Two-photon ionisation and absorption#

The two-photon processes of the noble gases He, Ne, Ar, Kr and Xe:

two-photon ionisation from the ground state, above half the ionisation energy;
two-photon excitation of bound even-parity levels below it;
the photoionisation of the levels so excited.

The first has a model, optmat.gas.two_photon (OEP 6, Phase 1; The model, below). The companion chapters describe Two-photon ionisation: source coverage and comparisons and Bound two-photon excitation. The reference curves are not fitted. Bound two-photon excitation has data but no model yet (OEP 6, Phase 2).

The tables are written by tools/two_photon_tables.py --update-docs from the records, and tests/test_two_photon_page.py checks that the page holds its output.

The model#

import numpy as np

from optmat.constants import BAR, K_B, C
from optmat.gas.two_photon import two_photon_ionisation_model

argon = two_photon_ionisation_model("Ar")  # "He", "Ne", "Ar", "Xe"; Xe also "circular"
wavelength = np.array([160e-9, 150e-9, 130e-9, 100e-9])  # m, one photon
omega = 2 * np.pi * C / wavelength  # rad/s
argon.cross_section(omega)  # m^4 s: 0 below threshold, NaN where unavailable
argon.available(omega)  # where it has a value
argon.region(omega)  # below_threshold, supported, unsupported_detail, unavailable
argon.rate(omega, 1e17)  # 1/s per atom at 1e17 W/m^2
argon.loss_coefficient(omega, BAR / (K_B * 293.15))  # beta, m/W: dI/dz = -beta I^2
argon.pair_cross_section(2 * omega)  # indexed by a photon pair's summed frequency
argon.thresholds, argon.e1  # the 2P3/2 and 2P1/2 thresholds, and E1
argon.reference, argon.uncertainty, argon.identity  # what it is and how well known

This grid samples below-threshold, supported and unavailable regions. Evaluating an unsupported_detail region also returns a value, with ModelRangeWarning as described below.

Each model is a curated reference curve approved by the author under data rule 8 (TC-943). It is interpolated linearly in \(\log\sigma^{(2)}\) between the curve’s own drawn vertices, and never fitted or extrapolated. Where a vector drawing’s staircase puts several vertices at one energy, they are one point at their geometric mean; partial cross sections are summed on their common coverage (TC-944).

Gas	Reference	Supported (nm)	Unsupported detail (nm)	Other calculations / reference	Status
He	Mihelič and Horvat 2021, ¹S + ¹D, velocity	100.80–60.13	60.13–58.47	0.75–1.19 (Saenz; Nikolopoulos; van der Hart)	validated
Ne	McKenna and van der Hart 2004	114.48–77.15	114.80–114.48; 77.15–74.39	none: single source	untested
Ar	McKenna and van der Hart 2004	155.59–112.47	156.71–155.59; 112.47–106.68	0.91–0.99 (Petrov et al., length and velocity)	validated
Kr	none (OEP 6, Q2)	—	—	—	—
Xe, linear	Lagutin et al. 2017, length	184.47–158.21	158.21–146.96	0.65–0.87 (Fink and Johnson, raster)	validated
Xe, circular	Lagutin et al. 2017, length	184.47–158.21	158.21–146.97	0.62–0.71 (Fink and Johnson, raster)	validated

The regions tile the frequency axis:

below_threshold: below \(I_p/2\), the direct continuum cross section is exactly 0. 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. These are the 0.6 eV below \(E_1\), where the first one-photon resonance’s wing is placed differently by each calculation. For an LS-coupled reference (Ne, Ar) they are also the band between the ion’s two thresholds.
unavailable: NaN. This is between the threshold and the reference’s first vertex (xenon: its two thresholds; Lagutin et al. compute nothing between them, OEP 6, Q3), and at and above \(E_1\).

Evaluation never raises on a frequency grid. It returns NaN where the model is unavailable and gives the mask (available); what to do there is the propagation code’s choice (the author’s decision, TC-903, TC-943). It raises only for frequencies that are not finite and positive, and for intensities or densities that are not finite and non-negative. Asking for krypton, or for circular polarisation of a gas other than xenon, raises UnavailableModelError with the reason.

The uncertainty is kept in three kinds that are never merged (OEP 6, Q6):

the source’s own accuracy: Lagutin et al.’s 7.7 %; unstated for the others, and then held as None, not zero;
the digitising uncertainty at a drawn vertex: 0.5 % for Mihelič and Horvat, 2 % for the others;
the spread of the other calculations over the supported band: an empirical envelope, not a confidence interval. A gauge variant of the reference is a diagnostic and does not enter it.

Composition.

With the strong-field rates: the model is the degenerate two-photon continuum. The strong-field rates (Ionisation rates) describe a pump at their carrier, and applying both to the same spectral component counts it twice. A band split is the propagation code’s.
Mixed pairs: a pump photon and a vacuum-ultraviolet one need nondegenerate amplitudes that no curated source gives (OEP 6, Q9).
With OEP 5’s support record: the lower threshold \(I_p/2\) lies above OEP 5’s band edge \(\min(E_1, E_2/2)\). Between them lie the bound two-photon levels.

The gallery example Two-photon ionisation of the noble gases plots the models and the loss of a prescribed field.

