Tight focusing Lorenz–Gaussian vortex beams modulated by power order space-variant phase

The aim of this paper is to investigate the focusing properties of linearly polarized Lorenz–Gaussian vortex beams modulated with power order space-variant (POSV) phase, which is based on vector diffraction theory. The results show that the shape of focal spot can be adjusted by changing the POSV parameter \(n\). In addition, the adjustable phase parameter \(C\) can flexibly control the position of the focus on the \(X\)-axis. Immediately after that, it is revealed that with the increase of topological charge number \(m\), the focused pattern can be separated along \(y = x\) direction. Even more interesting is the fact that the waist width \(\omega \) is inversely proportional to the length of the bar focusing peak. Specifically, when the value of waist width \(\omega \) is increased, the length of the strip-shaped focusing peak is continuously shortened. And the value of \(NA\) is inversely proportional to the size of the circular spot. A range of focal properties were acquired through the introduction of a space-variant phase with a power order. These new properties hold potential applications within various fields, including particle manipulation, optical modulation, and particle confinement.