  An extension of the input-output relation for a conventional Michelson
interferometric gravitational-wave detector is carried out to treat an
arbitrary coherent state for the injected optical beam. This extension is one
of necessary researches toward the clarification of the relation between
conventional gravitational-wave detectors and a simple model of a
gravitational-wave detector inspired by weak-measurements in [A.~Nishizawa,
Phys. Rev. A {\bf 92} (2015), 032123.]. The derived input-output relation
describes not only a conventional Michelson-interferometric gravitational-wave
detector but also the situation of weak measurements. As a result, we may say
that a conventional Michelson gravitational-wave detector already includes the
essence of the weak-value amplification as the reduction of the quantum noise
from the light source through the measurement at the dark port.
