J. Eur. Opt. Society-Rapid Publ. 21, 17( 2025) 177
Figure 2. Schematic illustration of an ideal optical system for laser beam shaping with subsequent amplification.
The experimental verification of these simulation results is currently in progress and will be reported in a forthcoming publication.
4 Phase mask design for tailored temperature distributions
Figure 3. Selected target intensity distribution for laser beam shaping with subsequent amplification.
Here, n denotes the index of a pixel in the target intensity distribution or the simulation result, N is the total number of pixels and I n and I target, n are the intensities in the n-th pixel in the simulation result and in the target distribution respectively. The RMSD value heavily depends on the area in the target plane that is used for the evaluation of the RMSD. For pixels outside of the actual target distribution, both the target and the actual pixel value are usually close to zero. As the RMSD calculates a mean deviation, a larger evaluation area generally leads to lower RMSD values. For the results shown here, the evaluation area for the RMSD is the same as the simulated and depicted field size, i. e. 1.5 mm 1.5 mm.
As most material processing applications are governed by the resulting temperature distribution in the work piece, it is crucial to determine intensity distributions that result in the desired temperature distributions for the respective process. The temperature distribution resulting from a given intensity distribution can be calculated by solving the direct heat conduction problem( DHCP). The calculation of the intensity distribution that results in a given temperature distribution, however, requires to solve the ill-posed and much more complex inverse heat conduction problem( IHCP). Bonhoff implemented an iterative algorithm to solve the IHCP for a given temperature distribution, material and process parameters and experimentally verified the performance of the algorithm for laser-based softening of steel and laser hardening [ 2, 43, 44 ].
Given a target intensity distribution from solving the IHCP, the IWPA, or a different phase retrieval algorithm, can then be used to calculate a phase mask that creates the desired intensity distribution. However, even in simulations, there always remains a finite difference between the target intensity distribution and the intensity distribution resulting from the calculated phase mask( cf. Fig. 4, right). This difference depends on the complexity of the target intensity distribution and is partly caused by the finite number of pixels of the used SLM. While this difference may be the global minimum that is achievable with the given number of SLM pixels, it does not necessarily result in the smallest achievable deviation from the target