Delayed-rate equations model for femtosecond laser-induced breakdown in dielectrics Jean-Luc D´eziel, 1 Louis J. Dub´e, 1, * and Charles Varin 1, 2, † 1 D´epartement de physique, de g´enie physique et d’optique, Universit´e Laval, Qu´ebec G1V 0A6, Canada 2 C´egep de l’Outaouais, Gatineau, Qu´ebec J8Y 6M4, Canada Experimental and theoretical studies of laser-induced breakdown in dielectrics provide conflicting conclusions about the possibility to trigger ionization avalanche on the sub-picosecond time scale and the relative importance of carrier-impact ionization over field ionization. On the one hand, current models based on single ionization-rate equations 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, models based on multiple rate equations that track the carriers kinetics rely on several free parameters, which limits the physical insight that we can gain from them. In this paper, we develop a model that overcomes these issues by tracking both the plasma density and carriers’ mean kinetic energy as a function of time, forming a set of delayed rate equations that we use to match the laser-induced damage threshold of several dielectric materials. In particular, we show that this simplified model reproduces the predictions from the multiple rate equations, with a limited number of free parameters determined unambiguously by fitting experimental data. A side benefit of the delayed rate equations model is its computational efficiency, opening the possibility for large-scale, three-dimensional modelling of laser-induced breakdown of transparent media. I. INTRODUCTION Computer modelling of strong-field optical phenomena in dielectrics driven by intense laser radiation is essential to understand the fundamental processes in play, e.g., during laser micro-machining, laser surgery, and high- harmonic generation in solids, to name a few. Mech- anisms for laser-induced breakdown were identified and studied in various contexts [1–8]. In the accepted picture, plasma formation in laser-driven dielectrics proceeds as follows. (1) Charge carriers are first created by field ionization (FI). (2) The charge carriers absorb energy from the laser field via inverse bremsstrahlung heating (IBH). (3) The hot charge carriers create new, cold ones through carrier-impact ionization (II). (4) The carriers created by II, in turn, gain energy from the laser field and create new carriers by II, and so on. This multi- plication of charge carriers via II leads to an exponen- tial growth of the plasma density, often referred to as an ionization avalanche. This picture applies well when the FI-II interplay has enough time to unfold, e.g., when the pulse duration is in the picosecond range or above. However, current experimental and theoretical studies of laser-induced breakdown in dielectrics provide conflicting conclusions about the relative importance of II over FI and the possibility to trigger ionization avalanche on the sub-picosecond time scale [9]. For example, a pumb-probe experiment in fused sil- ica [10] has shown that a significant amount of ionization can take place after the pump pulse, which cannot be de- scribed by FI alone and suggests a delayed II avalanche triggered by slowly-decaying hot plasmon excitations. In * Louis.Dube@phy.ulaval.ca † charles.varin@cegepoutaouais.qc.ca contrast, in another experiment in sapphire [11], there was no evidence of ionization avalanche. On the theory side, calculations based upon a Fokker-Planck equation in [12] lead to a strong dominance of II over FI while the simulations in [13] predict kinetic energies of the charge carriers that are too low for II to be significant. It was also suggested that the condition to trigger an avalanche should be given by the laser fluence instead of the pulse duration, but the predicted threshold values differ by more than an order of magnitude (see, e.g., [14, 15]). Actually, experiments involve different materials and laser parameters, which makes a direct comparison be- tween them difficult. Other challenges lie in the theo- retical models that are currently used to interpret the experimental observations. On the one hand, current models based on single ionization-rate equations (SRE) do not account for the gradual heating of the charge car- riers which, for short laser pulses, might not be suffi- cient to start an avalanche. On the other hand, models based on multiple rate equations (MRE) that track the carrier kinetics on discrete energy levels rely on several free parameters, which limits the physical insight that we can gain from them. While calculation of the FI rates with the Keldysh theory [16] is well established, models for IBH and II can vary significantly. For exam- ple in refs. [1, 4, 7, 17], plasma formation was modelled within similar theoretical frameworks, but assumed dif- ferent IBH rates, thus influencing directly the efficiency of II and leading to conflicting conclusions about the rel- ative importance of II over FI and the occurrence of ion- ization avalanche in short pulses. In this paper, we describe a model that overcomes the issues associated with the SRE and MRE models by tracking both the plasma density and carriers’ mean ki- netic energy as a function of time, forming a set of delayed rate equations (DRE) that we use to match the laser- induced damage threshold of several dielectric materials. arXiv:1906.08338v1 [physics.comp-ph] 19 Jun 2019