The experimental measurement of Berry's phase (or geometric phase) of adiabatic, cyclic and adiabatic, and noncyclic evolution curves are elaborated in this thesis. We have introduced experimental methods to observe Berry's phase appearing in adiabatic, cyclic and adiabatic, and noncyclic cases. And also we showed that there are two different observable geometrical phases for adiabatic and noncyclic case, which have the same value if the evolution is cyclic. The first is the relative phase difference between the phases for two different paths in the projective Hilbert Space, which have the same end points, and the second is an absolute value of Berry's phase for an evolution curve. In the later case the reference state to measure the phase change should be equal to the initial state. These consideration for noncyclic evolution curves has a number of applications in quantum optics. We report the experimental results of geometric phase in adiabatic and the noncyclic evolution curve by using single mode optical fibers in this thesis.
