Third-order conventions#

Read Third-order nonlinearity for the scope and first calculation. This chapter defines Taylor-series gamma, units, frequency arguments and the conversions needed to compare sources. Continue with Third-order models and fits. The API is optmat.gas.nonlinear.

Conventions#

The Taylor-series gamma#

The common currency is Shelton & Rice’s Taylor-series gamma (Chem. Rev. 94, 3, 1994), written \(\gamma_T\) or simply \(\gamma\). For a static field the dipole moment is

\[ p = \alpha E + \frac{\beta}{2} E^2 + \frac{\gamma}{6} E^3 + \dots \]

For fields at optical frequencies the same \(\gamma\) multiplies the same \(1/6\) and the product of the real fields, with the frequency arguments of the mixing channel. For an instantaneous (dispersionless) response the induced polarization of a gas of number density \(N\) is simply

\[ P(t) = \frac{N \gamma}{6} E(t)^3 . \]

Written with complex amplitudes, \(E(t) = \tfrac{1}{2}\left[\mathcal{E} e^{-i\omega t} + \text{c.c.}\right]\) and likewise for \(P\), this gives:

  • the Kerr (self-phase) amplitude \(P_\omega = N\gamma_\text{Kerr} |\mathcal{E}|^2\mathcal{E}/8\);

  • the third harmonic \(P_{3\omega} = N\gamma_\text{THG}\mathcal{E}^3/24\).

Units#

gamma is stored in atomic units:

\[ 1\ \text{au} = \frac{e^4 a_0^4}{E_h^3} = 6.235377\times10^{-65}\ \text{C}^4\,\text{m}^4\, \text{J}^{-3} = 5.0367\times10^{-40}\ \text{esu} \]

(optmat.gas.nonlinear.AU_HYPERPOLARIZABILITY; the esu factor is optmat.gas.nonlinear.ESU_PER_SI, \(c^4\times10^{-9}\) in SI numbers, as Shelton & Rice). Wavenumbers \(\nu\) are vacuum wavenumbers in cm\(^{-1}\), and wavelengths are vacuum wavelengths.

Susceptibility and n2#

The quantities a propagation code uses follow from gamma at the gas’s number density \(N\), in the dilute limit:

  • The third-order susceptibility of an instantaneous response, \(P = \varepsilon_0\chi^{(3)} E^3\), is \(\chi^{(3)} = N\gamma/(6\varepsilon_0)\).

  • The per-molecule coefficient is therefore \(\gamma_3 = \gamma/(6\varepsilon_0)\) in m\(^5\)/V\(^2\), with \(\chi^{(3)} = \gamma_3 N\) (gamma3()).

  • The self Kerr coefficient of linearly polarised light, \(n = n_0 + n_2 I\), is \(n_2 = N\gamma_\text{Kerr}/(8\varepsilon_0^2 c)\) (n2()). One atomic unit of gamma is \(8.91\times10^{-23}\) cm\(^2\)/W of \(n_2\) at 1 atm and 0 °C.

  • A weak probe sees twice the self coefficient for the same polarization (cross-phase modulation), with the pump-probe gamma.

n2 uses the real-gas density (the ideal gas for O3, which has no equation of state). The Lorentz local-field factors, \([(n^2+2)/3]^4 \approx 1 + 8(n-1)/3\), are neglected; they are 7e-4 for argon at 1 bar.

The sources’ conventions and their conversion#

The curated data keep the source’s own quantity, convention and conditions; see Quantity vocabulary, Third-order quantities. gamma_observation() converts them on the way into a fit. Every factor was verified on the page images and by reproducing a printed value (TC-313).

Source quantity

Convention

To \(\gamma_T\)

gamma in au, SI (C\(^4\) m\(^4\) J\(^{-3}\)) or esu

T (Taylor)

as is

Ward’s per-molecule chi in esu (Ward & New 1969, Ward & Miller 1979, Ward & Elliott 1984, Lundeen et al. 1983)

ward_1969

\(\times 6\)

Lehmeier et al.’s gamma, from \(P = 4\varepsilon_0\chi E E E\) (SI or esu)

lehmeier_1985

\(\times 24\)

CARS \(\chi^{(3)}\) per amagat, esu (Hahn & Lee, Farrow et al.)

chi_cars_per_amagat

\(2\chi/N_0\), with \(N_0\) Loschmidt’s number: \(10^{-18}\) cm\(^3\)/(erg amagat) is 147.8 au

\(n_2\), cm\(^2\)/W or m\(^2\)/W, self coefficient

quantity nonlinear_index

\(8\varepsilon_0^2 c\, n_2/N\) at the dataset’s state

\(n_2\), probe coefficient

coefficient = "probe"

halved first

\(n_2\) with a probe at another wavelength

parameters.probe_nm

as the self coefficient, with the pump-probe frequency sums

The dataset’s component must be the mean gamma (mean, parallel or unstated). For an atom the laboratory zzzz component is the mean. An n2’s density is the one its source meant, at the dataset’s state (real gas unless the source used the ideal gas). A dataset option factor covers a unit the conversion does not know. Its one use is Bishop & Pipin’s helium, tabulated in the reduced-mass atomic unit of 4He: 6.24136e-65 against 6.235377e-65, a factor of 1.00096 (their p. 3549).

