A new four-fold symmetry double-layer chiral structure is designed, of which the operating wavelength band is in optical region. The transmission and reflection coefficients are obtained by numerical simulation, then the circular dichroism, polarization azimuth rotation angle, chiral parameter, refractive index of the structure are calculated. The results show that giant optical activity is found at the resonance wavelength region, and when the ellipticity is zero, the transmitted light becomes linearly polarized light completely, the polarization azimuth rotation angle reaches 55°. It is found that the negative indexes of the left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) waves can be realized without negative permittivity and permeability simultaneously. Most importantly, the negative index of the LCP can be achieved in wide band, which is not limited at resonance wavelength region. When the ellipticity is zero, the negative index also can be achieved. Taking into account the experimental manufacture, the substrate is added in the chiral structure. The simulated results demonstrate that resonance wavelength regions shift to long wavelength, and the polarization azimuth rotation angle still has 40° when the transmitted light becomes linearly polarized light completely.