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DOI: 10.1038/NPHOTON.2017.73

In the format provided by the authors and unedited.

Supplementary Information
Vectorial optical field reconstruction by attosecond spatial
interferometry.

P. Carpeggiani1,2† , M. Reduzzi1,2† , A. Comby1† , H. Ahmadi1,3 , S. Kühn4 ,

F. Calegari2,5,6 , M. Nisoli1,2 , F. Frassetto7 , L. Poletto7 , D. Hoff8 , J. Ullrich9 ,
C. D. Schröter10 , R. Moshammer10 , G. G. Paulus8 , G. Sansone1,2,4,11 .1

1
(1) Dipartimento di Fisica, Politecnico Piazza Leonardo da Vinci 32, 20133 Milano Italy
(2) IFN-CNR, Piazza Leonardo da Vinci 32, 20133 Milano Italy
(3) Department of Physical Chemistry,
School of Chemistry, College of Science,
University of Tehran, P. O. Box 14155-6455, Tehran, Iran
(4) ELI-ALPS, ELI-Hu Kft., Dugonics ter 13, H-6720 Szeged, Hungary
(5) Deutsches Elektronen-Synchrotron,
Notkestrasse 85, Hamburg 22607, Germany
(6) Physics Department, University of Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
(7) Institute of Photonics and Nanotechnologies,
CNR via Trasea 7, 35131 Padova Italy
(8) Institut für Optik und Quantenelektronik,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
(9) Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Braunschweig, Germany
(10) Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany
(11) Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, 79106 Freiburg, Germany
† These authors contributed equally to this work

(Dated: 15th May 2017)

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Experimental setup

Isolated attosecond pulses were generated using CEP-stable few-cycle pulses centered
at λ0 = 744 nm, with an energy of 600 µJ at 10 kHz repetition rate. Broadband XUV
continua were generated in argon by means of the PG technique1 using a delay plate of
birifringent quartz with a thickness of 180 µm and zero-order broadband quarter-wave plate.
The intensity of the IR pulse in the temporal window of linear polarisation was estimated
in I ≃ 1.5 × 1014 W/cm2 . The XUV radiation was then focused by a toroidal mirror
in the interaction region of a Reaction Microscope (REMI) (Fig. 1a)2 , which was used to
determine the direction of polarisation of the attosecond pulses generated for α1 = 45◦
and α2 = −45◦ , by measuring the three dimensional photoelectron angular distribution in
helium. The measurements are shown in Fig. 1b,c. The main axis of the photoelectron
angular distributions, corresponding to the polarisation directions of the XUV radiation in
the interaction region of the REMI, were estimated in θ1 = 57◦ ± 3◦ and θ2 = 315◦ ± 3◦ .
The error was estimated by fitting the angular projections of the photoelectron distributions
with a cosine function.
We characterized the photoelectron angular distributions generated by the attosecond pulses
with and without metallic filter (aluminium filter with a thickness of 100 nm). We could
not observe any significant difference in the polarisation direction of the XUV radiation
in these two conditions. The influence of the two spatially separated XUV sources on the
photoelectron measurement was investigated by shifting (without removing) the binary plate
to avoid the generation of the two-foci in the focal plane. The polarisation of the XUV pulse
measured in this condition coincided with the polarisation of the twin XUV pulses within
our experimental uncertainty.
Considering the reflection on the surface of the toroidal mirror (gold at a grazing incidence
of 4◦ ) (see Fig. 1a), the polarisation directions of the isolated attosecond pulses in the
generation cell are given by: γ1 = π − θ1 = 123◦ ± 3◦ and γ2 = π − θ2 = 225◦ ± 3◦ . For a
multi-order quarter-wave plate, ∆γ = γ2 − γ1 = 90◦ . The deviation of ∆γ from this value
was taken into account in the pulse reconstruction.
The different complex reflectivity of the s and p components of the XUV attosecond pulse
on the toroidal mirror did not appreciably affect the estimation of the angles γ1,2 . Indeed,
the ratio of the amplitudes of the complex reflectivity of the s and p components was

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estimated in r ≃ 1.05. The deviation from the unitary value corresponds to a rotation of
the polarisation axis of the attosecond pulse from the specular direction of only 1.3◦ . This
value is definitely smaller than the error bar (±3◦ ) determined by the uncertainty of the
fit of the photoelectron angular distribution. The relative phase shift between the complex
reflectivity was estimated in ∆ϕr = 1.5◦ . Therefore, the reflection induces a very small
degree of ellipticity (ε ≤ 0.013) on the incident linearly polarised attosecond pulse that does
not affect the reconstruction procedure.
The diverging XUV pulse was analysed by an XUV spectrometer (see Fig. 1a) composed
by a cylindrical mirror and a concave grating, which dispersed the spectral components in
the focal plane (x-direction), where an MCP coupled to a phosphor screen was placed. In
the perpendicular direction (y-direction) the propagation of the two coherent XUV pulses
is not affected. A CCD camera acquired the signals at the back of the phosphor screen. A
piezoelectric translator was used to change and monitor in step of 20 nm the delay between
the driving and unknown fields.
The CEP of the driving pulses were characterized on a single-shot basis using a Stereo-ATI3 .
The measured values were used to pre-compensate the CEP drift using as feedback the
acoustic waveforms sent to the Dazzler present in the beam path before the laser amplifier4 .
The maximum intensity of the (linearly polarized) perturbing field used in the experiment
was estimated in Iunk = 1.5 × 1011 W/cm2 .

