## Abstract

Intense, mutually coherent beams of multiharmonic extreme ultraviolet light can now be created using seeded free-electron lasers, and the phase difference between harmonics can be tuned with attosecond accuracy. However, the absolute value of the phase is generally not determined. We present a method for determining precisely the absolute phase relationship of a fundamental wavelength and its second harmonic, as well as the amplitude ratio. Only a few easily calculated theoretical parameters are required in addition to the experimental data.

Original language | English |
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Article number | 213904 |

Journal | Physical review letters |

Volume | 123 |

Issue number | 21 |

DOIs | |

Publication status | Published - 2019 Nov 22 |

## ASJC Scopus subject areas

- Physics and Astronomy(all)

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*Physical review letters*,

*123*(21), [213904]. https://doi.org/10.1103/PhysRevLett.123.213904

**Complete Characterization of Phase and Amplitude of Bichromatic Extreme Ultraviolet Light.** / Di Fraia, Michele; Plekan, Oksana; Callegari, Carlo; Prince, Kevin C.; Giannessi, Luca; Allaria, Enrico; Badano, Laura; De Ninno, Giovanni; Trovò, Mauro; Diviacco, Bruno; Gauthier, David; Mirian, Najmeh; Penco, Giuseppe; Ribič, PrimoŽ Rebernik; Spampinati, Simone; Spezzani, Carlo; Gaio, Giulio; Orimo, Yuki; Tugs, Oyunbileg; Sato, Takeshi; Ishikawa, Kenichi L.; Carpeggiani, Paolo Antonio; Csizmadia, Tamás; Füle, Miklós; Sansone, Giuseppe; Kumar Maroju, Praveen; D'Elia, Alessandro; Mazza, Tommaso; Meyer, Michael; Gryzlova, Elena V.; Grum-Grzhimailo, Alexei N.; You, Daehyun; Ueda, Kiyoshi.

Research output: Contribution to journal › Article › peer-review

*Physical review letters*, vol. 123, no. 21, 213904. https://doi.org/10.1103/PhysRevLett.123.213904

**Complete Characterization of Phase and Amplitude of Bichromatic Extreme Ultraviolet Light**. In: Physical review letters. 2019 ; Vol. 123, No. 21.

}

TY - JOUR

T1 - Complete Characterization of Phase and Amplitude of Bichromatic Extreme Ultraviolet Light

AU - Di Fraia, Michele

AU - Plekan, Oksana

AU - Callegari, Carlo

AU - Prince, Kevin C.

AU - Giannessi, Luca

AU - Allaria, Enrico

AU - Badano, Laura

AU - De Ninno, Giovanni

AU - Trovò, Mauro

AU - Diviacco, Bruno

AU - Gauthier, David

AU - Mirian, Najmeh

AU - Penco, Giuseppe

AU - Ribič, PrimoŽ Rebernik

AU - Spampinati, Simone

AU - Spezzani, Carlo

AU - Gaio, Giulio

AU - Orimo, Yuki

AU - Tugs, Oyunbileg

AU - Sato, Takeshi

AU - Ishikawa, Kenichi L.

AU - Carpeggiani, Paolo Antonio

AU - Csizmadia, Tamás

AU - Füle, Miklós

AU - Sansone, Giuseppe

AU - Kumar Maroju, Praveen

AU - D'Elia, Alessandro

AU - Mazza, Tommaso

AU - Meyer, Michael

AU - Gryzlova, Elena V.

AU - Grum-Grzhimailo, Alexei N.

