**1. Introduction**


The well-known drawback of some laser material-processing technologies is nonuniform thermal conditions in the spot. The material is overheated in the center of the laser spot when an excess of the energy leads to intensive material evaporations and chemical decompositions [1–4], which is not characteristic of other additive technologies using alternative sources of concentrated energy flow [5,6]. Inversely, the material does not attain the necessary processing temperature at the periphery of the spot, and the energy is essentially lost by heat diffusion in the treated body (the target) [7–9]. Modern optics proposes shaping a laser beam that provides alternative laser power density distributions of transverse electromagnetic (TEM) mode:


These technical solutions have multiple laser powder bed fusion attempts but have never been researched theoretically with correction to the beam motion [10–12].

The lack of a reliable solution in terms of heat redistribution leads to the following disadvantages affecting the quality of parts obtained by laser-additive manufacturing and processing productivity (Figure 1) [13–17]:

Local overheating, capturing the underlying layers, creating additional stresses during metal solidification (partially solved by subsequent heat treatment and preliminary heating of the substrate) [18–20];

**Citation:** Grigoriev, S.N.; Gusarov, A.V.; Metel, A.S.; Tarasova, T.V.; Volosova, M.A.; Okunkova, A.A.; Gusev, A.S. Beam Shaping in Laser Powder Bed Fusion:Péclet Number and Dynamic Simulation. *Metals* **2022**, *12*, 722. https://doi.org/10.3390/ met12050722

Academic Editors: Antonio Riveiro and Matteo Benedetti

Received: 1 March 2022 Accepted: 18 April 2022 Published: 24 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).


**Figure 1.** The main consequences of the active interaction of powder material with atmosphere and the existing ways of solving them.

An obvious disadvantage of using optical means for redistributing laser energy into the beam can be its expansion by 150–350%, which may not allow for obtaining more precision parts, but can become a significant advantage in the production of products with dimensions of more than 100 mm, for which the width of the heat-affected zone will be significantly reduced [27,28]. Figure 1 is based on the results of optical diagnostics and video monitoring described in detail in [27].

There are many factors that influence the final surface quality (roughness, uniformity, and dimensional accuracy) [29–33] such as:


The conventional power (energy flux *q*, W/mm2) density distribution in radius *r* of the laser focus is the classical bell-like one approximated by the normal Gauss distribution (Laguerre–Gaussian mode, circularly symmetric beam profile TEM00) of the optical resonator as: 

$$q = \frac{P}{\pi r\_0^2} \exp\left(-\frac{r^2}{r\_0^2}\right),\tag{1}$$

where *P* is the laser beam power, W and *r*0 is the radius circle, mm.

In some laser-based technologies such as lithography (photo-activated processes) [34,35], laser scribing [36,37], and thin surface laser treatment (including medical purposes) [38–41], the optimal beam profile seems to be the flat-top (TEMFT) one that provides the energy flux's uniformity (uniform laser power density distribution). The typical powder consolidation mechanisms in laser powder bed fusion are thermo-activated [42]. Then the objective is transferred from the uniform power density distribution (energy flux *q*, W/mm2) to a radiation-induced uniform temperature field *T* ( ◦C).

Since the thermal energy is released on an adiabatic plane bounding a uniform conducting half-space inside a circle of radius *r*0 (mm), with radial distribution [43]:

$$q = \frac{P}{2\pi r\_0^2} \frac{1}{\sqrt{1 - r^2/r\_0^2}}.\tag{2}$$

the temperature rise over the circle:

$$T\_0 = \frac{P}{4\lambda r\_0},\tag{3}$$

where *λ* is the material thermal conductivity, W/mm·K. In this case, the laser radiation is absorbed by layered powder to heat a massive body with conduction as the principal heat transfer mechanism. Then profile (2) can be better for laser powder bed fusion and similar laser-based powder technologies. TEMFT profile (the cylindrical flat-top temperature distribution) is challenging to obtain because of a discontinuity at the beam boundary where *r* = *r*0. Then the airy distribution of the first harmonic, (donut of the first overtone) TEM01\*, seems to be a reasonable compromise [43]:

$$\eta = \frac{P}{\pi r\_0^2} \frac{r^2}{r\_0^2} \exp\left(-\frac{r^2}{r\_0^2}\right),\tag{4}$$

In the thermo-activated processes, the laser beam scans the powder surface, resulting in a non-uniform temperature distribution over the laser spot for various laser beam profiles [44,45]. An inverse problem of heat diffusion for the scanning laser beam can be solved to find the ideal power density distribution. Still, the solution mainly depends on the scanning speed factor—its value and direction. The influence of direction on the absorbed energy flux shows that the laser beam profile would be asymmetric. Moreover, the laser beam scans quite fast (up to 400 mm/s) and changes direction rapidly. Therefore, it can be an even more complicated scientific and technical task never solved before, since most of the published work on beam profiling considers the symmetric beam for their calculations.

This work aims to compare three types of abovementioned laser beam profiles, research the influence of the scanning speed in a linear medium, and develop a non-linear model, including the material evaporation factor.
