  To date, nearly two hundred planet-forming disks have been imaged with high
resolution. Our propensity to study bright and extended objects is however
biasing our view of the disk demography. In this work, we contribute to
alleviate this bias by analyzing fifteen disks targeted with VLT/SPHERE that
look faint in scattered light. Sources were selected based on a low far-IR
excess from the spectral energy distribution. The comparison with the ALMA
images available for a few sources shows that the scattered light surveyed by
these datasets is only detected from a small portion of the disk extent. The
mild anti-correlation between the disk brightness and the near-IR excess
demonstrates that these disks are self-shadowed: the inner disk rim intercepts
much starlight and leaves the outer disk in penumbra. Based on the uniform
distribution of the disk brightness in scattered light across all spectral
types, self-shadowing would act similarly for inner rims at a different
distance from the star. We discuss how the illumination pattern of the outer
disk may evolve with time. Some objects in the sample are proposed to be at an
intermediate stage toward bright disks from the literature with either no
shadow or with sign of azimuthally confined shadows.
