  Diverse protoplanetary disk morphology can result from planet-disk
interaction, suggesting planetary presence. To date, most scattered light
imaging campaigns have probed polarized light, which is only a fraction of the
total light and not very sensitive to planets. To observe and characterize
protoplanetary disk systems in the near-infrared in both polarized and total
intensity light, we carried out an unprecedented study of scattering properties
of disks, as well as of any planetary companions. Using SPHERE with
star-hopping at the Very Large Telescope, we observed 29 disk hosts and their
reference stars in $K_s$-band polarized light. We extracted disks in total
intensity by adopting the data imputation concept with sequential non-negative
matrix factorization (DI-sNMF). We obtained high-quality disk images in total
intensity for 15 systems and in polarized light for 23.
  For well-recovered disks in polarized light and total intensity, we
parameterized the polarization fraction phase functions using scaled beta
distribution: the peak of polarization fraction tentatively correlates with the
peak scattering angle, which could be reproduced using certain compact dust,
yet more detailed modeling studies are needed. We investigated the empirical
DI-sNMF detectability of disks using logistic regression: total intensity
detectability of disks primarily depends on host star brightness. For disks
with SPHERE data in $Y$-/$J$-/$H$-band, we summarized their polarized color at
~90 deg scattering angle: most of disks are blue in polarized $J-K_s$ color,
and they are relatively redder as stellar luminosity increases, indicating
larger scatterers. High-quality disk imagery in both total intensity and
polarized light thus allows for disk characterization in polarization fraction,
and reduces the confusion between disk and planetary signals.
