We demonstrate complete tunneling of light through large-scale mu-negative media, which has negative permeability but positive permittivity, by constructing a quasi-one-dimensional structure with side branches. For the structure with a single side branch, there always exists a transmission peak which can be easily tuned by varying the parameters of the side branch. For the structure with periodic array of side branches, the transmission peak is enlarged to a band, which exhibits left-handedness, and can be tuned by changing the distance between two neighboring side branches and the length of the side branch.
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