  A half-wave plate (HWP) is often used as a modulator to suppress systematic
error in the measurements of cosmic microwave background (CMB) polarization. A
HWP can also be used to measure circular polarization (CP) through its optical
leakage from CP to linear polarization. The CP of the CMB is predicted from
various sources, such as interactions in the Universe and extension of the
standard model. Interaction with supernova remnants of population III stars is
one of the brightest CP sources. Thus, the observation of the CP of CMB is a
new tool for searching for population III stars. In this paper, we demonstrate
the improved measurement of the leakage coefficient using the transmission
measurement of an actual HWP in the laboratory. We measured the transmittance
of linearly polarized light through the HWP used in \textsc{Polarbear} in the
frequency range of \SIrange{120}{160}{GHz}. We evaluate properties of the HWP
by fitting the data with a physical model using the Markov Chain Monte Carlo
method. We then estimate the band-averaged CP leakage coefficient using the
physical model. We find that the leakage coefficient strongly depends on the
spectra of CP sources. We thus calculate the maximum fractional leakage
coefficient from CP to linear polarization as $0.133 \pm 0.009$ in the
Rayleigh--Jeans spectrum. The nonzero value shows that \textsc{Polarbear} has
sensitivity to CP. Additionally, because we use the bandpass of detectors
installed in the telescope to calculate the band-averaged values, we also
consider systematic effects in the experiment.
