  Multi-wave band synchrotron linear polarization of gamma-ray burst (GRB)
afterglows is studied under the assumption of an anisotropic turbulent magnetic
field with a coherence length of the plasma skin-depth scale in the downstream
of forward shocks. We find that for typical GRBs, in comparison to the optical
polarization, the degree of radio polarization shows a similar temporal
evolution but a significantly smaller peak value. This results from differences
in observed intensity image shapes between the radio and optical bands. We also
show that the degree of the polarization spectrum undergoes a gradual variation
from the low- to the high-polarization regime above the intensity of the
spectral peak frequency, and that the difference in polarization angles in the
two regimes is zero or 90 degrees. Thus, simultaneous multi-wave band
polarimetric observations of GRB afterglows would be a new determinative test
of the plasma-scale magnetic field model. We also discuss theoretical
implications from the recent detection of radio linear polarization in GRB
171205A with ALMA and other models of magnetic field configuration.
