Micromachines 2023, 14, 989. https://doi.org/10.3390/mi14050989 www.mdpi.com/journal/micromachines Article Composite Diffraction-Free Beam Formation Based on Iteratively Calculated Primitives Pavel A. Khorin 1,2 , Alexey P. Porfirev 1,2 and Svetlana N. Khonina 1,2, * 1 Samara National Research University, Samara 443086, Russia; khorin.pa@ssau.ru (P.A.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 * Correspondence: khonina@ipsiras.ru Abstract: To form a diffraction-free beam with a complex structure, we propose to use a set of primitives calculated iteratively for the ring spatial spectrum. We also optimized the complex transmission function of the diffractive optical elements (DOEs), which form some primitive diffraction-free distributions (for example, a square or/and a triangle). The superposition of such DOEs supplemented with deflecting phases (a multi-order optical element) provides to generate a diffraction-free beam with a more complex transverse intensity distribution corresponding to the composition of these primitives. The proposed approach has two advantages. The first is the rapid (for the first few iterations) achievements of an acceptable error in the calculation of an optical element that forms a primitive distribution compared to a complex one. The second advantage is the convenience of reconfiguration. Since a complex distribution is assembled from primitive parts, it can be reconfigured quickly or dynamically by using a spatial light modulator (SLM) by moving and rotating these components. Numerical results were confirmed experimentally. Keywords: structured laser beams; diffraction optical elements; diffraction-free beam; holographic optical tweezers 1. Introduction The term “diffraction-free beam” was introduced to denote a laser beam propagating along the optical axis without changing the transverse distribution, i.e., without the influence of diffraction effects. The most famous among diffraction-free beams are the Bessel modes [1–3], which are the solution of the Helmholt equation in cylindrical coordinates. In addition, Mathieu beams [4,5] are famous for the elliptic coordinate system and parabolic beams [6,7] for the parabolic coordinate system, as well as various generalized beams [8–10]. A general property of classical diffraction-free beams is the concentration of the spatial spectrum on a narrow ring. Such a property is often used to generate diffraction-free beams [2,11]. There are also other beams with diffraction-free properties whose spatial spectrum differs significantly from a narrow ring. They include Airy beams, Olver beams, and their modifications [12–15]. Among the effective methods of forming diffraction-free beams are the applications of axicons [16–18], diffraction optical elements (DOEs) [19–21], or spatial light modulators (SLMs) [22–24]. With these approaches, in contrast to the focusing of a narrow ring [2,11], a significant part of the energy of the incident beam goes to the formation of a diffraction- free beam. However, in some cases, additional coding of the calculated complex amplitudes into a phase-only mask may be required [25,26]. Another simple method of energetically efficient formation of various diffraction-free beams is based on a partial diaphragm of the annular light distribution [27,28], formed, for example, by a tandem of an axicon and a lens [29,30], or a toroidal lens [31,32]. Citation: Khorin, P.A.; Porfirev, A.P.; Khonina, S.N. Composite Diffraction-Free Beam Formation Based on Iteratively Calculated Primitives. Micromachines 2023, 14, 989. htps://doi.org/10.3390/ mi14050989 Academic Editors: Giuseppe Bruneti, Muhammad Ali But Received: 26 March 2023 Revised: 27 April 2023 Accepted: 28 April 2023 Published: 30 April 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Atribution (CC BY) license (htps://creativecommons.org/license s/by/4.0/).