  We theoretically investigate the optomechanically induced transparency (OMIT)
phenomenon in a N-cavity optomechanical system doped with a pair of Rydberg
atoms with the presence of a strong pump field and a weak probe field applied
to the Nth cavity. 2N-1(N<10) number OMIT windows can be observed in the output
field when N cavities coupled with N mechanical oscillators, respectively. But,
the mechanical oscillators coupled with different even-odd label cavities lead
to different effect on OMIT. On the other hand, two additional transparent
windows (extra resonances) are presented, if two Rydberg atoms are coupled with
the cavity field. With the DDI increasing, it is interesting that the extra
resonances move to right and the left extra resonance moves slowly than the
right one. During this process, Fano resonance is also shown on the output
field.
