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J. Eur. Opt. Society-Rapid Publ. 22, 8( 2026)
Figure 1. Phase diagram and intensity distributions with different distances of RAVB where c = 7, l = 1, k = 632.8 nm.
f 1 ¼ðx 3 þ y 3 Þ = 3
Where x, y denote the values of the abscissa and ordinate, respectively. To effectively manipulate particles and achieve a uniform light spot distribution, the absolute value operation of the cubic phase is usually performed, and a vortex phase lh is added. l denotes the topological charge, and h represents the vortex angle.
A phase-controlled linear factor c is introduced, which also serves as a scaling factor for both the x and y coordinates. Adjusting c can control the size of the central rectangular region in the phase diagram. A larger value of c results in a smaller central rectangular region, and vice versa. So, a rectangular vortex cubic phase f can be expressed as follows:
x 3 f ¼ c þ
y 3
c = 3 þ l arctanðx þ iyÞ ð3Þ
Consequently, the angular spectrum / is expressed as: ð2Þ
/ ðx; yÞ ¼ exp½�aðx 2 þ y 2 ÞŠ expðif Þ ð4Þ
A RAVB is performed by a Fourier transform on this expression /. Theopticalfield distribution at the initial plane( z = 0) is described by:
wðx; y; z ¼ 0Þ ¼ 1 2p
Z 1 Z 1
�1 �1 exp½�aðx 2 þ y 2 ÞŠ expðif Þ exp½iðxx 0 þ yy 0 Šdx 0 dy 0
2.2 The general transmission situation of RAVB
RAVB not only addresses the issue of non-uniform intensity distribution but also enhances the beam peak intensity. The RAVB propagation characteristics of the three-dimensional space can be simulated using the Fresnel diffraction integral formula [ 11 ]:
wðx; y; z
Þ ¼ expðikzÞ ZZ ikz wðx; y; 0Þexp½ik
R 2
ðx � x00 Þ 2 þðy � y 00 Þ 2 Šdx 00 dy 00
2z ð5Þ
ð6Þ
In this study, we set a = 0.01. Where l = 1, k = 632.8nm, the RAVB field distribution characteristics at various propagation positions are numerically simulated. Figure 1a illustrates the spiral rectangular phase diagram. Figure 1b presents intensity distribution at the initial plane. RAVB intensity distribution exhibits a rectangular symmetric distribution, and each lobe is separated from the others. Due to OAM carried by the optical vortex, there is a rotation effect on the entire beam, resulting in a hollow focusing channel formation. Figures 1c – 1e depict intensity distributionsatpropagationdistancesofz = 28 cm, 35 cm and 77 cm, respectively.
As the propagation distance increases, the transverse self-acceleration property of the Airy beam results in a gradual convergence of its four main lobes, and the hollow structure continuously shrinks. They converge at a single point at some location( as z = 28 cm), where the peak intensity achieves its maximum value, we call the corresponding plane as the focus plane. The focusing effect formed as the result of Airy beams self-acceleration property is called as Airy beam self-focusing. The distance between the focus plane and the initial plane is defined as the focus depth.
As propagation distance continues to increase, the separation between the rectangular Airy beam and the on-axis vortex gradually expands. Consequently, the influence of vortex-carrying energy on the beam diminishes and RAVB rotational angle decreases. This rotational phenomenon gradually dissipates as the RAVB propagates. It is noteworthy that reformation of the four main lobes occurs while maintaining an intact structure throughout. Based on equation( 6) and the RAVB transmission characteristics, it can be known that the focusing characteristics of the rectangular Airy vortex beam can be adjusted via c, l and k. Here, we study the changes in the focusing characteristics of RAVB via c, l and k.
3 Experiments and simulations
3.1 The influence of phase modulation linear factor( c) on self-focusing characteristics of RAVB
In the computation of the symmetric cubic phase mask, c can adjust the dimensions of the central rectangular region within the phase mask, thereby affecting both beam’ s propagation and self-focusing behavior. We set l = 2, k = 632.8nm, Figures 2a1 – 2a4 depict the rectangular vortex cubic phase maps which calculated using equation( 3) for c = 4,5,6,8, respectively. Figures 2b1 – 2a4 and Figures 2c1 – 2c4 display the intensity distributions of RAVB in the initial plane and focus plane, respectively. Figures 2d1 – 2d4 present three-dimensional intensity distributions in the focus plane. As depicted in Figure 2, an