  In this paper we study cavity optomechanical systems in which the position of
a mechanical oscillator modulates both the resonance frequency (dispersive
coupling) and the linewidth (dissipative coupling) of a cavity mode. Using a
quantum noise approach we calculate the optical damping and the
optically-induced frequency shift. We find that dissipatively coupled systems
feature two parameter regions providing amplification and two parameter regions
providing cooling. To investigate the strong-coupling regime, we solve the
linearized equations of motion exactly and calculate the mechanical and optical
spectra. In addition to signatures of normal-mode splitting that are similar to
the case of purely dispersive coupling, the spectra contain a striking feature
that we trace back to the Fano line shape of the force spectrum. Finally, we
show that purely dissipative coupling can lead to optomechanically-induced
transparency which will provide an experimentally convenient way to observe
normal-mode splitting.
