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We report an unexpected result of the anisotropy of the nonlinear optical response of carbon nanotubes, inherent to their chirality. Using a model based on the resolution of the semiconductor Bloch equations, we theoretically demonstrate that, upon irradiation with an intense linearly polarized laser pulse along the axial direction, chiral nanotubes exhibit an oscillating azimuthal current that is absent in achiral species. This current induces a magnetic field parallel to the axis of the nanotube that radiates like a loop antenna.