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.

We present theoretical descriptions and measurements of optical beams carrying isolated polarization-singularity C-points. Our analysis covers all types of C-points, including asymmetric lemons, stars and monstars. They are formed by the superposition of a circularly polarized mode carrying an optical vortex and a fundamental Gaussian mode in the opposite state of polarization. The type of C-point can be controlled experimentally by varying two parameters controlling the asymmetry of the optical vortex. This was implemented via a superposition of modes with singly charged optical vortices of opposite sign, and varying the relative amplitude and phase. The results are in excellent agreement with the predictions.