We found what caused this problem, i.e. just RBW setting...
The value I used is too small, and here I updates the data and attaches the fine plots.
The set-up for the experiment is exactly same as the original one.
WIth this modification, we will do the measurements regarding modulation index which seems related to this mistake. 
 

Anyway, we still have the asymmetry b/w upper and lower sideband which is not negligible.
We need to consider this in the data analysis of subsequent measurements.