J. Eur. Opt. Society-Rapid Publ. 22, 8( 2026) 83
Figure 4. Intensity distribution on the focused planes of RAVB. Where c = 8 and l = 1,( a) k = 380 nm;( b) k = 457.9 nm;( c) k = 514.5 nm;( d) k = 632.8 nm.
increase in the value of c leads to a reduction in the dimensions of the central rectangular region within the phase mask. Consequently, the generated RAVB exhibits a larger spot size at the initial plane. In contrast, at the focus plane, RAVB dispalys a reduced focus spot size, an increased focus depth, an enhanced peak intensity, and a more uniform intensity distribution. Specifically, as c increases from 4 to 8, the focusing depth increases from 8 cm to 41 cm, and the corresponding peak intensity is amplified to 50 times.
As c increases, the diminished dimensions of the rectangular structure at the center of the phase mask lead to an improved beam focusing capability, and energy becomes increasingly concentrated toward this center. The increase in focus depth indicates that achieving complete self-focusing for RAVB requires longer propagation distances. This phenomenon can be primarily attributed to the self-bending property of Airy beams.
When an Airy beam propagates along the z-axis, its main lobe follows a parabolic trajectory within the x – y plane. This trajectory can be described as [ 25 ]:
x ¼ k2
16p 2 x 3
0 z 2; y ¼ k2
16p 2 y 3
0
The path of this main lobe is influenced by both transverse scale x 0 and wavelength k. For the same self-bending displacement x, as the transverse scale x 0 increases, it necessitates longer propagation distance z. The same holds true for the y-axis. When generating RAVB through phase modulation techniques, adjustments to transverse scale are achieved via c, thereby altering the dimensions of the central rectangular region within phase mask. Consequently, as c increases, the dimensions of the central rectangle in the phase mask decreases, and the focus depth z increases.
3.2 The influence of topological charge( l) on self-focusing characteristics of RAVB
z 2 ð7Þ
In this section, we set c = 8, k = 632.8 nm, the rectangular vortex cubic phase maps calculated using equation( 3) are presented in Figures 3a1 – 3a4 for l = 1, 3, 5, 7, respectively. It is observed that l does not affect the area of the central rectangular region within the phase map, but influences
Figure
5. The focus peak intensity and focus depth varying with k. Where c = 8 and l = 1.
beam rotation. The number of vortices can be distinctly identified in these phase maps. Figure 3b1 – 3b4 depict the RAVB intensity distributions at the initial plane. The hollow area of intensity distribution at the initial plane enlarges as l increases. Figures 3c1 – 3c4 present the RAVB intensity distributions at focus planes z = 35 cm, 41 cm, 44 cm and 46 cm, respectively. I 0 represents the maximum value of the light field intensity on the initial surface. As l increases, RAVB requires a longer propagation distance to achieve focusing, resulting in a gradual enlargement of the focus spot size. Figures 3d1 – 3d4 depict the three-dimensional intensity distributions at the focus plane. The peak intensity reveals that the intensity distribution of the focal spot is notably non-uniform, even as the parameter l varies only minimally.
The beam exhibits an increase in orbital angular momentum with higher l, which leads to greater energy dispersion. This phenomenon results in more pronounced energy spreading during propagation, thereby complicating the focusing at the focus plane. Due to the self-focusing capability of the beam during its propagation, this outward spreading is partially mitigated. Consequently, as l increases, both the focus depth and the size of focus spot