  With transfer matrix theory, we study the transmission of a distributed Bragg
reflector (DBR) with two dielectric gratings on top and on the bottom. Owing to
the diffraction of the two gratings, the transmission shows a comb-like
spectrum which red shifts with increasing the grating period during the
forbidden band of the DBR. The number density of the comb peaks increases with
increasing the number of the DBR cells, while the ratio of the average full
width at half maximum (FWHM) of the transmission peaks in the transmission comb
to the corresponding average free spectral range, being about 0.04 and 0.02 for
the TE and TM incident waves, is almost invariant. The average FWHM of the TM
waves is about half of the TE waves, and both they could be narrower than 0.1
nm. In addition, the transmission comb peaks of the TE and TM waves can be
fully separated during certain waveband. We further prove that the transmission
comb is robust against the randomness of the heights of the DBR layers, even
when a 15\% randomness is added to their heights. Therefore, the proposed
structure is a candidate for a multichannel narrow-band filter or a
multichannel polarizer.
