  Proton imaging is a powerful technique for imaging electromagnetic fields
within an experimental volume, in which spatial variations in proton fluence
are a result of deflections to proton trajectories due to interaction with the
fields. When deflections are large, proton trajectories can overlap, and this
nonlinearity creates regions of greatly increased proton fluence on the image,
known as caustics. The formation of caustics has been a persistent barrier to
reconstructing the underlying fields from proton images. We have developed a
new method for reconstructing the path-integrated magnetic fields which begins
to address the problem posed by caustics. Our method uses multiple proton
images of the same object, each image at a different energy, to fill in the
information gaps and provide some uniqueness when reconstructing caustic
features. We use a differential evolution algorithm to iteratively estimate the
underlying deflection function which accurately reproduces the observed proton
fluence at multiple proton energies simultaneously. We test this reconstruction
method using synthetic proton images generated for three different,
cylindrically symmetric field geometries at various field amplitudes and levels
of proton statistics, and present reconstruction results from a set of
experimental images. The method we propose requires no assumption of deflection
linearity and can reliably solve for fields underlying linear, nonlinear, and
caustic proton image features for the selected geometries, and is shown to be
fairly robust to noise in the input proton intensity.
