Four-dimensional investigation of the 2nd order volume autocorrelation technique
ABSTRACT The 2nd order volume autocorrelation technique, widely utilized in directly measuring ultra-short light pulses durations,
is examined in detail via model calculations that include three-dimensional integration over a large ionization volume, temporal
delay and spatial displacement of the two beams of the autocorrelator at the focus. The effects of the inherent displacement
to the 2nd order autocorrelation technique are demonstrated for short and long pulses, elucidating the appropriate implementation
of the technique in tight focusing conditions. Based on the above investigations, ahigh accuracy 2nd order volume autocorrelation
measurement of the duration of the 5th harmonic of a 50fs long laser pulse, including the measurement of the carrier wavelength
oscillation, is presented.
- [Show abstract] [Hide abstract]
ABSTRACT: The two basic approaches underlying the metrology of attosecond pulse trains are compared, i.e. the 2nd order Intensity Volume Autocorrelation and the Resolution of Attosecond Beating by Interference of Two photon Transitions (RABITT). They give rather dissimilar results with respect to the measured pulse durations. It is concluded that RABITT may underestimate the duration due to variations of the driving intensity, but in conjunction with theory, allows an estimation of the relative contributions of two different electron trajectories to the extreme-ultraviolet emission. Comment: 12 Pages, 4 Figures, Corresponding author email address: ptzallas@iesl.forth.gr05/2010; - SourceAvailable from: Dimitris Charalambidis[Show abstract] [Hide abstract]
ABSTRACT: The emission of above-ionization-threshold harmonics results from the recombination of two electron wavepackets moving along a "short" and a "long" trajectory in the atomic continuum. Attosecond pulse train generation has so far been attributed to the short trajectory, attempted to be isolated through targeted trajectory-selective phase matching conditions. Here, we provide experimental evidence for the contribution of both trajectories to the harmonic emission, even under phase matching conditions unfavorable for the long trajectory. This is finger printed in the interference modulation of the harmonic yield as a function of the driving laser intensity. The effect is also observable in the sidebands yield resulting from the frequency mixing of the harmonics and the driving laser field, an effect with consequences in cross-correlation pulse metrology approaches. Comment: 13 pages, 3 figuresPhysical Review A 07/2010; · 3.04 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: The two basic approaches underlying most of the metrology of attosecond pulse trains are compared in the spectral region {approx}14-24 eV, that is, the second-order intensity volume autocorrelation and the resolution of attosecond beating by interference of two photon transitions (RABITT). They give rather dissimilar pulse durations. It is concluded that for the present experimental conditions RABITT may underestimate the duration under measurement, due to variations of the driving intensity, but in conjunction with theory allows an estimation of the relative contributions of two different electron trajectories to the extreme-ultraviolet (XUV) radiation.Physical Review A 08/2010; · 3.04 Impact Factor
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Four-dimensional investigation of the 2nd order volume autocorrelation technique
O. Faucher1, P. Tzallas2*, E.P. Benis2, J. Kruse2, A. Peralta Conde2, C. Kalpouzos2
and
D. Charalambidis2,3
1Institut Carnot de Bourgogne, UMR 5209 CNRS-Université de Bourgogne, 9 Av. A.
Savary, BP 47 870, F-21078 Dijon Cedex, France
2Foundation for Research and Technology-Hellas, Institute of Electronic Structure and
Laser, PO Box 1527, GR71110 Heraklion, Crete, Greece
3Department of Physics, University of Crete, PO Box 2208, GR71003 Heraklion, Crete,
Greece
*Corresponding author: ptzallas@iesl.forth.gr
ABSTRACT
The 2nd order volume autocorrelation technique, widely utilized in directly
measuring ultra-short light pulses durations, is examined in detail via model calculations
that include three-dimensional integration over a large ionization volume, temporal delay
and spatial displacement of the two beams of the autocorrelator at the focus. The effects
of the inherent displacement to the 2nd order autocorrelation technique are demonstrated
for short and long pulses, elucidating the appropriate implementation of the technique in
tight focusing conditions. Based on the above investigations, a high accuracy 2nd order
hal-00429674, version 1 - 3 Nov 2009
Author manuscript, published in "Applied Physics B: Lasers and Optics 97 (2009) 505"
DOI : 10.1007/s00340-009-3559-z
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2
volume AC measurement of the duration of the 5th harmonic of a 50 fs long laser pulse,
including the measurement of the carrier wavelength oscillation, is presented.
