Geometric phases in twisted strips

We compute the optical phase shifts between the left and the right-circularly polarized light after it traverses non-planar cyclic paths described by the boundary curves of twisted closed strips of various widths. While the phase shift increases monotonically with the width of the strip for multi-twisted closed strips, it has a minimum at a certain critical width for the Moebius. In our detailed study of various twisted strips, this intriguing property singles out the Moebius geometry. Possible applications are pointed out.