Existence and Absence of Geometric Phases Due to Mode Transformations of High-Order Modes

We study the geometric phase that is acquired when the second-order mode of an optical beam undergoes a cyclic transformation. We ̄nd that a geometric phase appears when the initial and intermediate modes have di®erent quantities of orbital angular momentum. The phase is similar to the one measured previously for transformations of ̄rst-order modes. However, we ̄nd no accumulated geometric phase when the initial and intermediate modes have zero orbital angular momentum.