J. Eur. Opt. Society-Rapid Publ. 21, 17( 2025) 175
of an ideal lens and the focal plane behind the lens in the Fresnel approximation of wave-optics [ 18 ]. As( discrete) Fourier transforms can be calculated numerically very efficiently [ 19 ], this enables the simulation of the light diffraction behind the phase mask with limited computational effort. Since the initial publication of the IFTA, more sophisticated algorithms for phase retrieval have been published, mostly based on the original IFTA approach [ 6, 15, 17 ]. Still, the original IFTA remains the default approach in many applications in research and industry due to its simple implementation and its independence from additional algorithms for the simulation of wave-optical light propagation.
When considering optical systems for laser beam shaping, one disadvantage of this ongoing use of the IFTA and related algorithms is a limitation regarding the positioning of the target plane. Even though the focal plane of a lens is generally not an optimal choice for the position of the target plane for laser beam shaping, the focal plane of a focusing lens( or a plane conjugate to the focal plane) remains the typical choice for beam shaping systems [ 20 – 22 ]. Additionally, optical elements or non-linear effects between the phase mask and the target plane can also not be considered with IFTA-based approaches.
We here present an overview of our research on phase retrieval algorithms for laser beam shaping to overcome the limitations of IFTA-based approaches with a focus on beam shaping for laser material processing. In Section 2, we discuss our approach of replacing the Fourier transform in the IFTA with a full wave-optical propagation through arbitrary optical systems. This enables a flexible positioning of the target plane and the use of additional optical elements. In Section 3, we demonstrate the use of the new algorithm for the consideration of non-linear effects from optical amplifiers for laser beam shaping with subsequent amplification. In Section 4, we present our approach of coupling the aforementioned algorithm with a solver for the so-called inverse heat conduction problem for tailored temperature distributions within a work piece. And in Section 5, we give an overview of our research into diffractive neural networks( DNNs). DNNs allow to utilize the advances in machine learning of the last decade to optimize phase masks for, among others, high fidelity 2D and 3D laser beam shaping.
2 Iterative wave-optical propagation algorithm
To overcome the limitations of the single Fourier transform in IFTA-based phase retrieval algorithms, we implement a full wave-optical simulation of the light propagation between the phase mask and the target plane( cf. Fig. 1). We keep the remaining iterative structure of the IFTA and refer the reader to the literature for further details regarding the IFTA-like structure [ 15, 17, 21 ]. In reference to the IFTA, we denote our algorithm in the following as the iterative wave-optical propagation algorithm( IWPA).
For the simulation of the wave-optical propagations we use the OPT software package developed at the Fraunhofer Institute for Laser Technology [ 23 ]. OPT consists of a collection of C ++ and Python classes for physical-optical simulations of light propagation and optical amplifiers including non-linear amplification and thermally induced optical effects. The wave-optical free-space propagation of light is simulated with OPT by solving the Fresnel diffraction integral [ 24 ]. Propagation through optical elements, such as lenses or phase masks, is calculated in OPT in the thin element approximation [ 18 ]. The remaining IWPA is implemented in Python with a high-level interface for the calculation of the light propagation. This allows a simplified exchange of the simulation algorithm for e. g. comparisons of the results using OPT or a simple Fourier transform.
As OPT allows to model practically arbitrary optical systems, one main advantage of the IWPA is the ability to position the target plane in front or behind the focal plane of a focusing lens or do not use a focusing lens at all. This prevents the need for phase masks with large phase gradients to counteract the focusing power of the lens when creating extended intensity distributions [ 20, 21 ]. Large phase gradients are especially disadvantageous when using beam shaping elements like LCoS-SLMs that inherently suffer from pixel crosstalk [ 25 – 27 ].
Replacing the single Fourier transform of the IFTA with a Fresnel diffraction integral for arbitrary or multiple target planes is not a new concept and is discussed in the literature for a variety of different approaches [ 28 – 30 ]. It is also applied for phase retrieval in more complex optical systems consisting of multiple apertures and optical elements [ 31 ]. However, to our knowledge, these concepts are rarely or never used in the field of laser material processing. The IWPA also allows us to implement the compensation of non-linear effects in optical amplifiers as discussed in the following section.
3 Beam shaping with subsequent amplification
The IWPA also enables the calculation of phase masks for laser beam shaping with subsequent amplification. Beam shaping before passing the beam through an optical amplifier has two main advantages: Firstly, an adapted intensity distribution within the amplifier allows to improve the energy extraction from the active medium within the amplifier [ 32 ]. And secondly, moving the beam shaping element before the amplification step allows laser beam shaping at resulting laser powers and intensities beyond the damage threshold of the used beam shaping element. The damage thresholds of commercially available SLMs continue to rise but even“ high power handling” SLMs still have damage thresholds well below 1 kW [ 33 ], while the laser power used in many industrial applications lies in the multi-kilowatt range [ 34, 35 ]. However, for target planes behind the optical amplifier, the influence of the amplifier( e. g. the generally non-linear amplification) must be considered and precompensated for by the beam shaping element. To our knowledge, such a pre-compensation in an optical system with an SLM followed by an optical amplifier was only investigated for single homogeneous target distributions, whereby the beam shaping element was additionally