  $\textit{Aims.}$ We present a case study to demonstrate the potential of
multi-wavelength polarization measurements. The aim is to investigate the
effects that dichroic polarization and thermal re-emission have on tracing the
magnetic field in the interstellar medium (ISM). Furthermore, we analyze the
crucial influence of imperfectly aligned compact dust grains on the resulting
synthetic continuum polarization maps.$\\ \textit{Methods.}$ We developed an
extended version of the well-known 3D Monte-Carlo radiation transport code MC3D
for multi-wavelength polarization simulations running on an adaptive grid.We
investigated the interplay between radiation, magnetic fields and dust grains.
Our results were produced by post-processing both ideal density distributions
and sophisticated magnetohydrodynamic (MHD) collapse simulations with radiative
transfer simulations. We derived spatially resolved maps of intensity, optical
depth, and linear and circular polarization at various inclination angles and
scales in a wavelength range from 7 $\mu m$ to 1 $mm$.$\\ \textit{Results.}$ We
predict unique patterns in linear and circular polarization maps for different
types of density distributions and magnetic field morphologies for test setups
and sophisticated MHD collapse simulations. We show that alignment processes of
interstellar dust grains can significantly influence the resulting synthetic
polarization maps. Multi-wavelength polarization measurements allow one to
predict the morphology of the magnetic field inside the ISM. The interpretation
of polarization measurements of complex structures still remains ambiguous
because of the large variety of the predominant parameters in the ISM.
