grid at any carrier.

**The polynomial** (`"gas-polynomial"`). With $D$ the spectral multiplier
$d(\omega)=(\omega/2\pi c)^2$, expanding $S_2$ and $S_4$ over the four legs and
grouping the output factors gives (SEP 6, §4.5)

$$
\frac{6\widetilde P}{N\gamma_0}
=[1+Ad+(B+B')d^2]\,\mathcal F[E^3]
+[3A+6Bd]\,\mathcal F[E^2\,DE]
+\mathcal F\big[3(B+B')\,E^2D^2E+6B\,E(DE)^2\big].
$$

This is exact evaluation of the fit on the resolved grid: no approximation beyond the
fit itself. It costs three inverse transforms (of $E$, $DE$, $D^2E$) and three forward
transforms of products where the constant cubic needs one of each.

**The symmetric factorisation** (`"symmetric-kerr"`). One real, even multiplier
$g(\nu)=[\gamma_K(\nu)/\gamma_0]^{1/4}$ is applied to every leg:

$$
\chi^{(3)}_{\rm sym}=\chi_0\,g(\nu_o)g(\nu_1)g(\nu_2)g(\nu_3),
\qquad P_{\rm sym}=\varepsilon_0\chi_0\,G\big[(GE)^3\big],
$$

with $G$ the filter $g$: filter the field, cube it, filter the polarisation. It is exact
on the Kerr diagonal and agrees with the polynomial through second order in the
frequencies for every quartet; its leading error is
$(B-A^2/2)\sum_{j<k}(\nu_j^2-\nu_k^2)^2$, which grows with the spread of the quartet's
four frequencies. Over random quartets between 400 nm and 2 µm the median error is
0.006 % for helium and 0.05 % for argon and the largest 0.1 % and 0.9 %; xenon, the
most dispersive gas, reaches 3 % (SEP 6, §4.7). For a scalar field with no other
response it needs no transform beyond the constant cubic's, only two spectral
multiplications, since the filters can act on the spectra the cubic already
transforms.

Matching every degenerate value does not determine the general susceptibility: the
factorisation is a separability assumption, not a consequence of the data, and the
polynomial is the reference it is measured against.

**Cost.** The symmetric evaluator adds spectral multiplications to transforms
already needed for the cubic response. The polynomial requires derivative
fields and additional transforms; mixtures can share them where their leg
bounds agree. Whole-run cost also depends on the adaptive trajectory, so stage
cost alone does not predict the runtime. Use the [backend and performance
guide](../performance/index.md) for execution choices. The measured stage-time ratios
against the constant cubic, with their conditions, are in the
[benchmarks](../../dev/benchmarks.md#dispersive-electronic-response).

**The temporal collar.** A run's temporal collar $w(t)$, which keeps the
polarisation off the edges of the time window, multiplies each product in time
before its output multipliers and $\Pi_+$: $G[w\,(GE)^3]$, and likewise for every
term of the polynomial. The constant cubic's collar multiplies its polarisation, and
with no output multiplier the two orders coincide. For the dispersive evaluators the difference depends on the filtered fields and products near the collar.
Check it by enlarging the window while retaining resolution. The earlier comparison found a difference of a few parts in
$10^3$ for a pulse at nine tenths of the half-window. A statement of equivalence
for a well-contained pulse is conditional on the filtered field also remaining
clear of the collar.

**Lossless response.** With no active temporal collar or boundary term, both
kernels are real and symmetric in their four legs. The continuum response then
does no net work on a real field, $\int(\partial_tE)\,P\,\mathrm dt=0$, exactly as the
constant cubic. Discrete evaluation requires adequate sampling; an occupied
collar changes this identity. Energy conservation in a propagation therefore
requires checking absorbers, representation error and integration error
together. The test suite compares both evaluators with an explicit input-triplet
sum at $10^{-14}$ and checks the lossless identity.

## The harmonic sector: `thg`

The constant cubic's `thg=False` keeps "the fundamental band" by a narrowband rule
([one cubic, four forms](nonlinear.md#one-cubic-four-forms)). The dispersive responses
use a rule that holds at any bandwidth instead. Writing the real fields through their
analytic signals, each product of three fields is a sum over quartets whose four signed
legs have three, two or one positive members:

| `thg` | Sector | Quartets kept |
|---|---|---|
| `True` (the field-grid default) | full | all: self- and cross-phase, four-wave mixing, the third harmonic and its inverse |
| `False` (the envelope-grid default) | number-conserving | two positive and two negative legs, with $\Pi_+$, the projection onto positive absolute frequency |

The number-conserving sector is
$\mathrm{Re}\{\tfrac34\Pi_+[|u|^2u]\}$ for a field of analytic signal $u$, with the
filters of the evaluator applied to each leg. It keeps no harmonic quartet, so it
conserves photon number as well as energy. With the dispersion removed, and for any
band narrower than an octave, it is exactly the constant cubic's `thg=False` rule;
beyond an octave the constant rule, which lacks $\Pi_+$, does a little net work and the
dispersive sector does none. The general
modal route's `kerr_form` maps the same way — `"real"` to the full sector, `"analytic"`
to the conserving one — for a region that declares its gas fill
({class}`~solprop.modes.material.GasKerr`); a region with a constant
{class}`~solprop.modes.material.MaterialKerr` keeps it under every policy
([general modal propagation](../waveguides/general-modal.md)).

On an **envelope grid** the full sector carries two harmonic groups,
$\mathrm e^{+2\mathrm i\omega_0t}$ as the constant cubic's harmonic does, and
$\mathrm e^{-2\mathrm i\omega_0t}$, the one-into-three exchange that reaches positive
frequencies only for a field wider than an octave. An envelope grid built for harmonics
(`EnvelopeGrid.from_limits(..., thg=True)`) holds both for a field inside its active
band: a super-octave field agrees with a four-times finer evaluation to $10^{-13}$.
Like the constant cubic's harmonic, both groups can still wrap around the grid when the
field itself reaches the edges of the active band — when its collars hold content —
so keep the band wide enough that they stay empty.

## Beyond the gas's data: bounded, never refused

A run's field routinely extends far beyond any gas's data: a helium RDW run reaches
below 80 nm. The carrier policy runs there with a constant extrapolated from the
carrier. The rule for the dispersive policies is **never to be less sound than that,
and never to refuse a run for the frequencies its field occupies** (SEP 6, §4.10). It
rests on three facts optmat publishes in the support record of each hyperpolarisability
model:

- **The legs of the data.** The four signed legs of a quartet sum to zero, so no leg of
  a quartet within the fitted frequency sums $S_{2,\max}$ exceeds
  $\nu_c=\sqrt{3S_{2,\max}/4}$, the **leg bound**.
- **The resonances.** A single leg reaches the lowest dipole-coupled level $E_1$ and a
  pair of legs the lowest two-photon level $E_2$; a field stays clear of both while it
  occupies wavelengths longer than the **band edge**, $\min(E_1, E_2/2)$. A vector run,
  with two polarisation components, takes the band edge for any polarisation, which
  is 1–3 nm longer for Ne, Ar, Kr, Xe and CO.
- **Monotonicity.** Every fit rises with each $\nu_j^2$, inside its data and beyond.
  No theorem requires the physical $\gamma$ to do the same, but every curated
  calculation rises towards the first resonance, and faster than the fit.

solprop holds every leg of both evaluators at the leg bound,
$\nu_j\to\min(|\nu_j|,\nu_c)$: the polynomial's multiplier becomes
$\min(d(\omega),\nu_c^2)$ and the symmetric filter is evaluated at
$\min(|\nu|,\nu_c)$. Both stay real, even and the same on every leg, so the lossless
identity and the cost are unchanged, and no quartet within the fitted sums moves. The
run's active band (half its lowest frequency to 1.1 times its highest) is classified at
translation, per gas, into three regions:

| Region | Where | What the evaluators give | How to read it |
|---|---|---|---|
| supported | legs within $\nu_c$, inside the band edge | the fit itself | exact within the fit; beyond $S_{2,\max}$, the polynomial's own extrapolation |
| bounded | beyond $\nu_c$, inside the band edge | each such leg held at $\nu_c$ | a **bounded estimate**: clamping can underestimate a rising off-resonant response, but optmat does not certify a bound on the true coefficient or the observable |
| best effort | beyond the band edge, or beyond $\nu_c$ for a gas whose edge is undefined | the same bounded evaluation | the off-resonant fit is unqualified there; the response is held at the data edge rather than continuing its $\nu^4$ growth through a resonance |

A run whose band reaches the bounded or best-effort region logs that once, at INFO,
naming the gas and the band; nothing is refused and nothing warns per step. The leg
bounds and band edges are optmat's, read from its support record, never written into
solprop; at optmat 0.8.0 (unchanged since 0.4.0), for one linear polarisation:

| Gas | Leg bound (nm) | Band edge (nm) |
|---|---:|---:|
| He | 151.9 | 120.3 |
| Ne, Ar | 215.9 | 133.5, 189.4 |
| Kr, Xe | 230.1 | 216.7, 256.0 |
| H₂, D₂ | 215.9, 224.7 | 201.7, 201.1 |
| N₂, O₂ | 215.9, 224.7 | 290.1, 304.6 |
| CO | 231.5 | 308.9 |
| CO₂, CH₄, SF₆, N₂O, O₃ | 230.1, 224.7, 215.9, 383.4, 401.0 | undefined: no curated two-photon limit, so best effort beyond the leg bound |

Air has no fit of its own; each of its populations carries its own, and O₂ makes air
best effort below about 305 nm. CO₂, CO and O₃ have optmat fits but are not yet
built-in species.

## Pump dynamics and emission in one response

A resonant dispersive wave in the vacuum ultraviolet is emitted through four-wave
mixing: three components of the self-compressing soliton produce a VUV photon. Neither
dispersive evaluator gives a band one coefficient, so modelling that emission costs
nothing at the pump. In the helium RDW case of SEP 6:

- the soliton's self-phase modulation at 800 nm keeps $\gamma_K(800\,\mathrm{nm})$,
  exactly the carrier value, and its broadened wings, down to about 250 nm at maximum
  compression, lie inside helium's data and are evaluated exactly;
- a generating quartet such as $250+250-1500\to136.4$ nm takes a coefficient set by all
  four of its legs, 1.41 times the 800 nm value under the leg bound (1.49 unbounded),
  where the carrier policy is about 30 % low; in the symmetric model the single VUV leg
  enters only as a fourth root;
- the largest coefficients belong to quartets with two or more VUV legs — the
  cross-phase modulation of the dispersive wave by the pump, its own self-phase
  modulation — and they scale with the weak dispersive-wave field, so they hardly act
  back on the pump.

The dispersive wave's wavelength is set by phase matching to the soliton's nonlinear
wavenumber, which comes from the exactly evaluated self-phase modulation, so it moves
little between the policies; its energy, and its evolution after emission, are what
change. In the full-size case (the
[worked example](../../auto_examples/processes/plot_07_rdw_dispersive_kerr)) the
100–150 nm band at the exit carries about 30 % more energy under both dispersive
policies than under the carrier one with two or four modes, and about 53 % more with
one, and its centre moves by 2–3 nm; in the tutorial's shorter two-mode case, about
9 % and a nanometre. Below helium's 120.3 nm band edge that is a **sensitivity** to the
treatment of the VUV legs, not a qualified prediction: a resonant response, such as the
[helium TDSE response](tdse.md), can investigate those effects,
subject to its own atomic-model and finite-window limitations. It is not an
automatically qualified reference there. That
response owns helium's whole electronic response, so a dispersive policy is refused
beside it for the same gas.

## The demonstration cases in numbers

The coefficients each policy assigns to the quartets of SEP 6's two demonstration
cases, from optmat's fits; errors are signed relative to the polynomial, and
$S_2/S_{2,\max}$ above one is beyond the fitted sums. The leg-bounded column is what the
runs use, as a fraction of the unbounded polynomial.

| Quartet (nm) | Gas | $\gamma_{\rm poly}/\gamma_K(800)$ | Carrier error (%) | Symmetric (%) | Output-only (%) | $S_2/S_{2,\max}$ | Bounded / polynomial |
|---|---|---:|---:|---:|---:|---:|---:|
| 400 + 400 − 800 → 266.7 | Ar | 1.345 | −4.25 at 400; −25.65 at 800 | +0.556 | +50.00 | 0.98 | 1 |
| SPM of 266.7 | Ar | 2.018 | −36.17 at 400; −50.44 at 800 | 0 | 0 | 1.97 | 1 |
| XPM of 266.7 by 400 | Ar | 1.612 | −20.13 at 400; −37.98 at 800 | −0.025 | +25.13 | 1.42 | 1 |
| 250 + 250 − 1500 → 136.4 | He | 1.485 | −32.68 at 800 | −0.012 | +88.95 | 1.49 | 0.949 |
| SPM of 120 | He | 3.679 | −72.82 at 800 | 0 | 0 | 4.81 | 0.628 |
| SPM of 80 | He | 11.27 | −91.13 at 800 | 0 | 0 | 10.81 | 0.205 |

In **argon** every leg stays above the 215.9 nm leg bound, so the bounded evaluation is
the polynomial itself, and every frequency is inside the 189.4 nm band edge. The
generating quartet's carrier error at the 400 nm pump is small; the signal's
cross-phase modulation by the pump is 20 % stronger than the carrier says, and in the
tutorial that, not the generating coefficient, is what moves the conversion: the
dispersive runs convert faster at first and saturate earlier, about 30 % below the
carrier run at the exit of half a metre. In **helium** the VUV quartets are bounded and,
below 120.3 nm, best effort; the bound takes self-phase modulation at 80 nm from the raw
polynomial's 11.3 times the 800 nm value to 2.31 times.

The map below shows the symmetric factorisation's error over degenerate-pump four-wave
mixing, $2\omega_p-\omega_i\to\omega_s$, for argon and helium, with the boundaries of
the fitted sums and of the regions. The contours at 0.1, 1 and 5 % are descriptive
levels, not acceptance thresholds.

```{eval-rst}
.. plot::
   :filename-prefix: dispersive-kerr-1
   :alt: Contour maps of symmetric-factorisation error versus pump and idler wavelengths for argon and helium, with material-support boundaries marked.
   :caption: The symmetric factorisation against the polynomial for degenerate-pump four-wave mixing, both under the leg bound, as a function of the pump and idler wavelengths. Left of the dashed purple line the quartet is beyond the fitted frequency sums; left of the dotted line the signal's leg is held at the leg bound (bounded); left of the dash-dotted line the signal is beyond the band edge (best effort). The star is the argon DUV quartet of the tutorial. Where the bound holds a leg the two evaluators move together, and the error falls again.

   import matplotlib.pyplot as plt
   import numpy as np
   from matplotlib.lines import Line2D

   from solprop.physdata.nonlinear_data import OptmatKerr

   pump, idler = np.meshgrid(np.linspace(250e-9, 1000e-9, 241),
                             np.geomspace(520e-9, 4e-6, 241))
   with np.errstate(divide="ignore"):
       signal = 1.0 / (2.0 / pump - 1.0 / idler)
   kept = idler > pump

   fig, axes = plt.subplots(1, 2, figsize=(9.5, 3.9), layout="constrained",
                            sharey=True)
   for ax, gas in zip(axes, ("Ar", "He")):
       model = OptmatKerr(gas).dispersive_model()
       legs = np.stack([1 / signal, 1 / pump, 1 / pump, 1 / idler], axis=-1)
       polynomial = model.gamma3(legs)
       weights = model.kerr_ratio(np.minimum(legs**2, model.leg_bound**2))
       symmetric = model.gamma3_static * np.prod(weights**0.25, axis=-1)
       error = np.where(kept, 100 * np.abs(symmetric / polynomial - 1), np.nan)
       x, y = 1e9 * pump, 1e6 * idler
       filled = ax.contourf(x, y, np.log10(np.clip(error, 1e-4, 10)),
                            levels=np.linspace(-4, 1, 21), cmap="Blues")
       lines = ax.contour(x, y, error, levels=[0.1, 1.0, 5.0], colors="0.15",
                          linewidths=0.8)
       ax.clabel(lines, fmt=lambda value: f"{value:g} %", fontsize="small")
       sums = np.sum(legs**2, axis=-1) / (4 * model.leg_bound**2 / 3)
       ax.contour(x, y, np.where(kept, sums, np.nan), levels=[1.0], colors="C4",
                  linestyles="--", linewidths=1.3)
       ax.contour(x, y, np.where(kept, 1 / signal, np.nan),
                  levels=[model.leg_bound, model.band_edge(vector=False)],
                  colors=["C1", "C3"], linestyles=[":", "-."], linewidths=1.3)
       ax.set_title(f"{gas}: $2\\omega_p - \\omega_i$")
       ax.set_xlabel("pump wavelength (nm)")
       ax.set_yscale("log")
       ax.set_yticks([0.6, 1, 2, 3, 4], ["0.6", "1", "2", "3", "4"])
       ax.minorticks_off()
   axes[0].plot([400], [0.8], marker="*", color="C1", markersize=11,
                markeredgecolor="k", linestyle="none")
   axes[0].set_ylabel(r"idler wavelength ($\mu$m)")
   axes[1].legend(
       [Line2D([], [], color="C4", ls="--"), Line2D([], [], color="C1", ls=":"),
        Line2D([], [], color="C3", ls="-.")],
       ["fitted sums", "signal at the leg bound", "signal at the band edge"],
       loc="upper right", fontsize="small", framealpha=0.9)
   fig.colorbar(filled, ax=axes, label="log$_{10}$ symmetric error (%)")
```

## Mixtures and vectors

**The polynomial of a mixture is the exact per-species sum.** Each population of a
mixture is its own channel, with its own multipliers, leg bound and density path:
$P=\sum_s\varepsilon_0\gamma_{0,s}N_s\,\mathrm{cubic}_s(E)$. Species whose leg bounds
are equal share their filtered fields $DE$ and $D^2E$, and their coefficient tables add,
density-weighted, $\sum_sN_s\gamma_{0,s}(1,A_s,B_s,B'_s)$ — an exact rewriting, which
`numba_cpu` takes for every such mixture, so that its transforms grow with the number
of distinct leg bounds rather than of species. Krypton and xenon share one, as do
argon, H₂ and SF₆, neon and N₂, and D₂, O₂ and CH₄; air's three species have three.

**The symmetric evaluator of a mixture factorises the mixture's own curve.** Different
filters do not reduce to one summed coefficient as the constant cubic's do, so the
exact per-species symmetric sum, $\sum_s\varepsilon_0\gamma_{3,s}N_s\,
G_s[w(G_sE)^3]$, would cost one product per species. The symmetric evaluator instead
takes one filter from the mixture's own Kerr diagonal,
$\gamma_{K,\rm mix}(\nu)=\sum_sN_s\gamma_{K,s}(\tilde\nu_s)$, each species' leg
bounded at its own $\nu_c$, and factorises it exactly as it factorises one gas:

$$
P=\varepsilon_0\Big(\sum_sN_s\gamma_{3,s}\Big)\,G_{\rm mix}\big[w\,(G_{\rm mix}E)^3\big],
\qquad
g_{\rm mix}(\omega)=\left[\frac{\sum_sN_s\gamma_{3,s}\hat K_s(\omega)}
{\sum_sN_s\gamma_{3,s}}\right]^{1/4},
$$

with $\hat K_s=\gamma_{K,s}(\tilde\nu_s)/\gamma_{3,s}=g_s^4$ each species' own bounded
Kerr diagonal. On the Kerr diagonal the four legs give
$\varepsilon_0\sum_sN_s\gamma_{3,s}\,g_{\rm mix}^4=\varepsilon_0\gamma_{K,\rm mix}$, the
mixture's degenerate coefficient, exactly; the reference, the mixture's static
coefficient, cancels as it does for one gas. A mixture costs what one gas costs. The
filter depends on the composition, not on the total density: where the populations
have different density paths along z — independent partial pressures, say — it is
rebuilt from their densities at every stage; a fill of constant composition builds it
once. One species is its own operator exactly.

Off the diagonal the mixture's filter is an approximation of the exact per-species
symmetric sum, which remains the test reference. By the survey of SEP 6 §4.7 (quartets
on 400–2000 nm) it departs from that sum by at most 0.016 % for helium and argon and
0.045 % for air (medians 0.0005 %), fifty times less than the symmetric
factorisation's own departure from the polynomial, 0.9 % and 1.3 %: as an approximation
of the polynomial the mixture's filter is as good as the exact sum. In a 3 cm, 150 µJ
capillary run the exit field's nonlinear change departs from the polynomial's by 0.29 %
(helium and argon) and 0.36 % (air), against 0.27 % and 0.29 % for the exact sum and
5–6 % for the carrier policy ([SEP 6](../../dev/seps/archive/sep-0006-dispersive-nonlinearity.md)).

