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 <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/bourgeade_2012.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/physreve\.85\.056403 <https://doi.org/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
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.