AU - You, Daehyun

AU - Ueda, Kiyoshi

N1 - Funding Information: Di Fraia Michele 1 Plekan Oksana 1 https://orcid.org/0000-0001-5491-7752 Callegari Carlo 1 https://orcid.org/0000-0002-5416-7354 Prince Kevin C. 1,2 ,* Giannessi Luca 1,3 Allaria Enrico 1 Badano Laura 1 De Ninno Giovanni 1,4 Trovò Mauro 1 Diviacco Bruno 1 Gauthier David 1 ,‡ Mirian Najmeh 1 Penco Giuseppe 1 Ribič Primož Rebernik 1 Spampinati Simone 1 Spezzani Carlo 1 Gaio Giulio 1 Orimo Yuki 5 Tugs Oyunbileg 5 Sato Takeshi 5,6,7 https://orcid.org/0000-0003-2969-0212 Ishikawa Kenichi L. 5,6,7 Carpeggiani Paolo Antonio 8 Csizmadia Tamás 9 Füle Miklós 9 Sansone Giuseppe 10 Kumar Maroju Praveen 10 D’Elia Alessandro 11,12 Mazza Tommaso 13 Meyer Michael 13 Gryzlova Elena V. 14 Grum-Grzhimailo Alexei N. 14 You Daehyun 15 Ueda Kiyoshi 15 ,† 1 Elettra-Sincrotrone Trieste S.C.p.A , 34149 Basovizza, Trieste, Italy Centre for Translational Atomaterials, 2 Swinburne University of Technology , Melbourne 3122, Australia 3 INFN—Laboratori Nazionali di Frascati , 00044 Frascati, Rome, Italy Laboratory of Quantum Optics, 4 University of Nova Gorica , 5001 Nova Gorica, Slovenia Department of Nuclear Engineering and Management, Graduate School of Engineering, 5 The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Photon Science Center, Graduate School of Engineering, 6 The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Research Institute for Photon Science and Laser Technology, 7 The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan Institut für Photonik, 8 Technische Universität Wien , 1040 Vienna, Austria 9 ELI-ALPS , ELI-HU Non-Profit Ltd., H-6720 Szeged, Hungary Physikalisches Institut, 10 Albert-Ludwigs-Universität Freiburg , 79106 Freiburg, Germany 11 University of Trieste , Department of Physics, 34127 Trieste, Italy IOM-CNR, 12 Laboratorio Nazionale TASC , 34149 Basovizza, Trieste, Italy 13 European XFEL GmbH , D-22869 Schenefeld, Germany Skobeltsyn Institute of Nuclear Physics, 14 Lomonosov Moscow State University , Moscow 119991, Russia Institute of Multidisciplinary Research for Advanced Materials, 15 Tohoku University , Sendai 980-8577, Japan * prince@elettra.eu † kiyoshi.ueda@tohoku.ac.jp ‡ Present address: LIDYL, CEA, CNRS, Université Paris–Saclay, CEA–Saclay, 91191 Gif-sur-Yvette, France. 22 November 2019 22 November 2019 123 21 213904 27 May 2019 © 2019 American Physical Society 2019 American Physical Society Intense, mutually coherent beams of multiharmonic extreme ultraviolet light can now be created using seeded free-electron lasers, and the phase difference between harmonics can be tuned with attosecond accuracy. However, the absolute value of the phase is generally not determined. We present a method for determining precisely the absolute phase relationship of a fundamental wavelength and its second harmonic, as well as the amplitude ratio. Only a few easily calculated theoretical parameters are required in addition to the experimental data. Ministry of Education, Culture, Sports, Science and Technology 10.13039/501100001700 16H03881 17K05070 18H03891 19H00869 Network Joint Research Center Japan Science and Technology Agency 10.13039/501100002241 JPMJCE1313 JPMJCR15N1 Japan Society for the Promotion of Science 10.13039/501100001691 JP19J12870 Alexander von Humboldt-Stiftung 10.13039/100005156 Italian Ministry of Research 10.13039/501100003407 RBID08CRXK PRIN 2010 ERFKXL 006 Horizon 2020 Framework Programme 10.13039/100010661 H2020 Marie Sk?odowska-Curie Actions 10.13039/100010665 641789 Deutsche Forschungsgemeinschaft 10.13039/501100001659 SFB925/A1 Quantum mechanical processes, such as photoionization, are defined by the amplitudes and phases of the particles involved, and their description requires the determination of all of these, as, for example, in complete experiments [1–3] . While amplitudes can often be deduced from experimental intensities, the determination of phase is usually more challenging. Phase determination implies the concept of coherence, intrinsic in the nature of waves but not relevant for classical particles. Photoionization is one of the best showcases for such quantum mechanical concepts, as the