  The spectroscopy of faint planetary-mass companions to nearby stars is one of
the main challenges that new-generation high-contrast spectro-imagers are going
to face. In a previous work we presented a long slit coronagraph (LSC), for
which the presence of a slit in the coronagraphic focal plane induces a complex
distribution of energy in the Lyot pupil-plane that cannot be easily masked
with a binary Lyot stop. To alleviate this concern, we propose to use a pupil
apodization to suppress diffraction, creating an apodized long slit coronagraph
(ALSC). After describing how the apodization is optimized, we demonstrate its
advantages with respect to the CLC in the context of SPHERE/IRDIS long slit
spectroscopy (LSS) mode at low-resolution with a 0.12" slit and 0.18"
coronagraphic mask. We perform different sets of simulations with and without
aberrations, and with and without a slit to demonstrate that the apodization is
a more appropriate concept for LSS, at the expense of a significantly reduced
throughput (37%) compared to the LSC. Then we perform detailed end-to-end
simulations of the LSC and the ALSC that include realistic levels of
aberrations to obtain datasets representing 1h of integration time on stars of
spectral types A0 to M0 located at 10 pc. We insert spectra of planetary
companions at different effective temperatures (Teff) and surface gravities
(log g) into the data at angular separations of 0.3" to 1.5" and with contrast
ratios from 6 to 18 mag. Using the SD method to subtract the speckles, we show
that the ALSC brings a gain in sensitivity of up to 3 mag at 0.3" with respect
to the LSC, which leads to a much better spectral extraction below 0.5". In
terms of Teff, we demonstrate that at small angular separations the limit with
the ALSC is always lower by at least 100K, inducing an increase of sensitivity
of a factor up to 1.8 in objects' masses at young ages. [Abridged]