1. Introduction
The measurement of the duration of an ultra-short radiation pulse or
determination of the characteristic time of an ultra-fast process relies on a non-linear
process. During the last decade, the well established in femtosecond (fs) metrology 2nd
order autocorrelation (AC) technique has been successfully extended to the ultra-violet
(UV) and extreme ultra-violet (EUV) spectral regions, serving for the temporal
characterization of pulses in the attosecond (asec) time scale [1-3]. 2nd order AC is a
method that directly and safely determines the temporal duration of asec pulses. Once
established, it becomes straightforward provided that sufficient EUV intensity is
available.
Apart from the generation of asec EUV radiation and its applications to
attophysics and attochemistry [4,5], the availability of intense few-fs UV-EUV pulses, is
of potential importance to the study of the dynamics of photochemical reactions [6,7], in
view of the fact that the most of the organic compounds absorb in this spectral region.
Towards this target, sub-10 fs UV and EUV pulses can be for instance generated through
the interaction of a gas phase medium with high-peak-power few-cycle laser pulses [8] or
with many-cycle laser pulses of appropriately tailored polarization utilizing the
interferometric polarization gating technique [9,10]. In certain cases though, even UV-
EUV pulses of few tens of fs would be sufficiently short to study various photochemical
processes [11]. A prerequisite for a thorough quantitative study of the above dynamics is
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the most accurate knowledge of the duration of the EUV pulse, which is
straightforwardly accessible by the 2nd order AC technique.
In this work, we present numerical results for the applicability conditions of the
2nd order AC approach obtained by model calculations that includes not only three-
dimensional integration over a large ionization volume and temporal delay, but also
spatial displacement of the two beams of the autocorrelator at the focus. In contrast to all
previous works, the inclusion of the spatial displacement in our model is expected not
only to put under stringent tests the applicability of the 2nd order AC approach but also
provide with substantial information about the restrictions of the method. Indeed, the
inclusion of the spatial displacement becomes compulsory when the spatial length of the
pulse to be characterized becomes comparable to the confocal parameter of the foci.
While the effect of this displacement seems in principle negligible for few fs or sub-fs
pulses, it is not obvious how it may affect the 2nd order AC trace in tight focusing
geometries (indispensable to 2nd order AC of VUV-EUV pulses) of pulses longer than
several tens of fs. In addition, the 2nd order AC approach can be used in studying,
simultaneously with the pulse characterization, the dynamics of many ultra-fast processes
in photochemistry, since most of the organic compounds absorb in the VUV-EUV
spectral region [11]. In this approach, the AC trace is a convolution of both the pulse
duration and the characteristic decay times of the photochemical processes under
investigation and may well exceed a hundred of fs in duration, suffering thus the effects
of spatial displacement. Since no investigations have been reported on this displacement
issue up to now, at least to our knowledge, the introduced model is expected to provide
with the appropriate conditions for safely applying the 2nd order AC method to the
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characterization of UV-EUV pulses. Based on the above investigations, a high accuracy
measurement of the duration of the 5th harmonic of a 50 fs long laser pulse including the
carrier wavelength oscillation is presented. Earlier measurements of the 5th harmonic
through alternative approaches can be found in Refs 12 and 13.
2. Theoretical treatment
The theoretical description of the non-linear autocorrelator is based on the
description of the three-dimensional light distribution near focus presented in detail in
Ref 14, and adapted here for the geometry shown in Fig. 1. The harmonic field E,
reflected by the split spherical mirror, is obtained as the superposition of the two fields Ei,
with i=1 and 2, corresponding to the reflected parts from each half of the split mirror,
respectively. Assuming Gaussian temporal profiles and propagation along the ˆ z
direction, the fields are written
2
0
( , , , ) E x y z t ( , , )expx y z2ln2
−
exp( ( ))t
ii
p
t
Ei
ω
τ
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
,
where τp is the pulse duration, ω is the field carrier angular frequency and Ei0 (x,y,z) is the
spatial distribution of the complex field. The spatial distribution of the field amplitudes at
a certain space point P2 in the neighborhood of the focal point O (x = y = z = 0), is
obtained by the Debye integral [14], after applying the Huygens-Fresnel principle in a
small angles geometry, as
Ω−
−
λ
=
∫∫Ω
dRs
ˆ
ikPiPA
i
PE
i
ii
) exp())( exp()()(
1120
φ
.
k and λ are the wave vector and the wavelength of the radiation, respectively,
while Ω corresponds to the solid angle which one half of the split mirror subtends at the
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focal point O. As shown in Fig. 1, the position vector R
? specifies the position P2 relative
to the origin O, while ˆ s is the position unit vector that specifies the position P1 on the
surface of the split mirror relative to O.
)(1PA
is the incident field amplitude at P1. It is
considered Gaussian
2
0
2
( )exp()
( )
r
A rA
w r
=−
, with
222
rxy
=+
, w the beam radius and
A0 a constant factor).
iφ represents the phase of the incoming beam at the surface of the
split mirror.
FIGURE 1. Schematics of the model optical geometry. SSM: Split Spherical Mirror.
Considering all the above, the 2nd order volume AC trace of the harmonic field E
is obtained as a function of the delay
c /2δ τ =
,where δ is the displacement between the
two halves of the split mirror and c the speed of light , as
∫ ∫∫∫
−+=
4| )
τ
,,,(),,,(|)(
τ
tzyxEtzyxEdxdydzdtS .
S has been calculated over a rectangular parallelepiped volume of width 20 w0 and length
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6.5 b, with w0 and b being the beam waist and the confocal parameter of the Gaussian
harmonic field, respectively.
Finally, the intensity distributions at the focus of a split spherical mirror for the
displacement delays of zero, λ/4 and λ/2 are illustrated in Fig. 2.
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FIGURE 2. Intensity distributions at the focus of a split spherical mirror for the
displacement delays of: (top) zero, (middle) λ/4 and (bottom) λ/2. X and Z axis are both
in units of the wavelength λ while intensity I is in arbitrary units. Z axis is the axis of
propagation.
3. Experiment
The experimental set-up is shown in Fig. 3. A 10 Hz Ti:Sapphire laser system
delivering 50 fs pulses with energy up to 150 mJ and carrier wavelength at 800 nm was
used. An annular laser beam with 2 cm outer diameter and energy of 15 mJ/pulse was
focused by a 3.5 m focal length lens into a pulsed Xe gas jet, generating harmonics. After
the jet an iris stopped the IR beam while a Si plate was placed at the fundamental’s
Brewster angle of 72o blocking the residual IR radiation and reflecting the harmonics
towards the interaction chamber [15]. Directly after the Si wafer, an aperture blocked the
residual annular part of the fundamental laser beam. The harmonic beam was
subsequently focused into a Kr gas jet, by a wave-front splitting arrangement consisting
of a bisected gold spherical mirror of 5 cm focal length, the one half of which served as
the translation unit. The translation is succeeded via a dual system of piezoelectric
crystals (coarse and fine combination) allowing for a total stroke of 500 fs and a
minimum resolution of 10 asec. The 5th harmonic two-photon non-resonant ionization of
Kr was used as the non-linear detector.
Kr is ionized by 5th harmonic two-photon absorption (IP = 13.995 eV) [16]. In
order to avoid unwanted contributions from single-photon ionization by photons of the 9th
and higher harmonics, a 5 mm thick MgF2 plate was introduced in the harmonic beam
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filtering out all wavelength contributions shorter than 113 nm. The MgF2 plate acted also
as a temporal separator for the harmonics since the group velocity and the group velocity
dispersion are different for each harmonic. Since the various harmonics are temporally
separated and all possible multi-photon absorption channels are non-resonant, color-
mixing channels do not contribute to the excitation processes and the excited population
follows instantaneously the variations of the harmonic fields [17,18,19]. In addition, the
driving laser pulse was stretched properly in order to compensate for the chirp introduced
to the 5th harmonic by the MgF2 plate, thus leading to a Fourier transform limited (FTL)
5th harmonic pulse. As result, the duration of the surviving harmonics other than the 5th
become much longer than that of an FTL value, much too long to account for a second
order process. In a step further, it has been estimated that under the present experimental
conditions the intensities of the 3rd and 5th harmonic are approximately the same at the
interaction region. Since at such intensities the 3rd harmonic three-photon ionization rate
is much smaller than that of the 5th harmonic two-photon ionization, it can be safely
deduced that the contribution of the 3rd harmonic to the ionization signal is negligible.
FIGURE 3. Schematics of the non-linear volume AC apparatus.
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4. Results and discussion
Figure 4a illustrates an instructive example that sheds light into the effects of
displacement present in the autocorrelation method. Assuming a realistic tight focusing
condition of 5 cm focal length for a bisected spherical mirror and a CW laser pulse, we
performed calculations based on the presented model for the 2nd order volume AC. The
grey shaded area corresponds to the interferometric trace while the black solid line to the
intensity trace. A clear reduction of the signal from its maximum value, which is
supposed to be constant for a CW laser, is evident, and is due to the spatial separation of
the two foci of the two halves of the split mirror in the focal area. Specifically, a
reduction of 5 % is evident for a spatial displacement of 100 fs (or 30 μm) from both
sides of the zero delay (shadowed area in Fig. 4a). Even though the reduction for this
delay region is negligible, the displacement effects become increasingly important for
longer delays in such tight focusing conditions and should be considered accordingly.
