  Excitons, Coulomb-bound electron-hole pairs, are the fundamental excitations
governing the optoelectronic properties of semiconductors. While optical
signatures of excitons have been studied extensively, experimental access to
the excitonic wave function itself has been elusive. Using multidimensional
photoemission spectroscopy, we present a momentum-, energy- and time-resolved
perspective on excitons in the layered semiconductor WSe$_2$. By tuning the
excitation wavelength, we determine the energy-momentum signature of bright
exciton formation and its difference from conventional single-particle excited
states. The multidimensional data allows to retrieve fundamental exciton
properties like the binding energy and the exciton-lattice coupling and to
reconstruct the real-space excitonic distribution function via Fourier
transform. All quantities are in excellent agreement with microscopic
calculations. Our approach provides a full characterization of the exciton
properties and is applicable to bright and dark excitons in semiconducting
materials, heterostructures and devices.
