Complete retrieval of the field of ultrashort optical
pulses using the angle-frequency spectrum
F. Bragheri,1,5D. Faccio,2,5,* F. Bonaretti,2,5A. Lotti,2,5M. Clerici,2,5O. Jedrkiewicz,2,5C. Liberale,3
S. Henin,1L. Tartara,1,5V. Degiorgio,1,5and P. Di Trapani2,4,5
1Department of Electronics, University of Pavia, Via Ferrata 1, I-27100 Pavia, Italy
2CNISM & Department of Physics and Mathematics, University of Insubria, Via Valleggio 11, I-22100 Como, Italy
3Department of Experimental and Clinical Medicine, University of Magna Graecia, Viale Europa,
I-88100 Catanzaro, Italy
4Department of Quantum Electronics, Vilnius University, Sauletekio Avenue 9, Building 3, LT-2040,
5Virtual Institute for Nonlinear Optics, Centro di Cultura Scientifica Alessandro Volta, Villa Olmo,
Via Simone Cantoni 1, 22100 Como, Italy
* Corresponding author: firstname.lastname@example.org
Received July 28, 2008; revised October 28, 2008; accepted November 3, 2008;
posted November 10, 2008 (Doc. ID 99322); published December 5, 2008
We propose an experimental technique that allows for a complete characterization of the amplitude and
phase of optical pulses in space and time. By the combination of a spatially resolved spectral measurement
in the near and far fields and a frequency-resolved optical gating measurement, the electric field of the pulse
is obtained through a fast, error-reduction algorithm. © 2008 Optical Society of America
OCIS codes: 320.7100, 320.5550, 120.3940.
Ultrashort laser pulses are widely used in many labo-
ratories and are routinely adopted for many applica-
tions. In past decades, various techniques were devel-
oped to measure the electric field of optical pulses as
a function of time or frequency. The different ap-
proaches belong to two main categories: spectro-
graphic and interferometric techniques. The most
known examples of these approaches are frequency-
resolved optical gating (FROG)  and spectral inter-
ferometryfor direct electric-field
(SPIDER), respectively . The first is based on the
measurement of the temporally resolved spectrum
(spectrogram). An iterative inversion algorithm is ap-
plied to the measured spectrogram in order to re-
trieve the electric field. The second approach consists
of the measurement of the interference between a
pair of spectrally sheared replicas of the input pulse.
A direct inversion of the measured interferogram
yields the electric field of the pulse.
These techniques have an analog in the spatial co-
ordinate, and are often applied by assuming that
temporal and spatial features of pulses are indepen-
dent. This assumption is no longer valid for situa-
tions involving beam focusing , pulse shaping us-
ing zero-dispersion line  and compression , or
nonlinear interactions (see, e.g., [6,7]) leading to
space-time coupling effects. In the past few years
some techniques have been proposed to characterize
the amplitude and phase profile of pulses both in
space and time. Among these, the most interesting
are variants of FROG and SPIDER techniques with
extension to the spatial dimension characterization.
Indeed, in  a combination of FROG and digital ho-
lography is proposed to characterize the complete
(3D) electric field of a train of laser pulses possessing
at least one point in space where all the frequencies
are present. This method is based on the use of a tun-
able filter or of a series of bandpass filters. A varia-
tion of this technique is proposed in , which has
the advantage of being a single-shot measurement.
The same authors recently proposed a technique that
is based on crossed-beam spectral interferometry
. By exploiting the coherence between a reference
pulse and the one under investigation, this allows for
a characterization of the field both in time and space
with high spectral resolution, but requires a high-
resolution scan along the spatial dimension.
On the other hand, the extension to the spatial di-
mension of the SPIDER technique has been proposed
in [11–13]. Thanks to a direct inversion algorithm,
the technique is capable of a fast reconstruction of
the electric field as a function of one transverse spa-
tial coordinate and time. If high-energy interfering
pulses are available, this technique grants single-
shot operation, but it demands different experimen-
tal setups if pulses with largely different durations
have to be measured.
In this Letter we propose a new technique for the
complete retrieval of the optical amplitude and phase
using the ?k?,?? spectrum (CROAK) in one spatial
dimension and in time for ultrashort complex pulses
of arbitrary duration. The technique is ideally devel-
oped for pulses with cylindrical symmetry but may
also be applied, as in this work, to pulses that are
space–time coupled only along one transverse dimen-
sion. The method is based on the measurement of the
frequency-resolved near and far fields of the pulse,
i.e., ?r,?? and ?k?,?? spectra, respectively. From
these spectra we obtain the spatial phases for each
frequency of the pulse, which are finally linked to-
gether by a single FROG measurement. The tech-
nique exploits a sequence of two phase-retrieval algo-
rithms, already known in the literature, which allows
for a fast and accurate reconstruction of the ampli-
tude and phase of the pulse.
The proposed technique is not single shot, since
three different measurements are needed. However,
this does not significantly complicate the experimen-
OPTICS LETTERS / Vol. 33, No. 24 / December 15, 2008
0146-9592/08/242952-3/$15.00© 2008 Optical Society of America
tal setup, as slight modifications of the same layout
are sufficient to obtain all the necessary experimen-
tal data. The experimental layout is shown in Fig. 1.
The setup is very similar to that of the well-known
second-harmonic FROG (SH-FROG) with the addi-
tion of the lenses L that form an image of the input
pulse to be characterized onto the input facet of the
KDP crystal used for second-harmonic generation.
