Measurements of the geometric phase of first-order optical Gaussian beams

Geometric phase is ubiquitous in systems that undergo cyclic transformations in either parameter or state space. Manifestations of this phase have been found in many physical systems. In optics, geometric phase has had an important effect, enhancing the way optical systems are analyzed, and stimulating the discovery of interesting effects and new applications. Two well-investigated manifestations of geometric phase in optics are the coiled optics phase, produced by sending an optical beam in a three-dimensional (coiled) path; and Pancharatnam phase, produced by changes in the state of polarization. A third phase, produced by transformations in the phase structure or mode of a Gaussian beam, has just begun to be explored [1]. High-order Gaussian beams have received much attention in the last few years due to their ability to carry orbital angular momentum [2]. Previous experimental investigations of this phase have been done in the microwave regime [3,4]. Here we present the first measurements of this geometric phase in the optical regime. The space of modes for first-order Gaussian beams is the Poincare sphere [5]. This space is similar to the space of states of polarization: the north and south poles are the two eigenstates of the paraxial wave equation with cylindrical symmetry: the Laguerre-Gauss functions LG 0 1 and LG 0-1 , respectively, while points in the equator represent different orientations of the eigenstates with rectangular symmetry: the Hermite-Gauss functions HG 10 or HG 01. This mode-space has recently been shown to have a SU(2) structure [6]. Transformations between LG and HG modes are effected by π-and π/2-converters [7,8]. These consist pairs of cylindrical lenses that alter the phase structure of the beam via the Gouy phase. A π-converter transforms a LG 0 1 mode into a LG 0-1 mode and vice versa. Thus when a beam in a LG 0 1 mode is sent through two consecutive π-converters the beam returns to its initial mode possessing a geometric phase that depends on the path followed in mode space [5]. When the transverse axes of the converters form an angle α this path is given by two great semicircles that form an angle 2α in the Poincare sphere, as shown by the solid line in Fig. 1a. If the initial mode were a LG 0-1 , the path would be the one shown by the dashed line in Fig. 1a. The geometric phases for the LG 0 1 and LG …