Figures 4b and c show two realistic examples involving the model calculations of
a 30 fs and 200 fs duration Gaussian laser pulses, respectively, at the same geometrical
and focusing conditions as in Fig. 4a. The grey shaded area corresponds to the
interferometric trace while the black solid lines to the intensity trace. The black dots
correspond to the conventional 2nd order intensity volume AC trace, i.e., a 2nd order
intensity volume AC trace that does not take into account the displacement. As it is seen
in Fig. 4b, for the case of 30 fs pulse the conventional intensity volume AC trace is
exactly the same with the model calculations of the intensity volume AC trace. In
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addition, a Gaussian function (not shown in the graph) fits perfectly to these traces. This
is because the variation of the signal due to the foci displacement is negligible in the
scanning area of -100 fs to +100 fs (shadowed area in Fig. 4a), as it is inferred form Fig.
4a. However, in the case of 200 fs pulse, deviations start to appear at the tails of the trace,
as shown in Fig. 4c. Therefore, in the case of pulses longer than 200 fs, it is necessary to
de-convolve the AC trace from the function which describes the dependence of the ion
signal due to the foci displacement, in order to extract the duration of the pulse. This new
finding adds complementary to the studies of previous three-dimensional [2], two-
dimensional [20] and quasi-three-dimensional [21] works, in which the displacement of
the two foci was suitably ignored as it is now confirmed redundant for the given
experimental conditions.
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-500-400 -300-200 -1000 100200300 400500
0.4
0.6
0.8
1.0
-600-3000 300 600
1
2
2
nd order volume AC signal
Delay (fs)
Interfer. AC
Intens. AC
(a)
-100 -500 50100
1
2
3
2
nd order volume AC signal
Delay (fs)
Interfer. AC
Intens. AC
Conv. AC
(b)
Delay (fs)
Intens. AC
Conv. AC
(c)
FIGURE 4. Model calculations of the 2nd order volume interferometric (grey shaded
areas) and intensity (black solid lines) AC traces for a a) CW laser, b) 30 fs, and c) 200
fs pulses. Black dots: Conventional 2nd order intensity volume AC traces. The 200 fs
shadowed area in graph 3a depicts the area of negligible (< 5 %) deviation of the AC
signal due to spatial displacement for the tight focusing conditions considered here (see
text for detais).
Based of the results shown in Fig. 4b, the traces contrast, defined as the peak to
background ratios in the cases of intensity and interferometric volume 2nd order AC
traces, are obtained as 2.4 and 3.4, respectively, which are slightly higher than those
reported in [13]. However, it has been numerically verified that in the limit of infinite
integration volume the aforementioned ratios are reduced to 2.1 and 2.8, respectively, in
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agreement with the values given in [13]. In addition, the modulation depth, which is
defined as
) /() ( 2
min maxminmax
SSSSM
+−≡
, is measured
8 . 0
=
M . Finally, the
deconvolution factor, which is the ratio of AC width (FWHM) to the original pulse width
(FWHM), reads 1.51 for the interferometric AC while it is reduced to 1.41 for the
intensity AC case, in accordance to previous studies [20], as the inclusion of the
displacement should not affect the above results for such short pulses.
The same values are obtained from the 200 fs pulse duration case which in
principle shows that the displacement affects the measured duration negligibly. However,
the above measurement is valid only because we were precisely aware about the
background AC signal value. In realistic conditions, the background AC baseline would
not maintain a constant value, as clearly shown in Fig. 4c, but rather a reduced with delay
value. Therefore, an effortless Gaussian fit may very well lead to an inaccurate pulse
duration extraction. In such cases only the inclusion of displacement in a model
calculation would safely account for the pulse duration extraction from the AC trace.
