Pancharatnam–Berry Phase for Polarized Light

We study the Pancharatnam–Berry geometric phase for polarized light beams of a non-closed cycle on the Poincare sphere. A geometric phase in different states of polarization is of great importance. It is shown that the Pancharatnam in-phase transformation can be regarded as a transformation of basis vectors. Using spherical trigonometry, we provide a new concise method of proving the Pancharatnam theorem.