  We present a method of lensing reconstruction on the full sky, by extending
the optimal quadratic estimator proposed by Okamoto & Hu (2003) to the case
including the curl mode of deflection angle. The curl mode is induced by the
vector and tensor metric perturbations, and the reconstruction of the curl mode
would be a powerful tool to not only check systematics in the estimated
gradient mode but also probe any vector and tensor sources. We find that the
gradient and curl modes can be reconstructed separately, thanks to the
distinctive feature in the parity symmetry between the gradient and curl modes.
We compare our estimator with the flat-sky estimator proposed by Cooray et al
(2005). Based on the new formalism, the expected signal-to-noise ratio of the
curl mode produced by the primordial gravitational-waves and a specific model
of cosmic strings are estimated, and prospects for future observations are
discussed.
