Wigner representation and geometric transformations of optical orbital angular momentum spatial modes

Area of Science:

• Quantum Optics
• Mathematical Physics
• Information Optics

Background:

• Orbital angular momentum (OAM) in optical spatial modes is crucial for advanced optical applications.
• Wigner representations offer a phase-space description of quantum states, but exact forms for OAM modes are challenging.
• The SU(2) Lie group and Poincaré sphere are key mathematical frameworks in optics.

Purpose of the Study:

• To derive an exact closed-form Wigner representation for optical spatial modes carrying orbital angular momentum.
• To explore the properties and transformations of these modes within a phase-space framework.
• To elucidate the development of geometric phases during mode transformations.

Main Methods:

• Exploiting the SU(2) Lie-group algebra associated with the Poincaré sphere of OAM modes.
• Applying phase-space formalism to derive the Wigner representation.
• Analyzing orthogonality relations and observables within this representation.

Main Results:

• An exact, closed-form Wigner representation for optical spatial modes with orbital angular momentum was successfully derived.
• Orthogonality relations and key observables for these OAM states were obtained in the phase-space picture.
• The study elucidated the behavior of geometric phases during transformations of these optical modes.

Conclusions:

• The derived Wigner representation provides a powerful tool for analyzing OAM states in phase space.
• This work offers new insights into the mathematical structure and physical properties of OAM-carrying optical modes.
• The findings facilitate advancements in quantum information processing and optical communication technologies.