  Cloaking using complementary media was suggested by Lai et al. This was
proved in the quasistatic regime by H. M. Nguyen. One of the difficulties in
the study of this problem is the appearance of the localized resonance, i.e.,
the fields blow up in some regions and remain bounded in some others as the
loss goes to 0. To this end, the author introduced the technique of removing
localized singularity and used a standard three spheres inequality. This method
also works for the Helmholtz equation. However, it requires small size of the
cloaked region for large frequency due to the use of the (standard) three
spheres inequality. In this paper, we give a proof of cloaking using
complementary media in the finite frequency regime without imposing any
condition on the cloaked region; hence the cloak works for all frequency. To
successfully apply the approach of H.M. Nguyen, we establish a new three
spheres inequality. A modification of the cloaking setting to obtain illusion
optics is also discussed.
