Focusing of vortex beams: Lommel treatment

Area of Science:

• Optics and Photonics
• Mathematical Physics

Background:

• Vortex beams carry orbital angular momentum, enabling unique optical properties.
• Accurate modeling of focused vortex beams is crucial for applications in optical manipulation and microscopy.

Purpose of the Study:

• To analyze the focusing characteristics of vortex beams through a circular aperture lens.
• To express the amplitude and phase variations in the focal region using mathematical functions.

Main Methods:

• Paraxial scalar Debye regime analysis.
• Expression of focal region amplitude using higher-order Lommel functions of two variables.
• Utilizing recurrence relationships to simplify Lommel functions to lower orders.

Main Results:

• The amplitude in the focal region is naturally expressed using higher-order Lommel functions.
• These functions can be reduced to low-order Lommel functions via recurrence relations.
• Investigated phase variation reveals notable behavior of the Gouy phase anomaly.

Conclusions:

• The mathematical framework provides a simplified method for describing vortex beam focusing.
• The findings offer insights into the phase dynamics, particularly the Gouy phase anomaly, in focused vortex beams.