Complete characterization of phase and amplitude of bichromatic XUV light

The phase-contrast microscopy technique, a milestone in imaging techniques introduced in the ‘30s, attracted a wide interest in the optical imaging community. Such a technique pointed out the importance of considering also the phase of a propagating wave through matter, and not only its amplitude as usually detected by a camera or by the human eye. The possibility of gaining complementary information about the light passing through a specimen greatly increased the quality and resolution of the images.
Retrieving the amplitude and phase of light waves in the optical regime is nowadays a well-established technique, used in several fields from medicine to biology where the size of the object under analysis is of the order of several microns; however when the specimen under investigation is much smaller, like atoms or molecules, optical light is no longer useful and Ultraviolet (UV) or X-ray light is used instead. 
Thanks to the advances in computer technology, complex calculations can now be performed to simulate the temporal evolution of atoms and molecules upon light stimulation, i.e. tracking excitation or de-excitation processes with unprecedented precision. However in order to get an accurate simulation of the problem under investigation a complete knowledge of the experimental parameters is required, including the absolute phase of the incident light. 
Retrieving amplitude and phase in the Extreme UltraViolet (EUV) and X-ray regime is extremely challenging.  Even more challenging is the measurement of the phase of the produced light from large scale facilities like the seeded Free Electron Laser FERMI in Trieste. FERMI is able to produce intense, multi-color pulses in the X-ray and EUV regime with a well-defined phase relationship between them. Knowledge of the absolute phase of the two pulses is thus an important parameter that needs to be measured. 
In a recent experimental campaign, with the goal of extracting as much information as possible from an experiment, we have been able to measure the amplitude and phase relationship of a wavelength and its second harmonic imprinted onto the outgoing electron wave from a sample of He atoms. A direct comparison with the theoretical calculations was performed and excellent agreement found, validating the experimental technique and confirming the validity of the theoretical calculations, while opening a route for even more complex light – matter investigations. 
Previously it has been shown that FERMI can generate coherent, multi-color pulses to perform coherent control experiments and detect novel decay process. In such methods the phase between pulses of different colors was controlled and tuned with a precision of a few attoseconds (millionths of millionths of millionths of seconds), and precise knowledge of the amplitude and absolute phase is needed.
The method we present to extract such parameters relies on varying the relative phase of two pulses of commensurate frequency (wand 2w) and measuring the change in the photoelectron angular distribution (PAD) emitted from a gas target, specifically He, from two interfering pathways: a one-photon process and a two-photon process, as sketched in Fig. 1.
 

Figure 1.    Schematic process of interference between the partial photoelectron waves created by single- and two-photon ionizations. The (short) red arrows mark the fundamental photon, while the (long) blue one indicates the second harmonic. The horizontal lines show the lowest energy levels of helium. (PHYSICAL REVIEW LETTERS 123, 213904 (2019) © 2019 American Physical Society)
 

We used a velocity map imaging (VMI) spectrometer to measure the PAD, traditionally written as a series expansion of Legendre polynomials, with coefficients ß_i. From the VMI data, we extracted ß₁, ß₂, ß₃, and b₄as a function of the experimental relative phase between the two pulses. An example of the results at 14.3 eV photon energy is shown in Fig. 2, where ß₂, ß₃, ß₄, and ß₁– 2/3*ß₃ are reported. As expected from the theoretical calculations, ß₃ and ß₁– 2/3*ß₃ oscillate while ß₂ and ß₄ are constant, confirming the agreement with the simulations and experimental data. 

Figure 2.    β parameters as a function of ϕ. Markers, βparameters of dataset A as a function of phase; curves, cosine (constant) fit, odd (even) β; blue triangles, β₁− 2β₃/3; black circles, β₂; green inverted triangles, β₃; red squares, β₄. Error bars show standard errors of least-squares fitting using the model described in Eq. (3). Linearly polarized light, λ= 86.7 nm, λ/2= 43.4 nm (PHYSICAL REVIEW LETTERS 123, 213904 (2019) © 2019 American Physical Society).
 

This research was conducted by the following research team:

Michele Di Fraia,¹Oksana Plekan,¹Carlo Callegari,¹Kevin C. Prince,^{1, 2}Luca Giannessi,^{1, 3}Enrico Allaria,¹Laura Badano,¹Giovanni De Ninno,^1,4Mauro Trovò,¹Bruno Diviacco,¹David Gauthier,¹ Najmeh Mirian,¹Giuseppe Penco,¹Primoz Rebernik,¹Simone Spampinati,¹Carlo Spezzani,¹Giulio Gaio,¹Yuki Orimo,⁵Oyunbileg Tugs,⁵Takeshi Sato,^{5, 6,7}Kenichi L. Ishikawa,^{5, 6,7}Paolo Antonio Carpeggiani,⁸Tamas Csizmadia,⁹MiklósFüle⁹,Giuseppe Sansone,¹⁰ Praveen M. Kumar,¹⁰Alessandro D'Elia,^11,12Tommaso Mazza,¹³Michael Meyer,¹³ Elena V. Gryzlova,¹⁴Alexei N. Grum-Grzhimailo,¹⁴Daehyun You,¹⁵and Kiyoshi Ueda¹⁵.

¹ Elettra - Sincrotrone Trieste S.C.p.A., Trieste, Italy
² Centre for Translational Atomaterials, Swinburne University of Technology, Melbourne, Australia
³ INFN-Laboratori Nazionali di Frascati, Frascati, Rome, Italy
⁴ Laboratory of Quantum Optics, University of Nova Gorica, Nova Gorica, Slovenia
⁵ Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
⁶ Photon Science Center, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
⁷ Research Institute for Photon Science and Laser Technology, The University of Tokyo, Tokyo, Japan
⁸ Institut für Photonik, Technische Universität, Wien, Vienna, Austria
⁹ ELI-ALPS, ELI-HU Non-Profit Ltd., Szeged, Hungary
¹⁰ Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany
¹¹ Department of Physics, University of Trieste, Trieste, Italy
¹² IOM-CNR, Laboratorio Nazionale TASC, Basovizza, Trieste, Italy
¹³ European XFEL GmbH, Schenefeld, Germany
¹⁴ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia
¹⁵ Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai, Japan

Contact persons:

Kevin C. Prince, email: