optmat.gas.two_photon#

Two-photon ionisation of the noble gases: the continuum model (OEP 6, Phase 1).

The generalised two-photon ionisation cross section \(\sigma^{(2)}\) of a gas’s ground state is defined by the rate per atom

\[W = \sigma^{(2)} \Phi^2, \qquad \Phi = I / \hbar\omega,\]

with \(\Phi\) the photon flux; in SI, \(\sigma^{(2)}\) is in m^4 s (1 cm^4 s = 1e-8 m^4 s). A quasi-monochromatic wave in a gas of number density \(N\) then loses intensity as \(dI/dz = -\beta I^2\) with \(\beta = 2 N \sigma^{(2)} / \hbar\omega\), two photons per event.

Each model is a curated reference curve that the author approved under data rule 8 (TC-943), interpolated linearly in \(\log\sigma^{(2)}\) between the curve’s own drawn vertices and never fitted or extrapolated:

helium: Mihelič and Horvat (2021), the 1S and 1D partial cross sections summed;
neon and argon: McKenna and van der Hart (2004), R-matrix Floquet;
xenon: Lagutin et al. (2017), linear and circular polarisation.

Krypton has none: no vector-drawn or tabulated cross section is curated (OEP 6, Q2), and asking for it raises UnavailableModelError.

A model covers the window from the lowest two-photon threshold \(I_p/2\) to the first one-photon resonance \(E_1\), and states its regions (regions):

below_threshold: the direct continuum cross section is zero. Bound two-photon excitation and (2+1) ionisation there are not modelled (OEP 6, Phase 2).
supported: the reference’s values.
unsupported_detail: the reference’s values, with ModelRangeWarning. The source does not qualify them: an LS-coupled calculation between the ion’s two fine-structure thresholds, or the wing of the first one-photon resonance, which the sources place differently.
unavailable: no value. The reference does not reach it (between xenon’s two thresholds, for instance), or it is at or above \(E_1\).

Evaluation never raises on a frequency grid. It returns NaN where the model is unavailable, and available() gives the mask (the author’s decision, TC-943). A propagation code decides what to do there.

The model supplies the degenerate cross section. It does not supply a time-resolved loss for a broadband pulse, nor the mixed pairs of two bands. The pair-spectral yield of OEP 6, section 4, is an integral identity in a stated approximation, and its time dependence is a documented gap (TC-903). Nor does it double as a strong-field rate: applying it and ionisation_rate_model() to the same spectral component counts that component twice.

Examples#

>>> import math
>>> from optmat.constants import C
>>> argon = two_photon_ionisation_model("Ar")
>>> [round(2 * math.pi * C / t.omega * 1e9, 2) for t in argon.thresholds]
[157.34, 155.59]
>>> omega = 2 * math.pi * C / 120e-9  # 120 nm
>>> f"{float(argon.cross_section(omega)):.3e}"  # m^4 s
'1.146e-58'
>>> bool(argon.available(2 * math.pi * C / 100e-9))  # beyond E1 (106.7 nm)
False

Module Attributes

`EVALUATION_VERSION`	The version of the evaluation: interpolation, regions and conversions.
`DERIVATION_VERSION`	The version of the stored models' derivation, raised whenever it writes otherwise.
`REGION_LABELS`	The kinds of region a model's frequency axis is divided into.

Functions

`two_photon_ionisation_model`(gas[, polarization])	Return a noble gas's two-photon ionisation model (OEP 6, Phase 1).
`two_photon_ionisation_models`()	Return every two-photon ionisation model, by gas and polarisation.

Classes

`TwoPhotonIonisationModel`(gas, polarization, ...)	The two-photon ionisation cross section of a noble gas's ground state.
`TwoPhotonReference`(datasets, gauge, method, ...)	The approved reference curve of a model, and why it was chosen.
`TwoPhotonRegion`(label, lower, upper, reason)	A half-open band of photon angular frequency, `[lower, upper)`, and its label.
`TwoPhotonSpread`(label, datasets, form, role, ...)	Another calculation against the reference over their supported overlap.
`TwoPhotonThreshold`(name, omega, photons, ...)	A threshold or resonance as a photon's angular frequency, with its source.
`TwoPhotonUncertainty`(source, source_level, ...)	What is known of a model's uncertainty, kept as three separate kinds (Q6).

optmat.gas.two_photon.DERIVATION_VERSION: Final = 1#: The version of the stored models’ derivation, raised whenever it writes otherwise.

optmat.gas.two_photon.EVALUATION_VERSION: Final = 1#: The version of the evaluation: interpolation, regions and conversions.

optmat.gas.two_photon.REGION_LABELS: Final = ('below_threshold', 'supported', 'unsupported_detail', 'unavailable')#: The kinds of region a model’s frequency axis is divided into.

class optmat.gas.two_photon.TwoPhotonIonisationModel(gas, polarization, derivation_version, omega, sigma, thresholds, e1, regions, reference, uncertainty, input_digests=<factory>, *, status='user', status_reason='', modified=(), base='')[source]#

Bases: Identified

The two-photon ionisation cross section of a noble gas’s ground state.

