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.
