  Beam-helicity asymmetries and mass-differential cross sections have been
measured at the MAMI accelerator in Mainz for the photoproduction of neutral
and mixed-charge pion pairs in the reactions $\boldsymbol{\gamma}p\rightarrow
n\pi^0\pi^+$ and $\boldsymbol{\gamma}p\rightarrow p\pi^0\pi^0$ off free protons
and $\boldsymbol{\gamma}d \rightarrow (p)p\pi^0\pi^-$,
$\boldsymbol{\gamma}d\rightarrow (n)p\pi^0\pi^0$ and
$\boldsymbol{\gamma}d\rightarrow (n)n\pi^0\pi^+$,
$\boldsymbol{\gamma}d\rightarrow (p)n \pi^0\pi^0$ off quasi-free nucleons bound
in the deuteron for incident photon energies up to 1.4 GeV. Circularly
polarized photons were produced in bremsstrahlung processes of longitudinally
polarized electrons and tagged with the Glasgow-Mainz magnetic spectrometer.
The decay products (photons, protons, neutrons, and charged pions) were
detected in the $4\pi$ electromagnetic calorimeter composed of the Crystal Ball
and TAPS detectors. Using a full kinematic reconstruction of the final state,
excellent agreement was found between the results for free and quasi-free
protons, indicating that the quasi-free neutron results are also a close
approximation of the free-neutron results. Comparisons of the results to
predictions from model calculations suggest that especially the reaction
mechanisms in the production of the mixed-charge final states are still not
well understood, in particular at low incident photon energies in the second
nucleon resonance region.