phase of the photons is imprinted on the emitted electron wave function. Recently, coherent optical experiments have become possible at short wavelengths using a seeded free-electron laser (FEL), so that multiharmonic extreme ultraviolet (XUV) radiation can be used to coherently control the outcome of experiments [4,5] . The phase tuning of the light field does not involve the use of a traditional delay line but instead utilizes a technique based on accelerator physics in which the electron beam, rather than the light, is delayed to adjust the phase shift [6] . In the first group of experiments [4,5,7] , the relative phase between a fundamental wavelength and its second harmonic was tuned with a precision of a few attoseconds, but the absolute phase difference was unknown, that is, the zero of the phase scale was not determined. Further experiments are planned: coherent control experiments using bichromatic light like the examples above, the production of XUV pulse trains via a finite number of coherent harmonics, and other more exotic schemes. The key to all of these methods is control of the amplitude and phase of each harmonic component, so it is important to know both of these precisely. This knowledge is also indispensable for theoretical predictions and simulations of the experiments. The relative amplitudes of two wavelengths can be controlled (e.g., by gas and solid filters, or via accelerator parameters) and measured. The phase difference can be varied very precisely [6] , but it is experimentally difficult to measure it absolutely at short wavelengths. At long (optical) wavelengths, there are standard methods available for determining the phase difference between two harmonics, for instance, by frequency doubling of the fundamental and observing interference with the second harmonic [8] . Such methods are not available at short wavelengths because of the lack of suitable nonlinear materials. Only recently has soft x-ray second-harmonic generation at a surface been demonstrated using FEL radiation [9] , but this is far from a practical diagnostic. The method we describe employs gas phase targets and can be used at a range of wavelengths. It requires the ionization of an n s electron ( n is the principal quantum number), and we demonstrate the method for the He atom. For future coherent control experiments at soft x-ray FELs, it will be important to control and measure the phase at innershell excitation energies, and the method we present can be applied to this task, using other atomic subshells such as Ne 2 s , C 1 s , Ne 1 s , etc. The reason that it is difficult to measure phase at a FEL is as follows. Free-electron laser radiation is generated by relativistic electrons passing through several arrays of magnets known as undulators. The wavelength is selected by tuning the magnetic field of each undulator to the appropriate, resonant value for which the electrons lag the radiation by exactly one period every undulator period. Between each pair of undulators, there are fringe magnetic fields, which lengthen the path of the electron beam by a quantity that is not necessarily an integral value of the radiation wavelength and causes consecutive undulators to emit out of phase. For single-wavelength operation, this path length difference is compensated by the use of phase shifters [6] ; under the assumption that the output is maximal when all undulators emit in phase, a lookup table is generated for all phase shifters and all wavelengths. Bichromatic light is created by tuning the magnetic field of one or more undulators to the resonant condition for a harmonic wavelength, and in the following we consider only the fundamental plus second-harmonic configuration, i.e., wavelengths λ and λ / 2 (and the corresponding frequencies ω and 2 ω ). We first tried to use the single-wavelength lookup table to produce a new one for the bichromatic configuration, assuming that extra delays from undulator fringe fields are evenly distributed along the space between undulators. This method was not sufficiently precise to guarantee the accuracy required by the experiment (as an example: 10 as corresponds to a phase