  Quantum mechanically, photoionization can be fully described by the complex
photoionization amplitudes that describe the transition between the ground
state and the continuum state. Knowledge of the value of the phase of these
amplitudes has been a central interest in photoionization studies and newly
developing attosecond science, since the phase can reveal important information
about phenomena such as electron correlation. We present a new
attosecond-precision interferometric method of angle-resolved measurement for
the phase of the photoionization amplitudes, using two phase-locked Extreme
Ultraviolet pulses of frequency $\omega$ and $2\omega$, from a Free-Electron
Laser. Phase differences $\Delta \tilde \eta$ between one- and two-photon
ionization channels, averaged over multiple wave packets, are extracted for
neon $2p$ electrons as a function of emission angle at photoelectron energies
7.9, 10.2, and 16.6 eV. $\Delta \tilde \eta$ is nearly constant for emission
parallel to the electric vector but increases at 10.2 eV for emission
perpendicular to the electric vector. We model our observations with both
perturbation and \textit{ab initio} theory, and find excellent agreement. In
the existing method for attosecond measurement, Reconstruction of Attosecond
Beating By Interference of Two-photon Transitions (RABBITT), a phase difference
between two-photon pathways involving absorption and emission of an infrared
photon is extracted. Our method can be used for extraction of a phase
difference between single-photon and two-photon pathways and provides a new
tool for attosecond science, which is complementary to RABBITT.

Atomic Physics