  Active optical systems can give rise to intriguing phenomena and applications
that are not available in conventional passive systems. Structural rotation has
been widely employed to achieve non-reciprocity or time-reversal symmetry
breaking. Here, we examine the quasi-normal modes and scattering properties of
a two-dimensional cylindrical cavity under rotation. In addition to the
familiar phenomenon of Sagnac frequency shift, we observe the the hybridization
of the clockwise(CW) and counter-clockwise(CCW) chiral modes of the cavity
controlled by the rotation. The rotation tends to suppress one chiral mode
while amplify the other, and it leads to the variation of the far field. The
phenomenon can be understood as the result of a synthetic gauge field induced
by the rotation of the cylinder. We explicitly derived this gauge field and the
resulting Sagnac frequency shift. The analytical results are corroborated by
finite element simulations. Our results can be applied in the measurement of
rotating devices by probing the far field.