of 2 π / 10 at 30 nm). If the absolute phases were known at a reference pair of wavelengths λ r and λ r / 2 , one could consider keeping the undulator gaps fixed and tuning to another wavelength by changing the electron beam energy in the accelerator. However, experimental tests with a 3% electron energy change (corresponding to a 6% wavelength change) showed that the necessary readjustment of the accelerator in terms of trajectory, quadrupole strength, and undulator resonance could not guarantee the desired phase stability. Besides the insurmountable difficulties we just illustrated, these two unsuccessful methods cannot account for the phase uncertainty later introduced by the photon transport system through various optical elements (mirrors, filters, gas cell, etc). For this reason, the successful method demonstrated hereafter, which determines the absolute phase difference directly in the experimental chamber at the end of the beam line, is very appealing. It is based on nonlinear optics, and on the interference which is observed in the photoelectron angular distribution (PAD) between one- and two-photon ionization processes. The schematic process of ionization of an n s electron is shown in Fig. 1 . For two-photon ionization by linearly polarized light of frequency ω , there are two outgoing partial waves of s and d character, while for single-photon ionization by frequency 2 ω , there is a single outgoing p wave. These three outgoing waves interfere to give a PAD which depends on their relative phases. 1 10.1103/PhysRevLett.123.213904.f1 FIG. 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. The field is described by E ( t ) = I ω ( t ) cos ω t + I 2 ω ( t ) cos ( 2 ω t - ϕ ) , (1) where I ω ( t ) , I 2 ω ( t ) are the envelopes of the two pulses, and ϕ denotes the absolute ω - 2 ω relative phase (the larger the value of ϕ , the more delayed the 2 ω pulse). The experimental phase setting is ϕ ′ = ϕ + ϕ 0 , where ϕ 0 is an unknown phase offset. ϕ ′ is derived from a reading of the position of the magnetic structure creating the delay of the electrons [6] and can be controlled with a resolution of a few attoseconds [4] . ϕ 0 corresponds to additional delays introduced by magnetic stray fields and photon transport. Within perturbation theory and the rotating wave approximation, and for parallel, linearly polarized long pulses, the most general form of the PAD I e ( θ ) is the modulus squared of a combination of spherical harmonics Y ℓ , m with angular momentum ℓ up to 2, and m = 0 : I e ( θ ) = | c s e i η s Y 0 , 0 ( θ , φ ) + c p e i ( η p + ϕ ) Y 1 , 0 ( θ , φ ) + c d e i η d Y 2 , 0 ( θ , φ ) | 2 , (2) where c s , c p , and c d are amplitudes of partial waves, θ is the polar angle with respect to the electric vector (the azimuthal angle φ is redundant), and η s , η p , η d are scattering phase shifts. The volume of the interaction region is not taken into account. Equation (2) is traditionally written as a series expansion of Legendre polynomials P ℓ ( cos θ ) , I e ( θ ) = ( c s 2 + c p 2 + c d 2 ) 4 π ( 1 + ∑ ℓ = 1 4 β ℓ P ℓ ( cos θ ) ) , (3) and the expression of the asymmetry parameters β ℓ as a function of c s , c p , c d [Eqs. (4) – (7) , with h = 1 ] is found, after some tedious algebra, using the identities (S1) and (S5)–(S7) in the Supplemental Material [10] . Experimentally, there are a number of challenges to be faced, due to nonideal experimental conditions, such as variations of intensity across the excitation volume, incomplete coherence, small misalignments of the focal spots, etc. We define the decoherence parameter h ∈ [ 0 , 1 ] which we use to phenomenologically correct for these imperfections, and to scale the β 1 and β 3 oscillations. The resulting expressions for β l are β 1 = h × 4 15 c d c p cos ( - η d + η p + ϕ ) + 10 3 c p c s cos ( η p - η s + ϕ ) 5 ( c d 2 + c p 2 + c s 2 ) , (4) β 2 = 10 c d 2 + 14 5 c d c s cos ( η d - η s ) + 14 c p 2 7 ( c d 2 + c p 2 + c s 2 ) , (5) β 3 = h × 6 15 c d c p cos ( - η d + η p + ϕ ) 5 ( c d 2 + c p 2 + c s 2 ) ≡ [ β 3 ] 0 cos [ ϕ ′ - ( ϕ 0 + η d - η p ) ] , (6) β 4 = 18 c d 2 7 ( c d 2 + c p 2 + c s 2 ) , (7) β 1 - 2 3 β 3 = h × 2 3 c p c s cos ( η p - η s + ϕ ) c d 2 + c p 2 + c s 2 ≡ [ β 1 - 2 3 β 3 ] 0 cos [ ϕ ′ - ( ϕ 0 - η p + η s ) ] . (8) Equation (8) is derived from Eqs. (4) and (6) so that, like Eq. (6) , the right-hand side can be factored into a cosine containing the optical and scattering phase dependence, and an amplitude independent of phase (the prefactor in square brackets). We note that Eqs. (5) and (7) for β ℓ , where ℓ is even, do not depend on phase, in agreement with previous results [13,14] . Equations (6) and (8) are among the main results of this Letter. At fixed photon energy, a graph of these quantities against experimental phase ϕ ′ yields two oscillatory curves whose absolute phases are independent of photon intensity. For the particular case of He, the values of scattering phase shifts η s , η p , and η d for a wide range of electron energies are available in the literature [15–17] . We now show that we are able to extract two independent values of the phase offset ϕ 0 , whose excellent agreement attests to the robustness of the method. As well, further information can be extracted to benchmark the method; for example, the difference η d - η s can be determined from the phase difference of the curves of Eqs. (8) and (6) . The amplitude of the oscillatory curves depends on the coherent mixing of the two photon fields: the ratio c s / c d can also be extracted from Eqs. (8) and (6) . The ratio of the two amplitudes of oscillation, [ β 1 - 2 3 β 3 ] 0 and [ β 3 ] 0 , is equal to c s / c d multiplied by the numerical factor 5 / 3 . Further manipulation of all equations yields the ratio ( c s 2 + c d 2 ) / c p 2 , which in combination with theoretical calculations yields the relative intensities of the two fields. At fixed photon energy, a graph of these quantities against phase yields two oscillatory curves. Their relative phase is determined by the argument of the cosine functions and is independent of photon intensity. The amplitude of the oscillatory curves is independent of phase, but does depend on the amplitude of the photon field. From the relative amplitudes of the curves, we may extract information about the effective relative photon amplitudes, even in the case of imperfect experimental conditions. The photoionization processes can be accurately simulated using the time-dependent close-coupling (TDCC) method [18,19] or the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method [20–25] . We have performed TDCC simulations for a series of photon energies and list the values of c s , c p , and c d relevant for this experiment in Table I . In Table II of the Supplemental Material [10] , we compare the values of η s - η d and η p - η d from our simulations with those reported in Ref. [17] , and they agree very well. We have also confirmed that the PADs obtained from TDCC and MCTDHF simulations are in excellent agreement with each other. In the present TDCC calculations, the amplitudes are normalized so that c s 2 + c d 2 and c p 2 correspond to the degree of ionization by ω and 2 ω , respectively, for a 7 fs pulse duration. This value is much shorter than the experimental duration and was chosen for reasons of computational economy. So long as the bandwidth of the pulse is sufficiently far from any atomic resonance, one can safely scale the results to the present longer experimental pulses [18,19] . Note that, for a given photon energy, ℏ ω , c s 2 , c p 2 , and c d 2 scale linearly with pulse duration; c s and c d scale as I ω , whereas c p scales as I 2 ω . Thus, one can calculate β parameters for any intensity and pulse duration from these tabulated values as long as the perturbative treatment is valid, and processes of higher order than those considered here are negligible. I 10.1103/PhysRevLett.123.213904.t1 TABLE I. Ab initio results using the TDCC method. Both ω and 2 ω pulses are assumed to have a Gaussian temporal profile with 7 fs FWHM pulse duration. The peak intensity of ω is fixed at 1 0 13 W / cm 2 . I 2 ω max is the 2 ω peak intensity at which the ionization yields by ω and 2 ω pulses ( c s 2 + c d 2 and c p 2 , respectively) are equal to each other. The values of c s , c p , and c d are listed for this condition. σ ω ( 2 ) is the cross section for two-photon ionization by the ω pulse, σ 2 ω ( 1 ) is the cross section for single-photon ionization by the 2 ω pulse. ℏ ω (eV) I 2 ω max ( W / cm 2 ) c s c p c d η s - η d (rad) η p - η d (rad) σ ω ( 2 ) ( 10 - 52 cm 4 / s ) σ 2 ω ( 1 ) ( 10 - 18 cm 2 ) 14.3 1.34 × 10 10 3.22 × 10 − 3 1.14 × 10 − 2 − 1.09 × 10 − 2 5.36 2.26 12.9 5.92 15.9 1.24 × 10 10 1.77 × 10 − 3 9.44 × 10 − 3 − 9.28 × 10 − 3 5.07 2.12 11.0 4.92 19.1 1.09 × 10 10 − 4.76 × 10 − 4 6.81 × 10 − 4 − 6.79 × 10 − 4 4.76 1.985 8.24 3.44 We used a velocity map imaging (VMI) spectrometer [26] to measure the PAD described by Eq. (3) (alternative instruments include multidetector electron time-of-light spectrometers [27,28] ). The angular distributions and photoelectron spectra were determined by analysis after inversion of the velocity map images. The sample was irradiated with fundamental radiation at one of the wavelengths λ = 87.0 , 78.0, or 65.0 nm, and its second harmonic λ / 2 = 43.5 , 39.0, 32.5 nm, and the PAD was measured. The second-harmonic intensity was set so that the measured ratios of ionization rates due to single- and two-photon ionization were close to 2 ∶ 1 for datasets A , B , and C , while for dataset D , it was 4 ∶ 1 . The phase was tuned using the phase shifters installed at FERMI [6] . The intensities were sufficient to cause significant ionization, without danger of saturation effects; see Supplemental Material [10] . From the VMI data, we extracted β 1 , β 2 , β 3 , and β 4 as a function of the experimental relative phase ϕ ′ = ϕ + ϕ 0 between the two optical pulses of ω and 2 ω [see Eq. (1) ]. An example of the results for dataset A at 14.3 eV is shown in Fig. 2 , where β 2 , β 3 , β 4 , and β 1 - 2 3 β 3 are shown. As expected, β 3 and β 1 - 2 3 β 3 oscillate while β 2 and β 4 are constant. From least-squares fitting of the data with cosine curves and offsets, we extracted six parameters, β 2 , β 4 , [ β 3 ] 0 , ϕ 0 + η d - η p , [ β 1 - 2 3 β 3 ] 0 , and ϕ 0 - η p + η s . The results are given in Table II for four datasets that we recorded. 2 10.1103/PhysRevLett.123.213904.f2 FIG. 2. β parameters as a function of ϕ . Markers, β parameters of dataset A as a function of phase; curves, cosine (constant) fit, odd (even) β ; blue triangles, β 1 - 2 β 3 / 3 ; black circles, β 2 ; green inverted triangles, β 3 ; red squares, β 4 . 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 . II 10.1103/PhysRevLett.123.213904.t2 TABLE II. Results of analysis of four experimental datasets at three different photon energies. Dataset A Dataset B Dataset C Dataset D Photon energy (eV) 14.3 15.9 15.9 19.1 [ β 1 - 2 3 β 3 ] 0 0.141 ± 0.008 0.114 ± 0.008 0.057 ± 0.004 - 0.030 ± 0.003 ϕ 0 - η p + η s (rad) 1.70 ± 0.05 1.75 ± 0.06 6.28 ± 0.07 1.20 ± 0.11 β 2 0.856 ± 0.010 1.631 ± 0.009 1.691 ± 0.005 1.784 ± 0.005 [ β 3 ] 0 - 0.574 ± 0.013 - 0.810 ± 0.025 - 0.439 ± 0.017 - 0.723 ± 0.008 ϕ 0 + η d - η p (rad) 2.84 ± 0.02 3.17 ± 0.03 1.28 ± 0.04 2.28 ± 0.01 β 4 0.935 ± 0.013 1.028 ± 0.028 0.412 ± 0.009 1.010 ± 0.016 c s / c d (expt.) - 0.330 ± 0.021 - 0.189 ± 0.014 - 0.174 ± 0.013 0.055 ± 0.006 c s / c d (theory) − 0.295 − 0.191 − 0.191 0.070 η s - η d (expt.) (rad) 5.15 ± 0.06 4.87 ± 0.07 5.01 ± 0.08 5.21 ± 0.11 η s - η d (theory) (rad) 5.36 5.07 5.07 4.76 c p 2 : c s 2 + c d 2 0.82 ± 0.01 : 1 1.44 ± 0.05 : 1 3.58 ± 0.06 : 1 1.55 ± 0.03 : 1 h 0.262 ± 0.006 0.360 ± 0.010 0.226 ± 0.008 0.318 ± 0.004 I 2 ω / I ω ( 10 - 9 / W / cm 2 ) 10.5 ± 0.1 13.4 ± 0.2 21.1 ± 0.2 13.0 ± 0.1 ϕ 0 (rad) 5.07 ± 0.02 5.25 ± 0.03 3.38 ± 0.04 4.26 ± 0.01 To determine ϕ 0 from the measurements, we substituted the calculated values of phase (see Table II of the Supplemental Material [10] ) into Eqs. (6) and (8) and obtained two values of the phase offset ϕ 0 (see Table II ), which agree to within 0.03–0.04 rad, or 2 deg. Figure 2 illustrates this, where it can be seen that the choice of η p and either η s or η d yields the values of ϕ 0 and of the remaining η . There is a large difference in the values of ϕ 0 for different datasets, for example B and C : this is because the two sets were taken