Demagnifying gravitational lenses toward hunting a clue of exotic matter
  and energy
  We examine a gravitational lens model inspired by modified gravity theories
and exotic matter and energy. We study an asymptotically flat, static, and
spherically symmetric spacetime that is modified in such a way that the
spacetime metric depends on the inverse distance to the power of positive $n$
in the weak-field approximation. It is shown analytically and numerically that
there is a lower limit on the source angular displacement from the lens object
to get demagnification. Demagnifying gravitational lenses could appear,
provided the source position $\beta$ and the power $n$ satisfy $\beta >
2/(n+1)$ in the units of the Einstein ring radius under a large-$n$
approximation. Unusually, the total amplification of the lensed images, though
they are caused by the gravitational pull, could be less than unity. Therefore,
time-symmetric demagnification parts in numerical light curves by gravitational
microlensing (F.Abe, Astrophys. J. 725, 787, 2010) may be evidence of an Ellis
wormhole (being an example of traversable wormholes), but they do not always
prove it. Such a gravitational demagnification of the light might be used for
hunting a clue of exotic matter and energy that are described by an equation of
state more general than the Ellis wormhole case. Numerical calculations for the
$n=3$ and 10 cases show maximally $\sim 10$ and $\sim 60$ percent depletion of
the light, when the source position is $\beta \sim 1.1$ and $\beta \sim 0.7$,
respectively.