Built by two_photon_ionisation_model() from its stored derivation. SI throughout: photon angular frequency in rad/s, cross section in m^4 s.

Attributes:

gasGas: The species.
polarizationstr: "linear" or "circular".
derivation_versionint: DERIVATION_VERSION when it was derived.
omeganumpy.ndarray: The reference’s vertices, photon angular frequency, rad/s, increasing.
sigmanumpy.ndarray: The reference’s cross section at them, m^4 s.
thresholdstuple of TwoPhotonThreshold: The lowest ionic channel’s two-photon threshold first; the ion’s \({}^2P_{1/2}\) channel’s second, for Ne to Xe.
e1TwoPhotonThreshold: The first one-photon resonance, where the window ends.
regionstuple of TwoPhotonRegion: Bands covering every positive frequency, in order.
referenceTwoPhotonReference: The approved reference.
uncertaintyTwoPhotonUncertainty: Its uncertainty, by kind.
input_digestsMapping of str to str: SHA-256 digests of what it was derived from: the selection and each dataset.

EVALUATION_VERSION: ClassVar[int] = 1#

IDENTITY_KIND: ClassVar[str] = 'two_photon_ionisation'#

available(omega)[source]#

Return where the model has a value: below threshold, or on its reference.

Parameters:

omegaarray_like: Photon angular frequency, rad/s, finite and positive.

Returns:

numpy.ndarray: Booleans with the shape of omega: False where cross_section() is NaN.

Raises:

ArgumentError: If a frequency is not finite and positive.

cross_section(omega)[source]#

Return the two-photon ionisation cross section at photon frequencies.

Parameters:

omegaarray_like: Photon angular frequency, rad/s, finite and positive.

Returns:

numpy.ndarray: \(\sigma^{(2)}\), m^4 s, with the shape of omega: exactly 0 below the lowest threshold, NaN where available() is False.

Raises:

ArgumentError: If a frequency is not finite and positive.

Warns:

ModelRangeWarning: If a frequency lies in an unsupported_detail region.

derivation_version: int#

property digest: str#

The model’s digest, identity.digest.

Returns:

str: 64 hexadecimal digits.

e1: TwoPhotonThreshold#

gas: Gas#

input_digests: Mapping[str, str]#

loss_coefficient(omega, number_density)[source]#

Return the two-photon loss coefficient beta, dI/dz = -beta I^2.

Parameters:

omegaarray_like: Photon angular frequency, rad/s, finite and positive.
number_densityarray_like: The density of ground-state atoms, m^-3, finite and non-negative; broadcast with omega.

Returns:

numpy.ndarray: \(\beta = 2 N \sigma^{(2)} / \hbar\omega\), m/W: two photons per event; NaN where the model is unavailable.

Raises:

ArgumentError: If a frequency is not finite and positive, or a density is not finite and non-negative.

Warns:

ModelRangeWarning: If a frequency lies in an unsupported_detail region.

omega: Float[ndarray, 'n']#

pair_cross_section(pair_omega)[source]#

Return the cross section indexed by a photon pair’s summed frequency.

Parameters:

pair_omegaarray_like: The sum of the two photons’ angular frequencies, rad/s, finite and positive.

Returns:

numpy.ndarray: \(\sigma^{(2)}(\Omega/2)\), m^4 s, as cross_section().

Raises:

ArgumentError: If a frequency is not finite and positive.

Warns:

ModelRangeWarning: If a frequency lies in an unsupported_detail region.

polarization: str#

rate(omega, intensity)[source]#

Return the two-photon ionisation rate per atom of a monochromatic wave.

Parameters:

omegaarray_like: Photon angular frequency, rad/s, finite and positive.
intensityarray_like: Intensity, W/m^2, finite and non-negative; broadcast with omega.

Returns:

numpy.ndarray: \(W = \sigma^{(2)} (I / \hbar\omega)^2\), 1/s; NaN where the model is unavailable.

Raises:

ArgumentError: If a frequency is not finite and positive, or an intensity is not finite and non-negative.

Warns:

ModelRangeWarning: If a frequency lies in an unsupported_detail region.

reference: TwoPhotonReference#

region(omega)[source]#

Return the label of the region each frequency lies in.

Parameters:

omegaarray_like: Photon angular frequency, rad/s, finite and positive.

Returns:

numpy.ndarray: Labels (REGION_LABELS), with the shape of omega.

Raises:

ArgumentError: If a frequency is not finite and positive.

regions: tuple[TwoPhotonRegion, ...]#

sigma: Float[ndarray, 'n']#

thresholds: tuple[TwoPhotonThreshold, ...]#

uncertainty: TwoPhotonUncertainty#

class optmat.gas.two_photon.TwoPhotonReference(datasets, gauge, method, intensity_w_cm2, reason, registers)[source]#

Bases: object

The approved reference curve of a model, and why it was chosen.

