Fourier two-temperature model to describe ultrafast laser pulses interaction with metals: A novel mathematical technique
Introduction
The Fourier approach proved appropriate to describe the thermal field in laser-metal interaction [1] at intensities up to 1015 W/cm2 [2], as, major relativistic effects usually occur for higher intensities.
The irradiation of metals by ultra-short laser pulses became a fast rising area of research, especially in the case of laser processing. The applications domain is vast, extending from industry, medicine, fundamental research, to military utilizations. The subject also brings new achievements in micro- and nano-technologies, in electronics, mechanics, and biology [3].
Owing to the recent progress of ultra-short laser pulses generation based upon the Chirped Pulse Amplification (CPA) [4], the pulse duration could be shortened, from several nano-down to near atto-seconds. Different Fourier models have been proposed [5], [6], [7], [8], but they remain still unsatisfactory from complexity and accuracy points of view, and more profound theoretical approaches are therefore needed.
In the two-temperature model (TTM), firstly proposed by S. Anisimov et al. [9], the thermal fields are usually defined by two-coupled diffusion equations describing the electron and phonon (lattice) heat conduction, respectively. The two equations are connected by a “electron-phonon coupling constant” term.
There are three main physical processes induced and evolving during femtosecond laser pulse irradiation of metals. The first one, also used instead of a time scale, requires a few femtoseconds for electrons to reach the Fermi distribution, whilst the metallic lattice keeps undisturbed. From mathematical point of view, it corresponds to , where and are electron and phonon temperatures, respectively. In the second case, the energy is diffused throughout the lattice via electron-phonon collisions, in a time span of tens of picoseconds. The third case consists of energy diffusion inside the bulk metal through phonon-phonon heat transfer, which becomes prevalent.
The innovation with our contribution consists in the combination of the Zhukovsky mathematical formalism [10], [11], [12] with the TTM [13], [14], [15], which allows for the development of a general model, easy to apply and based upon simple physical parameters. Our approach considers an infinite speed of thermal waves.
An example of numerical simulation was conducted in case of an Au target, submitted to irradiation with 100 fs pulses, generated by a laser source.
Section snippets
Standard mathematical background
We proposed the following thermal model which is a combination of the TTM [9] (Eq. (1a)) and the Fourier equation [16] (Eq. (1b)): and Here, and are the electron heat capacity and heat conductivity, respectively. stands for the thermal phonon capacity, which is constant for metals, while Q is the heat transferred from laser to target and g is the electron-phonon coupling strength. x is the horizontal space
Zhukovsky's mathematical formalism
K. Zhukovsky proposed an entirely new set of solutions for Fourier and non-Fourier heat equations [10], [11], [12]. Let us consider the generalized linear heat equation in 1D in the form: Note that here and in the following α, δ, β, and γ are Zhukovsky unit-less parameters, providing basic information about the target. This stands for one major advantage of the Zhukovsky approach for description of laser interaction/heating of various materials.
Methodology
A combination of Eqs. (9) and (17), and of (9) and (18), respectively, was used for simulations. The thermo-physical properties of Au targets were selected for computations [18]. All calculations have been carried out via Mathematica software, using Intel i7 8th generation and 16 Gb Ram.
Simulation results
The advent of super-intense laser pulses produced by CPA technique in parallel with the progress of new materials appropriate to deliver high laser fluences, opened up new fields in optics. A CPA laser displays astonishing characteristics [4] as it can produce a broad field, extreme pressure, higher temperature and accelerating fields, acting as an omnipotent tool for high-energy particle irradiation. CPA technology allows for the generation of a wide range of intensities extending from 1014 to
Conclusions and outlook
A new theoretical model is proposed to describe the interaction between a laser beam and a semi-infinite metallic target via the generated temperature fields. The novelty of the approach resorts in the decoupling of the two heat equations in the Two-Temperature-Model.
An Au target submitted to irradiation by W/cm2 laser pulses was considered for simulations. A detailed analysis, including graphical representations of the electrons' temperature increment on surface and inside the Au target,
CRediT authorship contribution statement
Anca Nicarel: Conceptualization, Investigation, Methodology, Writing – original draft. Mihai Oane: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft. Ion N. Mihailescu: Conceptualization, Supervision, Writing – review & editing. Carmen Ristoscu: Conceptualization, Funding acquisition, Project administration, Resources, Validation, Visualization, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by Romanian Ministry of Education and Research, under Romanian National Nucleu Program LAPLAS VI – contract 16N/2019.
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