Dynamical rate equation model for femtosecond laser-induced breakdown in dielectrics

Jean-Luc Déziel, Louis J. Dubé, and Charles Varin

Phys. Rev. B 104, 045201 – Published 6 July 2021

Abstract

Experimental and theoretical studies of laser-induced breakdown in dielectrics provide conflicting conclusions about the possibility to trigger ionization avalanche on the subpicosecond time scale and the relative importance of carrier-impact ionization over field ionization. On the one hand, current models based on a single ionization-rate equation do not account for the gradual heating of the charge carriers, which, for short laser pulses, might not be sufficient to start an avalanche. On the other hand, kinetic models based on microscopic collision probabilities have led to variable outcomes that do not necessarily match experimental observations as a whole. In this paper, we present a rate-equation model that accounts for the avalanche process phenomenologically by using an auxiliary differential equation to track the gradual heating of the charge carriers and define the collisional impact rate dynamically. The computational simplicity of this dynamical rate-equation model offers the flexibility to extract effective values from experimental data. This is demonstrated by matching the experimental scaling trends for the laser-induced damage threshold of several dielectric materials for pulse durations ranging from a few fs to a few ps. Through numerical analysis, we show that the proposed model gives results comparable to those obtained with multiple rate equations and identify potential advantages for the development of large-scale, three-dimensional electromagnetic methods for the modeling of laser-induced breakdown in transparent media.

Received 19 June 2019
Revised 18 March 2021
Accepted 22 June 2021

DOI:https://doi.org/10.1103/PhysRevB.104.045201

Physics Subject Headings (PhySH)

Dielectric properties Electronic excitation & ionization Laser-plasma interactions Light-matter interaction Plasma production & heating

Dielectrics

Femtosecond laser irradiation

Plasma PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Jean-Luc Déziel^1,2, Louis J. Dubé^1,2,*, and Charles Varin^1,3,4,†

¹Département de physique, de génie physique et d'optique, Université Laval, Québec G1V 0A6, Canada
²Centre interdisciplinaire en modélisation mathématique, Université Laval, Québec G1V 0A6, Canada
³Département de physique, Cégep de l'Outaouais, Gatineau, Québec J8Y 6M4, Canada
⁴Centre collégial de transfert de technologie en cybersécurité, Cégep de l'Outaouais, Gatineau, Québec J8Y 6M4, Canada

^*Louis.Dube@phy.ulaval.ca
^†charles.varin@cegepoutaouais.qc.ca

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Vol. 104, Iss. 4 — 15 July 2021

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Figure 1
Relative contribution of collisional ionization ( $ρ_{ci}$ ) over the global ionization yield ( $ρ = ρ_{fi} + ρ_{ci}$ ) obtained with the SRE, MRE, and DRE models. In (a), impact ionization in SRE starts at $t = 0$ , while MRE and DRE show an initial time delay needed for the first charge carriers to be heated above the critical energy $E_{c}$ . (b) shows that the contribution from impact ionization drops more quickly below the avalanche threshold $F_{av}$ when a laser heating mechanism is included (MRE and DRE). For each model, the fluence $F$ is normalized by the respective $F_{av}$ value ( $F_{av}^{SRE} = 0.314 J / {cm}^{2}$ , $F_{av}^{MRE} = 0.554 J / {cm}^{2}$ , and $F_{av}^{DRE} = 0.411 J / {cm}^{2}$ , see text for details). Model parameters are laser wavelength $λ = 800 nm$ , band gap energy $E_{g} = 9 eV$ , effective masses $m_{e} = m_{h} = m_{0}$ (with the free electron mass $m_{0} = 9.1094 \times 10^{- 31} kg$ ), material density $ρ_{mat} = 2 \times 10^{28} m^{- 3}$ , molecular cross section $σ_{mat} = 10^{- 19} m^{2}$ , linear refractive index $n_{0} = 1.5$ , recombination rate $γ_{r} = 0$ , plasma damping rate $γ = 1 {fs}^{- 1}$ , and impact ionization coefficient (for SRE) $α = 4 {cm}^{2} / J$ . Electric field strengths $E$ are $3.974 GV / m$ (SRE), $5.276 GV / m$ (MRE), and $4.542 GV / m$ (DRE).
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Figure 2
DRE dynamical details. Left column [(a)–(c)], pulse duration $τ = 10$ fs and laser fluence $F = 1.75$ J/ ${cm}^{2}$ . Right column [(d)–(f)], $τ = 300$ fs and $F = 6.76$ J/ ${cm}^{2}$ . In both cases, the fluence was set to reach 10% of ionized molecules. (a) and (d) show the contribution of FI ( $ρ_{fi}$ ) and CI ( $ρ_{ci}$ ) to the total charge density $ρ = ρ_{fi} + ρ_{ci}$ . The density of electrons with kinetic energy greater than the critical energy $E_{c}$ ( $ξ ρ$ ) and the shape of the laser pulse (shaded area) are also shown. In (b) and (e) the kinetic energy of the electron in the conduction band obtained by the numerical integration of Eq. (8) is compared with the Fermi energy (see text) and the upper limit obtained analytically [Eq. (17)]. Finally, in (c) and (f) are shown the electron-neutral collisional rate [ $γ_{u}$ , Eq. (10)] and the photon absorption rate [ $γ_{lh}$ , Eq. (9)]. The electron-electron collision rate [ $γ_{e e}$ , Eq. (18)] is not part of the DRE model, but is shown for reference (see text). Parameters are laser wavelength $λ = 800$ nm, band gap energy $E_{g} = 9$ eV, effective masses $m_{e} = m_{h} = m_{0}$ (with the free electron mass $m_{0} = 9.1094 \times 10^{- 31} kg$ ), material density $ρ_{mat} = 2 \cdot 10^{28} m^{- 3}$ , molecular cross section $σ_{mat} = 10^{- 19} m^{2}$ , linear refractive index $n_{0} = 1.5$ , recombination rate $γ_{r} = 0$ and plasma damping rate $γ = 1 {fs}^{- 1}$ .
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Figure 3
Comparison between DRE calculations (solid curves) and experimental measurements (shapes) of fluence thresholds $F_{th}$ as a function of pulse duration $τ$ for various dielectric materials. The experimental data sets are from Ref. [35]. The parameters used for the DRE calculations are given in Table 1. The dashed line represents an alternative fit for ${TiO}_{2}$ (see text and Table 1).
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Figure 4
Laser-induced damage thresholds $F_{th}$ of ${SiO}_{2}$ as a function of pulse duration $τ$ . The experimental data sets from Refs. [7, 13, 35, 38, 39, 40, 41] are compared to three different DRE calculations (see also text for details). The blue dash-dotted curve is obtained with field ionization (FI) only [collisional ionization (CI) is turned off by setting the molecular cross section $σ_{mat} = 0$ ] and laser heating (LH) is neglected by using the density criterion [ $ρ ≳ ρ_{c}$ , see Eq. (19)]. The green dashed curve is obtained with FI, but includes LH by using the energy density criterion [ $E_{abs} ≳ 3$ kJ/ ${cm}^{3}$ , see Eq. (20)] (CI is still off with $σ_{mat} = 0$ ). Finally, the red curve is obtained with the full DRE model. Model parameters are given in Table 1.
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Figure 5
Insight into the continuum expressions obtained with the Drude model for the ponderomotive energy $E_{p}$ [Eq. (B5)] and laser-heating rate $γ_{lh}$ [Eq. (B7)]. Both curves are normalized by their peak amplitudes, $E_{0}$ and $E_{0} / ℏ$ , respectively.
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