We suggest a way to obtain a near-perfect nonmagnetic invisibility cloak structure. Within the particular example of cylindrical geometry and transverse magnetic polarization of light, we show that such a cloak can operate for a cylindrical object of very large diameter, keeping the total scattering cross section limited by 2λ/π. We also demonstrate that proposed cloak does not require extremely high values of dielectric permittivity, in contrast to the cloaks designed according to a transformation optics technique.
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