J. Eur. Opt. Society-Rapid Publ. 2026, 22, 8 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2026003 Available online at: https:// jeos. edpsciences. org
Recent Advances on Optics and Photonics 2026 Guest editors: Manuel Filipe P. C. M. Costa, Rogerio Nogueira and Alessandro Fantoni
RESEARCH ARTICLE
Journal of the European Optical Society-Rapid Publications Influence of modulation parameters on the self-focusing characteristics of rectangular Airy vortex beam
Liangqin Gan 1, 2,*
, Hongyi Lin 1, 2, Shanggong Yang 1, 2, and Dong Sun 1, 2
1 School of Optoelectronic and Communication Engineering, Xiamen University of Technology, Xiamen 361024, PR China 2 Fujian Key Laboratory of Optoelectronic Technology and Devices, Xiamen University of Technology, Xiamen 361024, PR China
Received 14 November 2025 / Accepted 12 January 2026
Abstract. This study investigates the self-focusing characteristics of a rectangular Airy vortex beam( RAVB) via phase modulation. Numerical simulations are conducted to examine the influence of modulation parameters, including linear factor( c), topological charge( l) and wavelength( k) on RAVB self-focusing characteristics. The focus depth, focus spot size and peak intensity of RAVB can be effectively controlled via the modulation parameters. Among these parameters, c has the most significant impact on RAVB self-focusing characteristics. As c increases from 4 to 8, the focus depth changes from 8 cm to 41 cm, while the peak intensity increases to 50 times. Although variations in l and k also contribute to increases in both focus depth and peak intensity, their effects are relatively minor compared to those of c.
Keywords: Self-focusing, Rectangular Airy vortex beam, Topological charge, Focus depth.
1 Introduction
The Airy beam has garnered significant attention in the field of optics due to its unique characteristics, including self-healing, non-diffraction and self-bending [ 1 – 3 ]. The Airy vortex beam is notable for carrying orbital angular momentum( OAM), which can greatly extend its application in various fields [ 4 – 8 ]. However, single Airy vortex beam faces challenges such as non-uniform intensity distribution and low intensity, which limit its application in efficient particle manipulation [ 9, 10 ]. RAVB not only significantly improves the uniformity of the intensity distribution, but also markedly increases peak intensity. Furthermore, RAVB exhibits a low intensity prior to focusing while rapidly achieving high peak intensity at the focus plane [ 11 ]. These characteristics offer a novel solution to mitigate potential thermal damage to cells that can arise from Gaussian beam used in laser medical treatment and laser cauterization applications [ 12 ]. The distinctive characteristics of RAVBs have attracted the attention of numerous researchers [ 13 – 18 ]. Yixian Qian et al. researched the propagation dynamics of generalized and symmetric Airy beam [ 13, 14 ]. Zhang et al. explored the propagation behavior of the rectangular symmetric Airy vortex beam within turbulent environments [ 17 ]. Wu et al. provided a detailed analysis of the propagation characteristics associated with the hollow RAVB [ 18 ]. Additionally, some researchers have
* Corresponding author: lqgan @ xmut. edu. cn investigated the propagation characteristics of the Airy vortex beam [ 8, 19 – 21 ]. However, to the best of our knowledge, there have been so far no references addressing the self-focusing characteristics of RAVB.
In this work, the RAVB self-focusing characteristics are numerically simulated by varying the linear factor( c), topological charge( l) of phase mask, and the wavelength( k) of the incident beam. Furthermore, our results demonstrate that RAVB self-focusing distribution can be controlled by appropriately selecting c, l and k. It is beneficial for applications that require laser beams with specialized profiles, such as beam shaping and microparticle manipulation.
2 Theory
2.1 The generation principle of RAVB
In the normalized space, the angular spectrum / 0 for a k-space Airy beam is given by [ 18 ]:
/ 0 ðkÞ ¼ expð�ak 2 Þ exp i 3 ðk3 � 3a 2 k � ia 3 Þ ð1Þ
Where a represents the attenuation factor, a 1, so the higher-order terms of a in equation( 1) can be neglected. The Airy beam can be generated by modulating a Gaussian beam through a cubic phase mask in a spatial light modulator( SLM) [ 13, 22 – 25 ]. In two dimensions, the cubic phase f 1 can be expressed as:
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