about 68 h apart, after numerous changes of undulator magnetic fields, and small corrections to the accelerator trajectory. During a scan, which lasted on the order of 2 h, no changes to the accelerator were made, other than a scanning of the phase. We checked that the conditions were sufficiently stable by repeating the first points of a scan at the end of the scan, and by scanning with increasing phase, followed by decreasing phase. Let us now return to the quantities c s / c d and η s - η d extracted from the fit parameters as explained above. These quantities do not depend on the pulse intensity and thus can be directly compared with the ab initio results in Table I . The results for c s / c d and η s - η d obtained using Eqs. (6) and (8) are also given in Table II . These two parameters agree well with theoretical values, confirming that the present methodology works well. Let us finally consider the other two parameters, c p 2 / ( c s 2 + c d 2 ) , which is proportional to I 2 ω / I ω 2 , and h , which scales the amplitudes of β 1 and β 3 . To extract c p 2 / ( c s 2 + c d 2 ) and h from our experimental results, we optimize those parameters by minimizing χ 2 given by χ 2 = ( [ β 1 exp - 2 3 β 3 exp ] 0 - [ β 1 th - 2 3 β 3 th ] 0 ) 2 α 1 2 + ( β 2 exp - β 2 th ) 2 α 2 2 + ( [ β 3 exp ] 0 - [ β 3 th ] 0 ) 2 α 3 2 + ( β 4 exp - β 4 th ) 2 α 4 2 , (9) where [ β 1 exp - 2 / 3 β 3 exp ] 0 , [ β 3 exp ] 0 , and β 2 , 4 exp are the present experimental values, and α 1 , α 3 , α 2 , 4 are their respective uncertainties (Table II ), and the β th are values calculated from Eqs. (4) – (8) with theoretical values in Table I , regarding c p / c d and h as fitting parameters. The resulting values are also given in Table II . As noted above, c p 2 / ( c s 2 + c d 2 ) is proportional to I 2 ω / I ω 2 , so we can determine I 2 ω / I ω , as given in Table II , by employing the theoretical ratio of c p 2 / ( c s 2 + c d 2 ) in Table I . In conclusion, we have demonstrated a method of determining the absolute phase between two wavelengths in a bichromatic XUV beam, as well as the coherent fraction of the relative intensity. The determination of phase is independent of the intensity of the two wavelengths. This is useful for several purposes: experiments at a fixed pair of wavelengths may require knowledge of the absolute phase relationship between the two in order to interpret the data, and this can be provided by adding He gas to the target. If multiple, phase locked wavelengths are used, the absolute phase can be extracted. Lastly, precise knowledge of the absolute phase difference and intensity ratio provides a far more rigorous basis for benchmarking theoretical simulations of experimental data. An advantage of the method is that it is applied at the experimental station, rather than at the exit of the FEL, so any alterations in phase difference introduced by beam transport are automatically included in the measurement. This work was supported in part by the X-ray Free Electron Laser Utilization Research Project and the X-ray Free Electron Laser Priority Strategy Program of the Ministry of Education, Culture, Sports, Science, and Technology of Japan (MEXT) and the IMRAM program of Tohoku University, and the Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials program. K. L. I. gratefully acknowledges support by the Cooperative Research Program of the Network Joint Research Center for Materials and Devices (Japan), the Grant-in-Aid for Scientific Research (Grants No. 16H03881, No. 17K05070, No. 18H03891, and No. 19H00869) from MEXT, JST COI (Grant No. JPMJCE1313), JST CREST (Grant No. JPMJCR15N1), and the Japan-Hungary Research Cooperative Program, JSPS and HAS. E. V. G. acknowledges the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.” D. Y. acknowledges supports from JSPS KAKENHI Grant No. JP19J12870, and a Grant-in-Aid of Tohoku University Institute for Promoting Graduate Degree Programs Division for Interdisciplinary Advanced Research and Education. T. M and M. M. acknowledge support by Deutesche Forschungsgemeinschaft Grant No. SFB925/A1. We acknowledge the support of the Alexander von Humboldt