Geometric phases in higher-order transverse optical modes

We study the geometric origin of generalized Gouy phases in paraxial optical modes of arbitrary order. We focus on the specific case of cyclic beam transformations of non-astigmatic vortex beams, thereby, generalizing the well-known geometric phase shift for first-order beams with orbital angular momentum to modes of arbitrary order. Our method involves two pairs of bosonic ladder operators, which, analogous to the algebraic description of the quantum-mechanical harmonic oscillator in two dimensions, connect transverse modes of different order. Rather than studying the geometry of the infinite-dimensional space of higher-order modes, we focus on the space underlying the ladder operators. We identify overall phases of the ladder operators, thereby obtaining the phases of all higher-order modes, and show that the variation of these phases under optical elements and transformations has a geometric interpretation in terms of the other parameters involved.