A lossless acoustic half-bipolar cylindrical cloak that has an exposed bottom is considered. Here, we show that a cloak that includes a complementary region including a negative-index medium inside of the cloaking shell works in the illumination direction independently even in the presence of the exposed bottom of the structure. This is due to the fact that the phase velocity of the wave in the normal direction can be cancelled in the presence of a boundary containing a negative-index medium that reduces scattering significantly.

While, it was shown that for the lossless acoustic cloak the illumination direction independent cloaking can be achieved by employing the structure with compressed geometry and complementary media, the effect of the material loss have not been fully explored yet. Here, we show that realistic cloaking materials with moderate loss still works when complementary media is introduced but with the attenuated back scattering waves.