Foundation (Project Tirinto), the Italian Ministry of Research Project FIRB No. RBID08CRXK and No. PRIN 2010 ERFKXL 006, the bilateral project CNR JSPS Ultrafast science with extreme ultraviolet Free Electron Lasers, and funding from the European Union Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 641789 MEDEA (Molecular ElectronDynamics investigated by IntensE Fields and Attosecond Pulses). We thank the machine physicists of FERMI for making this experiment possible with their excellent work in providing high quality FEL light. [1] 1 U. Becker and B. Langer , Phys. Scr. T78 , 13 ( 1998 ). PHSTBO 0031-8949 10.1238/Physica.Topical.078a00013 [2] 2 H. Kleinpoppen , B. Lohmann , and A. N. Grum-Grzhimailo , Perfect/Complete Scattering Experiments ( Springer , Berlin, 2013 ). [3] 3 P. Carpeggiani , E. V. Gryzlova , M. Reduzzi , A. Dubrouil , D. Faccialà , M. Negro , K. Ueda , S. M. Burkov , F. Frassetto , F. Stienkemeier , Y. Ovcharenko , M. Meyer , O. Plekan , P. Finetti , K. C. Prince , C. Callegari , A. N. Grum-Grzhimailo , and G. Sansone , Nat. Phys. 15 , 170 ( 2019 ). NPAHAX 1745-2473 10.1038/s41567-018-0340-4 [4] 4 K. C. Prince , Nat. Photonics 10 , 176 ( 2016 ). NPAHBY 1749-4885 10.1038/nphoton.2016.13 [5] 5 L. Giannessi , E. Allaria , K. C. Prince , C. Callegari , G. Sansone , K. Ueda , T. Morishita , C. N. Liu , A. N. Grum-Grzhimailo , E. V. Gryzlova , N. Douguet , and K. Bartschat , Sci. Rep. 8 , 7774 ( 2018 ). SRCEC3 2045-2322 10.1038/s41598-018-25833-7 [6] 6 B. Diviacco , R. Bracco , D. Millo , and M. M. Musardo , in Proceedings of the 2nd International Conference on Particle Accelerators (IPAC 2011), San Sebastián, Spain, 2011 , edited by C. Petit-Jean -Genaz ( IPAC’11 EPS-AG , Geneva, Switzerland, 2011 ), p. 3278 , http://accelconf.web.cern.ch/AccelConf/IPAC2011/index.htm . [7] 7 D. Iablonskyi , Phys. Rev. Lett. 119 , 073203 ( 2017 ). PRLTAO 0031-9007 10.1103/PhysRevLett.119.073203 [8] 8 Z.-M. Wang and D. S. Elliott , Phys. Rev. Lett. 87 , 173001 ( 2001 ). PRLTAO 0031-9007 10.1103/PhysRevLett.87.173001 [9] 9 R. K. Lam , S. L. Raj , T. A. Pascal , C. D. Pemmaraju , L. Foglia , A. Simoncig , N. Fabris , P. Miotti , C. J. Hull , A. M. Rizzuto , Phys. Rev. Lett. 120 , 023901 ( 2018 ). PRLTAO 0031-9007 10.1103/PhysRevLett.120.023901 [10] 10 See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.123.213904 , which includes Refs. [11,12], for further experimental details, results of calculations, and data samples. [11] 11 F. W. J. Olver , A. B. O. Daalhuis , D. W. Lozier , B. I. Schneider , R. F. Boisvert , C. W. Clark , B. R. Miller , and B. V. Saunders , Publisher Copyright: © 2019 American Physical Society.

PY - 2019/11/22

Y1 - 2019/11/22

N2 - Intense, mutually coherent beams of multiharmonic extreme ultraviolet light can now be created using seeded free-electron lasers, and the phase difference between harmonics can be tuned with attosecond accuracy. However, the absolute value of the phase is generally not determined. We present a method for determining precisely the absolute phase relationship of a fundamental wavelength and its second harmonic, as well as the amplitude ratio. Only a few easily calculated theoretical parameters are required in addition to the experimental data.

AB - Intense, mutually coherent beams of multiharmonic extreme ultraviolet light can now be created using seeded free-electron lasers, and the phase difference between harmonics can be tuned with attosecond accuracy. However, the absolute value of the phase is generally not determined. We present a method for determining precisely the absolute phase relationship of a fundamental wavelength and its second harmonic, as well as the amplitude ratio. Only a few easily calculated theoretical parameters are required in addition to the experimental data.

UR - http://www.scopus.com/inward/record.url?scp=85075601979&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075601979&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.123.213904

DO - 10.1103/PhysRevLett.123.213904

M3 - Article

C2 - 31809175

AN - SCOPUS:85075601979

VL - 123

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 21

M1 - 213904

ER -