  Abridged. In the double detonation scenario for Type Ia supernovae (SNe Ia) a
detonation initiates in a shell of He-rich material accreted from a companion
star by a sub-Chandrasekhar-mass White Dwarf (WD). This shell detonation drives
a shock front into the carbon-oxygen (C/O) WD that triggers a secondary
detonation in the core. The core detonation results in a complete disruption of
the WD. Earlier studies concluded that this scenario has difficulties in
accounting for the observed properties of SNe Ia since the explosion ejecta are
surrounded by the products of explosive He burning in the shell. Recently, it
was proposed that detonations might be possible for much less massive He shells
than previously assumed. Moreover, it was shown that even detonations of these
minimum He shell masses robustly trigger detonations of the C/O core. Here we
present time-dependent multi-wavelength radiative transfer calculations for
models with minimum He shell mass and derive synthetic observables for both the
optical and {\gamma}-ray spectral regions. These differ strongly from those
found in earlier simulations of sub-Chandrasekhar-mass explosions in which more
massive He shells were considered. Our models predict light curves which cover
both the range of brightnesses and the rise and decline times of observed SNe
Ia. However, their colours and spectra do not match the observations. In
particular, their B-V colours are generally too red. We show that this
discrepancy is mainly due to the composition of the burning products of the He
shell of our models which contain significant amounts of Ti and Cr. Using a toy
model, we also show that the burning products of the He shell depend crucially
on its initial composition. This leads us to conclude that good agreement
between sub-Chandrasekhar-mass explosions and observed SNe Ia may still be
feasible but further study of the shell properties is required.
