  We present a novel approach for the numerical solution of problems of
diffraction by open arcs in two dimensional space. Our methodology relies on
composition of {\em weighted versions} of the classical integral operators
associated with the Dirichlet and Neumann problems (TE and TM polarizations,
respectively) together with a generalization to the open-arc case of the well
known closed-surface Calder\'on formulae. When used in conjunction with
spectrally accurate discretization rules and Krylov-subspace linear algebra
solvers such as GMRES, the new second-kind TE and TM formulations for open arcs
produce results of high accuracy in small numbers of iterations and short
computing times---for low and high frequencies alike.