Other conventions in the literature, recorded and converted in the curated data or its notes (TC-313), include:

  • Rosasco’s \(\sigma = N\gamma_T/3\);

  • Hellwarth’s \(\sigma\) in esu;

  • Bishop’s dispersion coefficients A and A’, which map to Shelton’s A and B through the square and fourth power of 219474.63 cm\(^{-1}\) per hartree.

Processes and their frequencies#

A process is named by its frequency arguments \((-\omega_\sigma; \omega_1, \omega_2, \omega_3)\), with \(\omega_\sigma = \omega_1 + \omega_2 + \omega_3\). The fundamental (or pump) wavenumber is \(\nu\); \(\nu_s\) is a Stokes and \(\nu_r\) a probe wavenumber. The model uses the two symmetric sums

\[ W_2 = \nu_\sigma^2 + \nu_1^2 + \nu_2^2 + \nu_3^2, \qquad W_4 = \nu_\sigma^4 + \nu_1^4 + \nu_2^4 + \nu_3^4 , \]

(frequency_sums()). A negative frequency enters as its square, so its sign does not matter.

Process (process)

Arguments

\(W_2\)

\(W_4\)

Where it enters

static

\((0; 0, 0, 0)\)

0

0

the static limit, \(\gamma_0\)

kerr

\((-\omega; \omega, \omega, -\omega)\)

\(4\nu^2\)

\(4\nu^4\)

\(n_2\), SPM; DFWM

pump_probe

\((-\omega_r; \omega_r, \omega_p, -\omega_p)\)

\(2\nu^2 + 2\nu_r^2\)

\(2\nu^4 + 2\nu_r^4\)

XPM; a probe’s index

thg

\((-3\omega; \omega, \omega, \omega)\)

\(12\nu^2\)

\(84\nu^4\)

the third harmonic

eshg

\((-2\omega; \omega, \omega, 0)\)

\(6\nu^2\)

\(18\nu^4\)

the ESHG data

dc_kerr

\((-\omega; \omega, 0, 0)\)

\(2\nu^2\)

\(2\nu^4\)

the dc Kerr data

cars

\((-\omega_{as}; \omega_p, \omega_p, -\omega_s)\)

\((2\nu-\nu_s)^2 + 2\nu^2 + \nu_s^2\)

likewise

CARS, four-wave mixing

The wavelength of a stored value is always the fundamental (input) wavelength, or the pump for CARS and pump-probe.

Parts: electronic, vibrational, rotational#

A measured third-order response contains:

  • the electronic gamma, \(\gamma^e\), instantaneous on the scale of the pulse;

  • a vibrational part, \(\gamma^v\);

  • for a molecule that is not spherical, the rotational (orientational) part.

Which of these a measurement contains depends on the process and on the pulse. The data record it (part: electronic, vibrational, rotational, total, instantaneous and so on). The model is electronic:

  • H2, D2, N2, O2: Shelton’s ESHG includes a vibrational and rotational part, \(\gamma^{vR}\). It is subtracted before fitting: for H2 and D2 from Shelton’s tabulated values at the same wavelengths (option subtract_dataset), for N2 and O2 from his formula \(\gamma^{vR} = G\nu^{-2} + H\nu^{-4}\) (Table V; option subtract_formula).

  • CO2, CH4, SF6: the fits are to the total ESHG gamma. Its vibrational part is small but not separated (“we are unable to accurately determine \(\langle\gamma^e \rangle\) or its static limit for these molecules”, Shelton 1990, p. 13). It is process-dependent, which the model cannot represent, and it is in the 2 % floor of the accuracy page.

  • Atoms: gamma is purely electronic.

  • Femtosecond n2 of molecules: the rotational response is delayed and depends on the pulse. The Maryland group separate it and report the electronic n2; values called “instantaneous” or measured with longer pulses contain part or all of it, and are 1.3 to 6 times higher. In propagation the rotational and vibrational Raman responses are separate terms, with their own data. The gamma here is the instantaneous electronic term they add to.

  • N2O: its only value, Wahlstrand et al.’s n2, contains the vibrational part.