Simulations and field reconstruction

The effect of the perturbing (in principle unknown) field on the properties of the high-
order harmonic spectra was simulated by solving the Lewenstein integral5 with the stationary
phase method6 , taking into account the total electric field given by the sum of the driving
and perturbing fields: Etot = Edr + Eunk . The total field was used to solve the saddle point
equations7 :
[ps − Atot (ts )] · [ps − Atot (ts )]
ω− − Ip = 0
2
[ps − Atot (t′s )] · [ps − Atot (t′s )]
+ Ip = 0 (1)
2
where Ip is the ionisation potential of the atom, ω is the harmonic frequency, Atot (t) is the
vector potential associated to Etot (t), and ts and t′s are the stationary values of the recom-

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bination and ionisation instants of the electronic wave packet, respectively. The stationary
momentum ps is given by:
∫ t
1 ′′ ′′
ps = Atot (t )dt (2)
t − t′ t′

The complex solutions (ps , ts , t′s ) of Eqs. (1-2) identify the most relevant contributions to
the XUV spectrum emitted by a single atom.
For each temporal delay τ between the driving and unknown fields, we considered only the
solution corresponding to the short path with the highest cut-off frequency6 . The harmonic
spectrum E(ω) is then given by:

i 2π [ π ]3/2
E(ω) = √ ′ )/2
d∗ [ps − Atot (ts )] ×
det(S ) ′′ ϵ + i(t s − t s
× Etot (ts ) · d[ps − Atot (ts )] exp[−iS(ps , ts , t′s ) + iωts ]
′ ′

(3)

where S is the dipole phase, det(S ′′ ) is the determinant of the matrix of the second time
derivative of the phase (ωt − S) with respect to t and t′ evaluated in correspondence of the
saddle points solutions (ps , ts , t′s ), ϵ is a regularisation constant, and d is the gaussian form of
the dipole matrix element5 . The component of the XUV spectrum along the polarisation of
the driving field dominates the total signal and we neglect the perpendicular XUV component
in the following discussion.
The binary 0-π plate introduced a π-shift between the two halves of the pump beam, thus
leading to two focal spots that will be indicated as a and b. The distance between the two
spots was measured in 2d ≃ 100 µm. The XUV spectra generated in the focal spot without
ref
(spot a; reference field EXU V (ω)) and with the influence of the unknown, perturbing field
pert
(spot b; perturbed field EXU V (ω)) are given by:

ref ref
EXU V (ω) = E (ω) exp[iφref (ω)]
pert ref
EXU V (ω) = [E (ω) + ∆E(ω)] exp[i(φref (ω) + ∆φ(ω))] (4)

where ∆E(ω) and ∆φ(ω) indicate the variations in the amplitude and phase introduced by
the perturbing field, respectively.
The total XUV electric field EXU V (x, y, z = 0, ω) in the high-order harmonic generation
plane (z = 0) was described as the sum of identical gaussian beams, spaced by 2d, multiplied

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by the spectra given in Eq. 4:

EXU V (x, y, z = 0, ω) = Ea (x, y, 0, ω) + Eb (x, y, 0, ω) =

ref
= exp(−x2 /w0x
2
) exp[−(y − d)2 /w0y
2
]EXU V (ω) +
pert
+ exp(−x2 /w0x
2
) exp[−(y + d)2 /w0y
2
]EXU V (ω) (5)

where w0x,y indicate the beam waists in the x and y direction, respectively. In the simula-
tions we considered w0x = 40 µm and w0y = 20 µm.
The XUV spectrometer dispersed and focused the harmonic components along the x-
direction on its imaging plane (placed at a distance z0 from the source point), while the two
XUV fields freely propagates in the y-direction. In this plane the intensity distribution is
characterised by an interference term between the two sources, which is proportional to:

E ref (ω)[E ref (ω) + ∆E(ω)] exp[i(2kyz0 /Ry ) − ∆φ(ω)) (6)

2
where k = 2πc/ω (c speed of light) and Ry = z0 [1 + (kw0y /(2z0 ))2 ].
The Fourier transform of the signal along the y-direction is characterised by a peak with
spatial frequency 2kz0 /Ry . The amplitude and phase modulations of this peak as a function
of the delay τ correspond to ∆E(ω; τ ) and ∆φ(ω; τ ) and were used for the reconstruction
of the perturbing field from the simulations and experimental data (see Fig. 1 and Fig. 3 of
the manuscript, respectively).
We implemented also a different algorithm based on the analysis of a single interference
fringe. In this case, we evaluated the centre of mass of a single fringe for each delay step τ .
The results are presented in Fig. 2, which reports the phase retrieved by the Fourier (solid
line) and the centre of mass analysis (dashed line) for the measurement shown in Fig. 3 of
the manuscript. The two methods present an overall good agreement.

