  We performed wave-optics-based numerical simulations at mid-infrared
wavelengths to investigate how the presence or absence of entrance slits and
optical aberrations affect the spectral resolving power $R$ of a compact,
high-spectral-resolving-power spectrometer containing an immersion-echelle
grating. We tested three cases of telescope aberration (aberration-free,
astigmatism and spherical aberration), assuming the aberration budget of the
Space Infrared Telescope for Cosmology and Astrophysics (SPICA), which has a
20-$\mathrm{\mu m}$-wavelength diffraction limit. In cases with a slit, we
found that the value of $R$ at around 10--20 $\mathrm{\mu m}$ is approximately
independent of the assumed aberrations, which is significantly different from
the prediction of geometrical optics. Our results also indicate that
diffraction from the slit improves $R$ by enlarging the effective illuminated
area on the grating window and that this improvement decreases at short
wavelengths. For the slit-less cases, we found that the impact of aberrations
on $R$ can be roughly estimated using the Strehl ratio.
