A sounding-rocket program called the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) is proposed to be launched in the Summer of 2014. CLASP will observe the upper solar chromosphere in Ly-alpha (121.567 nm), aiming to detect the linear polarization signal produced by scattering processes and the Hanle effect for the first time. The CLASP needs a rotating half-waveplate and a polarization analyzer working at the Ly-alpha wavelength to measure the linear polarization signal. We select Magnesium Fluoride (MgF2) as a material of the optical components because of its birefringent property and high transparency at UV wavelength. We have confirmed that the reflection at the Brewster's Angle of MgF2 plate is a good polarization analyzer for the Ly-alpha line by deriving its ordinary refractive index and extinction coefficient along the ordinary and extraordinary axes. These optical parameters are calculated with a least-square fitting in such a way that the reflectance and transmittance satisfy the Kramers-Kronig relation. The reflectance and transmittance against oblique incident angles for the s-polarized and the p-polarized light are measured using the synchrotron beamline at the Ultraviolet Synchrotron Orbital Radiation Facility (UVSOR). We have also measured a retardation of a zeroth-order waveplate made of MgF2. The thickness difference of the waveplate is 14.57 um.This waveplate works as a half-waveplate at 121.74 nm. From this measurement, we estimate that a waveplate with the thickness difference of 15.71 um will work as a half-waveplate at the Ly-alpha wavelength. We have developed a rotating waveplate - polarization analyzer system called a prototype of CLASP polarimeter, and input the perfect Stokes Q and U signals. The modulation patterns that are consistent with the theoretical prediction are successfully obtained in both cases.