Conventions#

The generalised two-photon ionisation cross section \(\sigma^{(2)}\) is defined by the rate per atom,

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

with \(\Phi\) the photon flux in cm⁻² s⁻¹, so \(\sigma^{(2)}\) is in cm⁴ s (1 cm⁴ s = 10⁻⁸ m⁴ s). This is the convention of Uiterwaal et al. and of McKenna and van der Hart (TC-856), and of every calculation curated here, each record stating its own definition. No factor 2! enters it.

Exception, McCown et al.: they print a coefficient \(\alpha\) in cm⁴ W⁻¹ defined by \(dn/dt = \alpha[\mathrm{Xe}]I^2/\hbar\omega\) (their Eq. 9), so \(\sigma^{(2)} = \alpha\hbar\omega\) (TC-884).
Intensity dependence: a non-perturbative calculation (R-matrix Floquet) or a measurement gives \(\Gamma/\Phi^2\) at a stated intensity. Off resonance it equals the lowest-order cross section.

Each event removes two photons. A quasi-monochromatic wave in a gas of number density \(N\) therefore loses intensity as

\[ \frac{dI}{dz} = -2\hbar\omega N W = -\beta I^2,\qquad \beta = \frac{2N\sigma^{(2)}}{\hbar\omega}. \]

Of the \(2\hbar\omega\) absorbed, \(I_p\) ionises the atom and the rest is the electron’s kinetic energy.

Bound two-photon excitation is curated in two conventions (TC-872):

two_photon_transition_strength: the rotationally averaged strength \(\delta^{TP}\) in atomic units (Hättig et al.; Henriksson et al.), with the rotational average’s convention and the polarisation stated.
two_photon_excitation_cross_section: the frequency-integrated cross section \(\sigma_0^{(2)}\) in cm⁴ or m⁴. The rate is \(W = \sigma_0^{(2)} g\,\Phi^2\), with \(g\) the line profile normalised in angular frequency (every curated source: TC-889 to TC-892), and with the photon statistics \(G^{(2)}(0)\) of the measurement stated.

A cross section in cm⁴ s at a line’s peak depends on the line’s width, and is not curated as data. The ionisation of the excited level is excited_state_photoionization_cross_section, in cm².

Energy axes. Every curve is held against the photon energy as its source drew it, and no axis is shifted:

LS-coupled calculations (McKenna and van der Hart; Petrov et al.) have one threshold, with resonances at their own computed positions.
Saenz and Lambropoulos scaled their photon-energy axis so that the energy above threshold matches experiment (their ground state is 0.69 eV off; TC-873).
Karamatskou and Santra shifted their Fig. 3 by 0.33 eV (TC-881).
L’Huillier, Tang and Lambropoulos plot the total two-photon energy, held as the photon wavenumber, half of it (TC-882).

Near a resonance, curves are compared only at their own resonance positions, never at NIST’s.

Thresholds#

Direct two-photon ionisation of the ground state needs \(2\hbar\omega \ge I_p\). For Ne, Ar, Kr and Xe the ion has two fine-structure levels, ²P₃/₂ and ²P₁/₂. Between the two thresholds, the Rydberg series converging on ²P₁/₂ autoionise.

Above the first one-photon resonance \(E_1\) a single photon is resonantly absorbed; the window of two-photon ionisation without one-photon resonance is therefore \([I_p/2, E_1)\). Below \(I_p/2\), down to OEP 5’s \(E_2/2\) (the lowest two-photon level), lie the bound two-photon resonances.

From nist_asd_2026, the ion’s ²P₁/₂ level added to the ionisation energy for the upper threshold; photon energy in eV, vacuum wavelength in nm:

Gas	I_p/2, ²P₃/₂ (eV (nm))	²P₁/₂ threshold (eV (nm))	E₁ (eV (nm))	E₂/2 (eV (nm))
He	12.2937 (100.85)	—	21.2180 (58.43)	10.3079 (120.28)
Ne	10.7823 (114.99)	10.8307 (114.48)	16.6708 (74.37)	9.2879 (133.49)
Ar	7.8798 (157.34)	7.9686 (155.59)	11.6236 (106.67)	6.5474 (189.36)
Kr	6.9998 (177.13)	7.3327 (169.08)	10.0324 (123.58)	5.7223 (216.67)
Xe	6.0649 (204.43)	6.7181 (184.55)	8.4365 (146.96)	4.8428 (256.02)

Some consequences:

Argon at the F₂ laser lines (157.52 and 157.63 nm; TC-900) is just below its threshold.
Krypton at 193 nm and xenon at 248 nm are below theirs. Their ionisation there is two-photon resonant (2+1) ionisation (Geohegan et al., Kröll and Bischel).
Xenon at 193 nm, McCown et al.’s measurement, lies between its two thresholds, in the autoionising region.

See Two-photon ionisation: source coverage and comparisons.

See Bound two-photon excitation.

What optmat does not supply#

A time-resolved two-photon loss. The cross section, and the integrated yield and energy of a broadband pulse in the pair-spectral approximation, are OEP 6’s (section 4). How that loss is distributed in time within a pulse is not determined by \(\sigma^{(2)}\). It needs the nondegenerate two-photon amplitudes (OEP 6, Q9), which no curated source supplies. It is a documented gap, for a new OEP (OEP 8, Q6) or for a propagation code’s own model (TC-903).

The continuum model and the bound-level records are what OEP 6 proposed and implemented: its theory, decisions and phase reports are the record. The bound resonances’ response for propagation is OEP 8’s.