A supplied species, which carries no optmat model, is refused under a dispersive policy
by name rather than given a borrowed curve; air resolves through its populations.

**Vectors use the isotropic Kleinman tensor.** Two polarisation components contract
through $T_{ijkl}=(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})/3$,
so that three filtered fields give $\tfrac13[(X\cdot Y)Z+(X\cdot Z)Y+(Y\cdot Z)X]$, which
is $(E\cdot E)E$ when they coincide. This is a named approximation: Kleinman's ratio
$\chi_{xxyy}=\chi_{xxxx}/3$ is held at every frequency. optmat's $\gamma$ is the mean
over parallel linear polarisations, and the distinct dispersion of the tensor's
components is not represented. The run records it as `"isotropic-kleinman"`.

## When the carrier is enough, and which evaluator to choose

### What the demonstration cases measured

Both of SEP 6's demonstration cases were run at full size, under all three policies on
the default backend, in the worked examples
[the dispersive wave under a frequency-dependent response](../../auto_examples/processes/plot_07_rdw_dispersive_kerr)
and [deep-ultraviolet four-wave mixing in argon](../../auto_examples/processes/plot_08_duv_fwm_dispersive_kerr).
Each difference below is the carrier policy's against the polynomial's, and the symmetric
evaluator's against the polynomial's. Values are rounded from the recorded example output.

| Case | Observable | Carrier | Symmetric |
|---|---|---:|---:|
| Helium dispersive wave, four modes | 100–150 nm energy at the exit | −23 % | +0.1 % |
| | the same, two modes; one mode | −23 %; −35 % | +0.1 %; +0.4 % |
| | its centre wavelength | +2.3 nm | −0.05 nm |
| | the pump's largest peak power; the energy leaving the fibre | +4 %; −8 % | 0 %; +0.2 % |
| | the same without plasma | +8 %; 0 % | not run |
| Argon four-wave mixing, 1 m | the best pressure | +10 mbar (+5 %) | same |
| | the largest signal energy at each policy's own best pressure | +1.7 % | +0.4 % |
| | the signal at the carrier's best pressure; at the linear phase matching | +11 %; +20 % | 0 %; 0 % |

Two kinds of result appear. In **argon**, the quartets the case drives have every leg
within the gas's leg bound and inside its band edge, so the dispersive responses are
exact relative to optmat's fit and the carrier column is the carrier approximation's
error. It sits in the phases: the
signal's cross-phase modulation by the pump is a quarter stronger than the carrier
coefficient says, which moves the phase matching, while the best attainable conversion
hardly changes. In **helium** the dispersive wave's quartets reach past the leg bound and
the band edge, so the carrier column is a **sensitivity** of the observable to the
coefficients of its vacuum-ultraviolet legs, which no off-resonant model fixes; it
depends on the mode set, by a factor of 1.5 here.

### By observable

- **Self-phase modulation and the pump near the carrier.** The carrier coefficient is
  exact for self-phase modulation at the carrier under every policy. What it misses is
  the variation of the Kerr diagonal over the occupied band: $\gamma_K$ rises by 12 %
  from 2 µm to 400 nm in helium, 37 % in argon and 87 % in xenon (SEP 6, §4.7). For a
  band where that variation is small against what a result needs, the carrier policy is
  adequate. In the helium case the pump's peak power, which oscillates along the fibre
  in step under every policy, is 4–8 % lower under the dispersive responses; with
  plasma on the lower peak ionises less, and about 9 % more energy leaves the fibre.
- **A band generated far from the carrier** — a dispersive wave, a four-wave-mixing
  signal, ultraviolet light from a near-infrared pump. Its energy depends on quartets
  with legs far from the carrier, and on its cross- and self-phase modulation, whose
  coefficients differ from the carrier's by tens of per cent (the demonstration-case
  table above). Use a dispersive policy; where the band is bounded or best effort, run
  the carrier policy beside it and report the spread.
- **A phase-matched wavelength** — a dispersive wave's centre, a signal's wavelength. Set
  by linear phase matching to a nonlinear wavenumber that the carrier mostly gets right,
  it moves by a few nanometres.
- **An operating point** — the pressure or energy that maximises a conversion. The
  nonlinear phases move it; the carrier policy can misplace it by several per cent and
  misjudge the yield at a given setting by tens of per cent while agreeing on the best
  yield.

