Citation: Khonina, S.N.; Porfirev, A.P.; Volotovskiy, S.G.; Ustinov, A.V.; Karpeev, S.V. Simple Method of Light Field Calculation for Shaping of 3D Light Curves. Photonics 2023, 10, 941. https://doi.org/10.3390/ photonics10080941 Received: 8 July 2023 Revised: 28 July 2023 Accepted: 14 August 2023 Published: 17 August 2023 Copyright: © 2023 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/). photonics hv Article Simple Method of Light Field Calculation for Shaping of 3D Light Curves Svetlana N. Khonina 1,2 , Alexey P. Porfirev 1,2 , Sergey G. Volotovskiy 2 , Andrey V. Ustinov 2 and Sergey V. Karpeev 1,2, * 1 Scientific Research Laboratory of Automated Systems of Scientific Research, Samara National Research University, Samara 443086, Russia; khonina@ipsiras.ru (S.N.K.); porfirev.alexey@ipsiras.ru (A.P.P.) 2 Image Processing Systems Institute of RAS—Branch of the FSRC “Crystallography and Photonics” RAS, Samara 443001, Russia; sv@ipsiras.ru (S.G.V.); andr@ipsiras.ru (A.V.U.) * Correspondence: karp@ipsiras.ru Abstract: We propose a method for generating three-dimensional light fields with given intensity and phase distributions using purely phase transmission functions. The method is based on a generalization of the well-known approach to the design of diffractive optical elements that focus an incident laser beam into an array of light spots in space. To calculate purely phase transmission functions, we use amplitude encoding, which made it possible to implement the designed elements using a single spatial light modulator. The generation of light beams in the form of rings, spirals, Lissajous figures, and multi-petal “rose” distributions uniformly elongated along the optical axis in the required segment is demonstrated. It is also possible to control the three-dimensional structure of the intensity and phase of the shaped light fields along the propagation axis. The experimentally generated intensity distributions are in good agreement with the numerically obtained results and show high potential for the application of the proposed method in laser manipulation with nano- and microparticles, as well as in laser material processing. Keywords: laser beam shaping; intensity; phase; Whittaker integral; Lissajous figures; trans- mission function 1. Introduction The generation of optical fields with a given amplitude and phase distribution is a very popular task in optical manipulation [1–5] and laser material processing [6–8]. In these areas, structured laser beams with a predetermined amplitude and/or phase gra- dient are widely used for the realization of guiding of nano- and micro-objects [9], as well as for transfer of molten material [10,11]. For example, annular optical vortex beams with helical wave fronts are used to rotate optically trapped nano- and microparticles [5] and to implement the spiral-shaped mass transfer of a temporarily molten mater in thin films of polymers [12], metals [10], and semiconductors [13]. Full control of the ampli- tude and phase using computer-generated holograms made it possible to implement the guiding of the trapped particles along complex two-dimensional trajectories. Because of this, various manipulation techniques and applications have been demonstrated in recent decades [2,5,8,13,14]. A further transition to the design of elements generating the required three-dimensional light fields led to the demonstration of the three-dimensional controlled guiding of microparticles in space [15]. In the field of laser material processing, light fields with a desired three-dimensional structure have been used to fabricate three-dimensional chiral microstructures [16]. There are many different techniques for the generation of structured laser beams as well as their superpositions and arrays—the use of spatial light modulators, mode convertors, and the use of nonlinear media [17–20]. In addition, there are special types of structured laser beams that are curvilinear, such as Airy beams [21]. However, to implement Photonics 2023, 10, 941. https://doi.org/10.3390/photonics10080941 https://www.mdpi.com/journal/photonics