Experimental electric field reconstruction of pulses with time-dependent polarisa-

tion and calibration procedure

In the PG scheme, the angles of polarisation of the isolated attosecond pulses (γ1,2 )
with respect to the input polarisation direction of the driving field, for the rotation angles
α1,2 = ±45◦ of the birefringent quartz delay plate, depend on its thickness and on the central
wavelength of the pulse. If the plate corresponds to a multiple-order quarter-wave plate of

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the fundamental radiation, the polarisation angles are γ1,2 = ±45◦ .

In our experimental conditions, the thickness of the delay plate was determined as the best
compromise between the generation of an isolated attosecond pulse with optimal charac-
teristics (in terms of photon flux and absence of satellite pulses), and the requirement to
generate XUV pulses with (or close to) perpendicular polarisations when switching between
α1 and α2 .
The field components along the x and y directions (Ex (t), Ey (t)) were retrieved from the
fields projections E1 (t) and E2 (t), measured along γ1 and γ2 , using the equations:
    
E 1 sin γ2 − sin γ1 E
 x =    1 (7)
Ey cos γ1 sin γ2 − sin γ1 cos γ2 − cos γ2 cos γ1 E2
The time-dependent ellipticity ε(t) of the pulse was evaluated as:
(1 [ 2|Ẽ ||Ẽ | sin(φ − φ ) ])
x y x y
ε(t) = tan arcsin (8)
2 2
|Ẽx | + |Ẽy | 2

where φx and φy indicate the phases of the complex components of the electric field Ẽx and
Ẽy , respectively, with Ex,y = 12 (Ẽx,y + Ẽx,y
∗ 8
).
The XUV spectra generated for α1 and α2 presented small differences, which we attributed
to the small incidence angle on the focusing and steering mirrors between the PG-unit and
the generation cell, and on the finite grazing incidence angle of the attosecond pulses on
the XUV optics. As a result, the two polarisation directions γ1 and γ2 were not perfectly
equivalent and a small amplitude difference and phase shift for the two reconstructed electric
field components E1 (t) and E2 (t) was introduced.
In order to minimise these effects, before each measurement, we performed a calibration us-
ing a linearly polarised perturbing field (along the x direction, see Fig. 3a of the manuscript).
The calibration allowed to determine the correction factor for the ratio of the amplitudes
a = E1 /E2 (typically on the order of unit) and for the relative temporal shift t0 between the
two measurements (typically on the order of a few tens of attoseconds). These parameters
were then used in the reconstruction of the fields with time-dependent polarisation. The
results of a typical calibration are shown in Fig. 3. The small y-component of the field was
attributed to a small misalignment of the delay plate.

1. Sansone, G. et al. Isolated single-cycle attosecond pulses. Science 314, 443-446 (2006).

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DOI: 10.1038/NPHOTON.2017.73
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2. Ullrich, J. et al. Recoil-ion and electron momentum spectroscopy: reaction-microscopes. Rep.

Prog. Phys. 66, 1463-1545 (2003).
3. Wittmann, T. et al. Single-shot carrier-envelope phase measurement of few-cycle laser pulses.
Nature Phys. 5, 357-362 (2009).
4. Feng, C. et al. Complete analog control of the carrier-envelope-phase of a high-power laser
amplifier. Opt. Exp. 21, 25248-25256 (2013).
5. Lewenstein, M., Balcou, P., Ivanov, M. Y., L’Huillier, A. & Corkum, P. B. Theory of High-
Harmonic Generation by Low-Frequency Laser Fields. Phys. Rev. A 49, 2117-2132 (1994).
6. Sansone, G., Vozzi, C., Stagira, S. & Nisoli, M. Nonadiabatic quantum path analysis of high-
order harmonic generation: Role of the carrier-envelope phase on short and long paths. Phys.
Rev. A 70, 013411 (2004).
7. Sansone, G. Quantum path analysis of isolated attosecond pulse generation by polarisation
gating. Phys. Rev. A 79, 053410 (2009).
8. Strelkov, V. et al. Generation of attosecond pulses with ellipticity-modulated fundamental.
Appl. Phys. B 78, 879-884 (2004).

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Figure 1. Experimental beamline and photoelectron angular distributions measured

for α1 and α2 .
a, Schematic view of the experimental beamline including the REMI, the XUV spectrometer, and
the Stereo-ATI. b,c, Photoelectron angular distributions measured by the REMI for the rotation
angles of the delay plate in the PG unit of the driving pulses in the plane perpendicular to the
propagation direction α1 = 45◦ and α2 = −45◦ , respectively. The two axis represent the x and y
components of the photoelectron momentum. The white solid lines indicate the polarisation of the
two attosecond pulses in the interaction region of the REMI.

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Figure 2. Data analysis.

Phase modulation retrieved from the experimental data (Fig. 3 of the manuscript) using the Fourier
transform algorithm (solid line) and the single-fringe analysis (short dotted line).

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Figure 3. Calibration for the reconstruction of linearly polarised pulses.

Electric field components along the x (solid line) and y (short dotted line) directions for nominal
linear polarisation along the x direction.

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