A coefficient error of one per cent need not mean a one per cent error in a
phase-matched conversion. Where in doubt, run the observable under the carrier policy
and under a dispersive one: the difference is the carrier approximation's cost for that
observable.

### Which evaluator

**`"symmetric-kerr"`** is often close in cost to the carrier response; see
[the two evaluators](#the-two-evaluators) for the work each requires. In both
demonstration cases it stays close to the polynomial, within half a per cent on the
dispersive wave and about a per cent across the argon scan: it is the one to use for
production runs and scans. Its error is largest for quartets of widely spread
frequencies in the most dispersive gases — xenon, krypton, O₂, CH₄ and SF₆ reach 1.7–3.2 %
on random quartets between 400 nm and 2 µm (SEP 6, §4.7) — so check it against the
polynomial there.

**`"gas-polynomial"`** is exact relative to optmat's fit and is the reference the
symmetric evaluator is measured against. Its extra transforms usually make it more expensive, particularly for a
mixture with distinct leg bounds, such as air, whose species cannot share the
same derivative fields. Use it to check the symmetric evaluator, for
a result that must be exact relative to the fit, and for mixtures whose species share a
leg bound (krypton and xenon cost what one gas costs).

### The symmetric error against the fit's own uncertainty

The fits are not exact either. For the quartets of §4.7 of SEP 6 (random frequencies
between 400 nm and 2 µm inside each fit's data), the table sets the symmetric
evaluator's departure from the polynomial beside the one-sigma statistical uncertainty
of the same quartet's dispersion, $\gamma(q)/\gamma_0$, propagated from the covariance
of optmat's $A$, $B$ and $B'$ (scaled by the reduced $\chi^2$ where that exceeds one).
Both columns are median / maximum in per cent; the last is the fraction of quartets
whose symmetric error exceeds that uncertainty (SEP 6's fit-uncertainty study).

| Gas | Symmetric error | Fit's one sigma | Error above sigma |
|---|---|---|---:|
| He | 0.006 / 0.124 | 0.015 / 0.109 | 0.26 |
| Ne | 0.007 / 0.153 | 0.141 / 0.274 | 0.00 |
| Ar | 0.053 / 0.913 | 0.162 / 0.366 | 0.21 |
| Kr | 0.115 / 1.714 | 1.066 / 3.197 | 0.00 |
| Xe | 0.260 / 3.212 | 0.481 / 2.682 | 0.32 |
| H₂, D₂ | 0.047 / 0.718 | 0.063 / 0.499 | 0.42 |
| N₂ | 0.047 / 0.837 | 0.209 / 0.437 | 0.10 |
| O₂ | 0.183 / 2.908 | 0.500 / 1.415 | 0.23 |
| CH₄ | 0.133 / 1.831 | 0.404 / 1.353 | 0.17 |
| SF₆ | 0.105 / 2.141 | 0.491 / 1.019 | 0.11 |
| N₂O | 0.048 / 0.656 | 2.265 / 5.109 | 0.00 |

The symmetric evaluator's typical error is below the fit's statistical uncertainty of
the dispersion in every gas; its largest errors, at the most widely spread quartets,
exceed it in a tenth to two fifths of the quartets for most gases. The uncertainty is
measured here relative to the static $\gamma_0$; relative to the pump's own Kerr value
the correlated part cancels and it is several times smaller, so the comparison depends
on what a result is normalised to. A covariance is not a bound on the expansion's
truncation, on borrowed priors (N₂O's dispersion is CO₂'s) or on the systematic errors
optmat's accuracy page lists.

### What the regions mean for a result

The run's manifest records, per gas, the part of its active band in each region; the
region table above says what the evaluators give there. For a result:

- for **supported** quartets, the polynomial evaluates the declared fit and the
  symmetric policy approximates it. Accuracy of an observable also depends on
  the fit's physical validity, the propagation model and numerical convergence;
- a result depending on **bounded** legs is a model estimate. The coefficient
  may underestimate a rising off-resonant response, but this does not bound the
  generated energy or conversion efficiency;
- one that depends on **best-effort** legs uses an unqualified continuation:
  report it as a sensitivity, with the carrier result beside it, never as a correction
  of the carrier result. A resonant response requires its own atomic-model and
  numerical qualification before it can provide a reference.

