  Radar and optical simultaneous observations of meteors are important to
understand the size distribution of the interplanetary dust. However, faint
meteors detected by high power large aperture radar observations, which are
typically as faint as 10 mag. in optical, have not been detected until recently
in optical observations, mainly due to insufficient sensitivity of the optical
observations. In this paper, two radar and optical simultaneous observations
were organized. The first observation was carried out in 2009 to 2010 using
Middle and Upper Atmosphere Radar (MU radar) and an image-intensified CCD
camera. The second observation was carried out in 2018 using the MU radar and a
mosaic CMOS camera, Tomo-e Gozen, mounted on the 1.05-m Kiso Schmidt Telescope.
In total, 331 simultaneous meteors were detected. The relationship between
radar cross sections and optical V-band magnitudes was well approximated by a
linear function. A transformation function from the radar cross section to the
V-band magnitude was derived for sporadic meteors. The transformation function
was applied to about 150,000 meteors detected by the MU radar in 2009--2015,
large part of which are sporadic, and a luminosity function was derived in the
magnitude range of $-1.5$ to $9.5$ mag. The luminosity function was well
approximated by a single power-law function with the population index of $r =
3.52{\pm}0.12$. The present observation indicates that the MU radar has
capability to detect interplanetary dust of $10^{-5}$ to $10^{0}$ g in mass as
meteors.
