Geometric interpretation of the Pancharatnam connection and non-cyclic polarization changes

If the state of polarization of a monochromatic light beam is changed in a cyclical manner, the beam acquires-in addition to the usual dynamic phase-a geometric phase. This geometric or Pancharatnam-Berry phase equals half the solid angle of the contour traced out on the Poincaré sphere. We show that such a geometric interpretation also exists for the Pancharatnam connection, the criterion according to which two beams with different polarization states are said to be in phase. This interpretation offers what is to our knowledge a new and intuitive method to calculate the geometric phase that accompanies non-cyclic polarization changes.