The label is per frequency of the active band, not per quartet the field drives: a run
whose band reaches the bounded region is labelled so even where the field there is
negligible. Look at where the observable's band and its generating quartets lie.

### Backends

Both evaluators run on `numpy_ref`, `numba_cpu` and the native backend's CPU build on
every route but the GNLSE, agreeing with `numpy_ref` to a few times machine epsilon, and
on the native backend's device builds, qualified on one AMD Instinct generation and one NVIDIA Hopper
device. `backend="auto"` therefore runs them on the native backend where it is installed, as the
carrier policy runs; nothing falls back.

## Reading what a run recorded

A run's material manifest records the law it used. Its
`policies["electronic_response"]` names the law, the evaluator, the sector, the tensor
approximation and the leg-bound rule, with each population's support-record digest and
regions; for a symmetric mixture, `mixture_filters` names each mixture filter, its
rule, the general modal region it acts in (`None` elsewhere) and the composition it is
built from — each population's label and the SHA-256 identity of its density path. Its
`electronic` entries hold, per population, the model's name and digest, the leg bound
and both band edges in m⁻¹, the regions as wavelength intervals in m, and, in a
symmetric mixture, the mixture filter it belongs to (`filter`). The policies enter the
resume key, so a run never resumes under another law, another model, another band or
another composition. Inspecting a run before propagating it costs only its
translation:

```{code-block} python
from solprop.driver import Simulation
from solprop.physdata import MaterialSelection

simulation = Simulation.from_capillary(
    50e-6,
    0.5,
    "Ar",
    0.2,
    lambda0=400e-9,
    wavelength_limits=(210e-9, 2e-6),
    trange=250e-15,
    fwhm=30e-15,
    energy=10e-6,
    backend="numba_cpu",
    materials=MaterialSelection(electronic="gas-polynomial"),
)
manifest = simulation.material_manifest()
manifest.policies["electronic_response"]["law"]  # 'gas-polynomial'
(entry,) = manifest.electronic
(
    entry["evaluator"],
    entry["sector"],
    entry["tensor"],
)  # polynomial, full, isotropic-kleinman
entry["regions"]["bounded"]  # [1.909e-07, 2.159e-07]: 190.9 to 215.9 nm
entry["regions"]["best_effort"]  # None: the band stays inside argon's band edge
```

After a run the same manifest is `output.metadata.materials`, and the
[tutorial](../../auto_tutorials/capillaries/plot_11_dispersive_kerr) shades the regions on a
spectrum from it.

## Limitations

- **No resonant response.** Both evaluators are off-resonant expansions. In the
  best-effort region they give a held constant, not physics; nonlinear absorption and
  the imaginary part of the susceptibility near a resonance are not modelled, and the
  real kernel is not a causal material law there.
- **No tensor dispersion.** The isotropic Kleinman tensor holds one ratio at every
  frequency (above).
- **Gases only.** Fused silica keeps its carrier-evaluated cubic: its degenerate $n_2$
  curve does not determine its nondegenerate response, and no curated nondegenerate
  silica data exist to factorise. A GNLSE run's fitted $\gamma$ is a constant
  coefficient, and refuses a dispersive policy by name; on the general modal route only
  a region declaring its gas fill takes the dispersive response.
- **The fits' own uncertainties remain.** Being exact relative to a fit does not remove
  its absolute-scale and dispersion uncertainty, the vibrational contributions some
  molecular fits retain, or N₂O's dispersion borrowed from CO₂ (optmat's records say
  which).
- **The region check is per frequency of the active band.** It does not identify which
  quartets a field actually drives; a band that reaches the bounded region is labelled
  so even if the field there is negligible.
- **Backends.** `numpy_ref`, `numba_cpu` and the native backend's CPU build on every route but the
  GNLSE (single-mode, multimode, radial, Cartesian and line free space, planar, general
  modal); the native backend's device builds on one AMD Instinct generation (HIP) and one NVIDIA
  Hopper device (CUDA).

This page documents the functionality proposed and implemented by
[SEP 6](../../dev/seps/archive/sep-0006-dispersive-nonlinearity.md).