It is of significant importance to emphasize at this point that all the above analysis
is valid for pulses with durations close to their Fourier-transform limited (FTL) value. For
non-FTL pulses, the presence of the chirp is responsible for the appearance of wings at
the tails of the AC trace, a fact that complicates the situation and is not included in our
study. This point indicates that in order to observe non-negligible effects due to the
inherent displacement in the 2nd order volume AC approach, FTL pulses of durations
longer than 200 fs for tight focusing geometries are needed. Considering the fact that
nowadays the majority of the fs laser installations worldwide involve laser FTL pulses of
less than 50 fs durations, it becomes unrealistic to meet the conditions for observing the
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displacement effects in 2nd order AC traces, except maybe in the XFEL installations that
still involve pulses longer than 100 fs. However, inverting the argument, the above
reasoning implies that the 2nd order volume AC approach can be safely applied by the
laboratories worldwide to directly and accurately obtain the generated harmonic pulses
duration as long as the intensity required is available.
Following this path, which is establishing the negligibility of the displacement
effects on the 2nd order AC trace for pulses of few tens of fs duration, the duration of the
5th harmonic of a Ti:Sapph laser has been accurately determined utilizing the 2nd order
volume AC measurements shown in Fig. 5. The two-photon ionization signal of Kr is
plotted as a function of the delay τ between the two 5th harmonic pulses of a Ti:Sapph
laser pulse with 50 fs duration over an interval of 250 fs with a step of 3.3 fs (open
circles). For each delay step 125 laser shots were accumulated. A Gaussian fit (dash line)
to the data results in a duration of 27.3 ± 0.7 fs, which is a value close to the FTL value of
22.4 fs. A model calculation for the 2nd order volume AC trace involving a pulse of 27.3
fs duration and accounting for the measured trace contrast is shown in Fig. 5 (grey solid
line) depicts no essential deviation from the Gaussian fit, as it was expected. The
measured peak to background ratio is found to be 1.8, which is close to the calculated
value of 2.1, the deviation attributed primarily to the imperfect overlap of the two beams
at the focal area.
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-100 -500 50 100150200
1.0
1.2
1.4
1.6
1.8
2.0
2.2
-1.0 -0.5 0.0 0.5 1.0
Delay (τ / T5th)
1.6
1.8
2.0
2.2
2.4
Kr
+ Signal (arb. u.)
Delay τ (fs)
125 shots/point
Intens. AC
Gaussian fit
τ5th=27.3 fs
Kr
+ Signal (arb. u.)
25 shots/point
20 points average
Sinusoidal fit
FIGURE 5. Open circles: Measured 2nd order volume AC trace of the 5th harmonic.
Grey solid line: Simulated 2nd order volume AC trace. Dash line: Gaussian fit. The
measured 2nd order interferometric volume AC trace is shown in the inset. Grey dots:
Experimental data. Black dots: Running average over 20 data points. Black solid line:
Sinusoidal fit to the averaged data.
In a step further, the oscillation of the ion yield at the 5th harmonic field frequency
was recorded and is shown in the inset of Fig. 5. In this trace the delay step is 10 asec,
while for each point of the trace 25 shots were accumulated (grey dots). A running
average of 20 data points is shown in blue dots. The averaged data were fitted to a
sinusoidal function (black line) resulting in a period of 540 ± 30 asec. This result
establishes access to sub-500 asec resolution for our detection apparatus in a spectral
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region of high interest to ultra-fast photochemical processes. The modulation depth of the
oscillations in the interferometric AC trace reads M = 0.22 which is 3.6 times smaller
than the calculated value of 0.8. This divergence is attributed to laser pulse energy
instabilities and/or the spatial chirp of the 5th harmonic across the beam cross section at
the focus. It is worth noticing here that the coarse and fine scan traces correspond to
different set of data and thus at possibly slightly different beam stability conditions. For
this reason the noise level of the two traces is not directly comparable. However we
should clarify that the fluctuations in the coarse scan are not purely due to noise, but to a
large extent, because the measured points correspond to quasi randomly chosen points of
an interferometric trace. Thus, the modulation inherent in an interferometric trace appears
as a strongly fluctuating trace in a coarse scan.
4. Conclusions
Model calculations for the 2nd order AC approach that include three-dimensional
integration over a large ionization volume, temporal delay and spatial displacement of the
two beams of the autocorrelator at the focus were presented. The effects of the inherent to
the 2nd order AC approach displacement were exposed by certain examples elucidating
the restrictions of the 2nd order AC approach in tight focusing conditions. In general, the
2nd order AC approach can be safely applied for pulses with duration up to few tens of fs,
however, special care should be taken in the pulse duration extraction in cases where the
pulse duration exceeds the order of 100 fs. The same applies to 2nd order AC traces that
involve convolution of both the pulse duration and the characteristic decay times of
photochemical processes. The negligibility of the displacement effect was demonstrated
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