Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his construction seems to be very unclear because of heavy use of coördinates. Are there any more coördinate-free expositions of this topic?