Attributes:

datasetstuple of str: Its datasets, summed where they are partial cross sections.
gaugestr: The gauge, as the record states it.
methodstr: The calculation.
intensity_w_cm2float or None: The intensity of a non-perturbative calculation, W/cm^2; None for a lowest-order one.
reasonstr: Why it is the reference (the author’s approval, TC-943).
registerstuple of str: The register entries.

datasets: tuple[str, ...]#

gauge: str#

intensity_w_cm2: float | None#

method: str#

reason: str#

registers: tuple[str, ...]#

class optmat.gas.two_photon.TwoPhotonRegion(label, lower, upper, reason)[source]#

Bases: object

A half-open band of photon angular frequency, [lower, upper), and its label.

Attributes:

labelstr: One of REGION_LABELS.
lowerfloat: The lower edge, rad/s.
upperfloat: The upper edge, rad/s; inf for the last region.
reasonstr: Why the band is so labelled.

label: str#

lower: float#

reason: str#

upper: float#

class optmat.gas.two_photon.TwoPhotonSpread(label, datasets, form, role, lower, upper, points, lower_ratio, upper_ratio)[source]#

Bases: object

Another calculation against the reference over their supported overlap.

Attributes:

labelstr: The calculation.
datasetstuple of str: Its datasets, summed where they are partial cross sections.
formstr: "vector" or "raster": how its values were read.
rolestr: "independent", another calculation, which enters the envelope; or "gauge", the reference’s other gauge, a diagnostic only.
lowerfloat: The lowest photon angular frequency compared, rad/s.
upperfloat: The highest, rad/s.
pointsint: The number of reference vertices compared.
lower_ratiofloat: The smallest ratio of it to the reference.
upper_ratiofloat: The largest.

datasets: tuple[str, ...]#

form: str#

label: str#

lower: float#

lower_ratio: float#

points: int#

role: str#

upper: float#

upper_ratio: float#

class optmat.gas.two_photon.TwoPhotonThreshold(name, omega, photons, energy, datasets)[source]#

Bases: object

A threshold or resonance as a photon’s angular frequency, with its source.

Attributes:

namestr: What it is, e.g. "Ar+ 2P3/2" or "E1".
omegafloat: The photon’s angular frequency, rad/s: the level’s energy over the number of photons that reach it, over hbar.
photonsint: 2 for an ionisation threshold, 1 for \(E_1\).
energyfloat: The energy above the ground state, J.
datasetstuple of str: The curated datasets it is computed from.

datasets: tuple[str, ...]#

energy: float#

name: str#

omega: float#

photons: int#

class optmat.gas.two_photon.TwoPhotonUncertainty(source, source_level, source_provenance, digitising, digitising_provenance, spreads)[source]#

Bases: object

What is known of a model’s uncertainty, kept as three separate kinds (Q6).

Attributes:

sourcefloat or None: The source’s own relative accuracy, None where it states none (never zero).
source_levelstr: Its level, as stated, or "not stated".
source_provenancestr: Where it is stated.
digitisingfloat: The relative digitising uncertainty at a drawn vertex.
digitising_provenancestr: Its record and register entry.
spreadstuple of TwoPhotonSpread: The other calculations against the reference, over the supported region.

digitising: float#

digitising_provenance: str#

property envelope: tuple[float, float] | None#

The smallest and largest ratio of the independent calculations.

It is an empirical envelope of other calculations, not a confidence interval or a bound on the true cross section.

Returns:

tuple of float or None: (lower, upper) ratios to the reference; None with no independent calculation.

source: float | None#

source_level: str#

source_provenance: str#

spreads: tuple[TwoPhotonSpread, ...]#

optmat.gas.two_photon.two_photon_ionisation_model(gas, polarization='linear')[source]#

Return a noble gas’s two-photon ionisation model (OEP 6, Phase 1).

Parameters:

gasGas or str: The gas: He, Ne, Ar or Xe (Kr has none).
polarizationstr, optional: "linear" (default), or "circular" where an approved reference gives it (Xe).

Returns:

TwoPhotonIonisationModel: The model.

Raises:

UnavailableModelError: For a gas or polarisation with no approved reference, with the reason.
ArgumentError: For a polarisation other than linear or circular.

Examples

>>> xenon = two_photon_ionisation_model("Xe", "circular")
>>> xenon.reference.datasets
('lagutin_2017/fig8_cipfcp_circular',)

optmat.gas.two_photon.two_photon_ionisation_models()[source]#

Return every two-photon ionisation model, by gas and polarisation.

Returns:

dict: {(gas, polarization): model} for every approved reference.

Examples

>>> sorted(f"{g.value}/{p}" for g, p in two_photon_ionisation_models())
['Ar/linear', 'He/linear', 'Ne/linear', 'Xe/circular', 'Xe/linear']