bourgeade_2012

A. Bourgeade and G. Duchateau, “Time-dependent ionization models designed for intense and short laser pulse propagation in dielectric materials”, Phys. Rev. E 85, 056403 (2012)

Source record · Catalogue · Review register

DOI: 10.1103/physreve.85.056403

Source kind: theory

Recorded technique: Two multiphoton-ionization models driven by the instantaneous electric field, for coupling to a Maxwell-equation propagation code: BVkP (Bloch-Volkov final states, matrix elements from k.P theory at the zone centre, Eqs. 1-16) and an optical-Bloch density-matrix model with two valence and N_C conduction levels (Eqs. 17-22), each with an added non-radiating ‘free’ state reached by relaxation (Sec. II C). Applied to 100-fs pulses at 413 nm (three photons across a 9-eV gap), a pulse with 4 % of its second harmonic, and chirped pulses (300-600 nm). No measurements; impact ionization excluded.

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 2 formulas.

Curation notes

Journal pages 056403-1 to -11 = PDF pages 1-11. Identity validated.

NO MATERIAL, NO DATASET. The models describe a generic direct-gap dielectric: the introduction names silica, LBO, BBO, KDP, KTP and sapphire, and frequency-converting crystals such as KDP motivate the work, but no calculation is assigned to a material. The parameters (atomic units in Sec. II unless stated): band gap ‘set to 9 eV’ (p. 056403-5); the valence-electron density N_0 = 2.2 x 10^22 cm^-3 (p. 056403-3); the BVkP localization parameter alpha = 1.55 a.u., chosen so that the model reproduces Keldysh’s formula for the conditions of interest; the optical-Bloch dipole elements mu_vv = 1.68 x 10^-29 C m and mu_vc = 1.34 x 10^-29 C m, chosen to mimic the BVkP rate early in the pulse (p. 056403-5); the relaxation time into the free state, 3 ps, ‘according to these relaxation time measurements’: conduction electrons in KDP relax ‘on a timescale ranging from roughly 300 fs to tens of ps [39]’ (Duchateau et al., Phys. Rev. B 83, 075114 (2011)), ‘the same order of magnitude for other wide-band-gap dielectric materials, such as NaCl, MgO, or Al2O3’ [40-42] (p. 056403-5): ranges for other materials, not extracted (the vocabulary would need a material for each; the ranges are not values). With no material, no dataset can be written; the numbers are in the formulas’ coefficients.

RESULTS (figures only): the density of conduction and free electrons against time, photon energy and intensity (Figs. 1-6); the fitted multiphoton slopes 2.88-3.07 at 413 nm (2.52 for the two-level optical-Bloch model) against the expected 3 (p. 056403-7); a 4 %-intensity second harmonic raising the free-electron density by an order of magnitude at a 3-eV fundamental (Fig. 8); chirped pulses changing the multiphoton order from 3 to 4 near 400 fs (Fig. 9). Final free-electron density ~2e18 cm^-3 (three-photon case) and ~1e20 cm^-3 (two-photon), compared with measurements on various dielectrics [39-42] and first-principles results for alpha-quartz [43,44]. Impact ionization is estimated below 5 % for 100 fs and < 5e13 W/cm2 from Rethfeld’s Fig. 6 (Phys. Rev. B 73, 035101 (2006), not in the library; p. 056403-5).

Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

No datasets are recorded.

Formulas

bourgeade_2012/bvkp_rate

{
  "id": "bvkp_rate",
  "equation": "Eqs. (9), (15), (16), pp. 056403-3",
  "printed": "P = <phi_c|P_Z|phi_v> = sqrt(m_0^2 E_g/(2 m*_vc)), 1/m*_vc = 1/m*_v + 1/m*_c (9); T_cv(t) = -(P/(m_0 (E_c - E_v))) int_0^t dt exp{i (E_c - E_v) t} E(t)/(1 - i A(t)/alpha)^2 (15); W_MPI(t) = N_0 d|T_cv(t)|^2/dt = 2 N_0 Re{T_cv*(t) dT_cv(t)/dt} (16)",
  "variable": "E(t): the laser field; A(t) = -int_0^t E dt' the vector potential; alpha the localization parameter of the (hydrogen-like) valence state; atomic units",
  "coefficients": {
    "alpha": "1.55 a.u.",
    "N_0": "2.2e22 cm^-3",
    "E_g": "9 eV"
  },
  "check": "The model is checked against Keldysh's formula by the choice of alpha (p. 056403-5); no rate value is printed to reproduce."
}

Valid while |A(t)/alpha| < 1 (perturbative), Eq. (14). Several final conduction states c_i are summed numerically. The relaxation to a free state (Sec. II C) makes the ionization irreversible; tau_r = 3 ps.

bourgeade_2012/optical_bloch

{
  "id": "optical_bloch",
  "equation": "Eqs. (19)-(22), p. 056403-4",
  "printed": "i d rho/dt = [H, rho] (19); i d rho_jk/dt = (E_j - E_k) + E(t) . sum_l rho_jl mu_lk - E(t) . sum_l rho_lk mu_jl (20); rho(t + dt) = [Id + (i/2) H(t + dt/2)][Id - (i/2) H(t + dt/2)]^-1 rho(t) [Id - (i/2) H(t + dt/2)][Id + (i/2) H(t + dt/2)]^-1 (22)",
  "variable": "rho: density matrix of two valence and N_C conduction levels; mu_jl = <psi_j|r|psi_l>",
  "coefficients": {
    "mu_vv": "1.68e-29 C m",
    "mu_vc": "1.34e-29 C m",
    "tau_r": "3 ps"
  },
  "check": "Nothing numeric to reproduce."
}

Eq. (20) is printed with (E_j - E_k) not multiplied by rho_jk (as printed). The conduction-valence elements are equal and scaled as 1/sqrt(N_C); two valence levels are needed for even photon orders (Appendix B).

Corrections

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.