  In this paper, we consider interaction-free measurement (IFM) with imperfect
interaction. In the IFM proposed by Kwiat et al., we assume that interaction
between an absorbing object and a probe photon is imperfect, so that the photon
is absorbed with probability 1-\eta (0\leq\eta\leq 1) and it passes by the
object without being absorbed with probability \eta when it approaches close to
the object. We derive the success probability P that we can find the object
without the photon absorbed under the imperfect interaction as a power series
in 1/N, and show the following result: Even if the interaction between the
object and the photon is imperfect, we can let the success probability P of the
IFM get close to unity arbitrarily by making the reflectivity of the beam
splitter larger and increasing the number of the beam splitters. Moreover, we
obtain an approximating equation of P for large N from the derived power series
in 1/N.
