  The critical properties of functional materials and nanoscale devices often
originate from the coexistence of different thermodynamic phases and / or
oxidization states, but sample makeup is seldom completely known a priori.
Coherent diffractive imaging (CDI) provides the spatial resolution needed to
observe nanoscale coexistence while returning the full amplitude and phase
information of an object, but to date lacks the spectral information necessary
for composition identification. Here we demonstrate CDI spectroscopy (CDIS),
acquiring images of the prototypical quantum material vanadium oxide across the
vanadium L2,3 and oxygen K X-ray absorption edges with nanometer scale
resolution. Using the hyperspectral X-ray image we show coexistence of multiple
oxidization states and phases in a single sample and extract the full complex
refractive index of V2O5 and the monoclinic insulating and rutile conducting
phases of VO2. These results constrain the role of hidden phases in the
insulator-to-metal transition in VO2.
