  We propose a three-mode optomechanical system to realize optical
nonreciprocal transmission with unidirectional amplification, where the system
consists of two coupled cavities and one mechanical resonator which interacts
with only one of the cavities. Additionally, the optical gain is introduced
into the optomechanical cavity. It is found that for a strong optical input,
the optical transmission coefficient can be greatly amplified in a particular
direction and suppressed in the opposite direction. The expressions of the
optimal transmission coefficient and the corresponding isolation ratio are
given analytically. Our results pave a way to design high-quality nonreciprocal
devices based on optomechanical systems.
