Third-order nonlinearity#

The Kerr terms (self- and cross-phase modulation) and third-harmonic generation of a propagation need the gas’s third-order response at the frequencies involved. optmat provides it as the electronic second hyperpolarizability gamma of each gas, a function of the process and of all four frequencies. It is fitted to the curated third-order data (TC-224 to TC-314) and stored with its covariance.

This page introduces the response and its use. Continue with Third-order conventions, Third-order models and fits, Third-order support and resonance limits and Third-order accuracy for the conversions, theory, data support and uncertainty. The Quick start introduces the material interfaces; The Raman response describes the delayed response.

What is modelled#

  • The electronic second hyperpolarizability \(\gamma(-\omega_\sigma; \omega_1, \omega_2, \omega_3)\) of an isolated molecule, orientationally averaged (the mean gamma, the one an isotropic gas shows for parallel linear polarizations).

  • Below resonance: from the static limit to a fundamental of 400 nm (700 nm for O3), and to 20 µm.

  • For every third-order process at once: the Kerr effect, third-harmonic generation, ESHG, the dc Kerr effect, CARS-type four-wave mixing and a nondegenerate pump-probe index.

What is not modelled here:

  • The vibrational and rotational (Raman) response: in propagation it is a separate, delayed term. The one exception is the small vibrational part of the ESHG of CO2, CH4 and SF6, which could not be separated; see Third-order conventions.

  • Ionisation and plasma.

  • Pair (collision-induced) contributions: gamma is per molecule, and the third-order susceptibility is linear in density.

  • Higher-order Kerr terms: the claims of such terms were refuted (TC-313).

Detailed chapters#

See Third-order conventions.

See Third-order models and fits.

For configuration and regeneration procedures, see Running fits and robustness checks.

See Third-order support and resonance limits.

Using the models#

from optmat.conditions import State
from optmat.gas.gases import Gas
from optmat.gas.nonlinear import hyperpolarizability_models

argon = hyperpolarizability_models()[Gas.Ar]
argon.gamma_au("kerr", 800e-9)  # 1258 au: SPM and XPM at 800 nm
argon.gamma_au("thg", 800e-9)  # 1480 au: the third harmonic of 800 nm
argon.gamma_au("pump_probe", 800e-9, 400e-9)  # the index a 400 nm probe sees
argon.gamma_sigma_au("kerr", 800e-9)  # statistical one sigma, au
argon.gamma3("kerr", 800e-9)  # gamma_T / (6 epsilon_0), m^5/V^2
argon.n2(800e-9, State(1e5, 293.15))  # self n2 at 1 bar, 20 C, m^2/W
argon.parameters  # gamma0, A, B, B'
argon.record["validation"]  # the validators against the fit

gamma_au takes arrays of wavelengths, inf for the static limit, and for cars and pump_probe the second wavelength. For refitting commands and a Python fitting example, see Running fits and robustness checks.

Limitations#

  • Resonance: the form holds below resonance, within the fitted \(W_2\). It says nothing near a resonance (O3’s third harmonic, the vacuum-ultraviolet harmonics of short fundamentals, two-photon resonances), nor past one: a leg beyond \(E_1\) or a pair sum beyond \(E_2\) (Third-order support and resonance limits). Xenon’s n2 at 248.4 nm, 1 nm above the two-photon resonance with 6p[1/2]_0 at 249.6 nm, is negative and large: -(2.06 +- 0.14) and -(1.82 +- 0.08) x 10^-14 esu measured, -(1.54 +- 0.30) calculated, about -8e-21 m2/W at 1 atm (lehmberg_1995, TC-854); a fundamental near 250 nm in xenon is outside the model.

  • The level: it rests on the ESHG scale, which Shelton himself states to about 1 %. The Maryland n2 lie up to 16 % lower (0.84-1.03 at 800 nm), an open question (accuracy page).

  • The unseparated vibrational parts of CO2, CH4 and SF6.

  • N2O and O3 are known to 15 %; CO to 10 %.

  • Dilute limit: no local-field factors, and no pair or density dependence.

  • Temperature: the electronic gamma is taken as independent of temperature.

The support record is what OEP 5 proposed and implemented: its theory, decisions and phase report are the record.