The noncollinear second-harmonic signal is then im-
aged by the lens LFonto the entrance slit of an imag-
ing spectrometer and recorded with a 16 bit CCD
camera (Andor Technologies). When repeated for
each value of the delay ?, this measurement gives the
SH-FROG spectrogram S as a function of the radial
position r, i.e., S=S?r,?,??. The mirror M may then
be moved in order to intercept beam 2 instead of
beam 1 and allow measurement of the time-
integrated near-field spectrum I?r,??. Finally, by
moving the lens LFto a distance from the spectrom-
eter equal to its focal length, the time-integrated far-
field spectrum I??,?? is measured and subsequently
expressed in the function of the variables ?k?,??.
Once all data are collected, we use an error-reduction
algorithm to retrieve the spatial phase for each wave-
length of the pulse. The retrieval algorithm is based
on the error-reduction, or Gerchberg–Saxton, algo-
rithm . It consists of four steps: (1) Fourier trans-
form the electric field profile E?r,??, whose ampli-
tude is obtained by the experimental intensity while
the phase is an initial guess; (2) replacement of the
modulus of the resulting computed Fourier transform
with the measured ?E?k?,??? profile; (3) inverse Fou-
rier transform; and (4) replacement of the modulus of
the resulting computed field E?r,?? with the mea-
sured one to form a new estimate of the object. Con-
vergence is verified by comparing the retrieved
?k?,?? spectrum with the measured ?k?,?? spec-
trum. This procedure gives us the amplitude and
phase in r for each frequency. The electric field may
be written as
E?r,?? = ?E?r,???ei???r,??+?0????.
We note that error reduction is applied only along the
spatial dimension so that each frequency slice is in-
dependent of the other. In other words, there is a ran-
dom phase jump between each frequency slice ?0???;
this can be eliminated by renormalizing the phase at
each ? for a fixed spatial position r (e.g., r=0) to the
phase obtained at the same coordinate r from the
FROG measurement S?0,?,??. The phase profiles at
all other incidents of r will therefore also be correctly
aligned, and we obtain the full ?r,t? amplitude and
phase of the pulse by performing a final temporal
We verified the technique by characterizing a tilted
pulse . Experiments were performed with a
1.2 ps, 1055 nm laser pulse delivered by a 2 Hz, am-
plified Nd:glass laser (Twinkle, Light Conversion
Ltd.). The tilted pulse was obtained by means of a
grating that induces angular dispersion on the in-
coming radiation. In particular, the laser pulse was
directed on an 1800 lines/mm grating, producing a
front tilt of 82.5°, as shown schematically in Fig. 2(a).
The imaging lenses L were placed so as to image a
plane 2 cm after the grating onto the KDP crystal in-
The measured spectrogram at r=0 and intensity
profiles of the near- and far-field spectra are shown in
Figs. 2(b)–2(d), whereas the spatiotemporal profile of
the pulse reconstructed through the retrieval algo-
rithm is shown in Fig. 3. In particular, Fig. 3(a)
shows the normalized amplitude of the retrieved
pulse as a function of time t and space r, whereas Fig.
3(b) reports the retrieved phase profile shown only in
correspondence of the pulse. Optimal convergence
was obtained with an initial guess function having a
quadratic spatial phase curvature, and repeated com-
putational trials showed convergence to the same
spatial phase distribution or its phase-conjugated
one. From the first plot we observe that the algorithm
has correctly retrieved the tilted pulse with a space–
time coupling given by r=?·t. The constant param-
eter ? is related to the tilt angle, which is found to be
82°, in extremely good agreement with the expected
value. On the other hand, the second plot shows a
parabolic behavior of the phase along the pulse with
a maximum in correspondence of the central spatial
and temporal coordinates. The curvature of the spa-
tial phase, which may also be derived directly from
used in the three steps of the technique. Lenses L form a
telescope that images the object to be measured onto the
nonlinear KDP crystal. Lens LFis moved so as to image the
output facet of KDP or to give the object’s far field at the
entrance of the spectrometer. M, mirror.
(Color online) Experimental layout of the setup
Fig. 2. (Color online) (a) Tilted pulse produced by the grat-
ing. Experimental data: (b) measured spectrogram, (c) far-
field spectrum, (d) near-field spectrum.
December 15, 2008 / Vol. 33, No. 24 / OPTICS LETTERS
the output of the error-reduction algorithm, has a ra- Download full-text
dius of 56 m along the tilt direction and is given by
the input laser beam. We note that a parabolic phase
in the transverse dimension may be expected after
reflection from a grating . Evaluating this phase
curvature analytically , we find that at a distance
of 2 cm after the grating (corresponding to the plane
imaged onto the nonlinear crystal) we obtain a cur-
vature with a radius of 215 m, which is relatively
large and is not visible in our measurements.
Our data show that the proposed technique can
strongly coupled space-time pulses. It is worth men-
tioning that simple pulses such as a collimated
Gaussian beam also can be characterized by choosing
an appropriate lens to perform the far-field measure-
ment. We emphasize that although it is not single
shot, the technique requires a simple setup and
shares the same advantages of standard FROG, such
as the possibility of characterizing even very low-
intensity pulses with largely different durations
without the need to modify any of the optical ele-
ments. We also note that the measurements de-
scribed here use only a single FROG measurement at
a given r, although use of the CCD gives the FROG
traces at all r. This is acceptable only if, at this r, all
of the pulse frequencies are present simultaneously.
In any case, this limitation may be easily removed by
using FROG measurements at more than one r by
taking care that the overlap of the obtained spectra
covers the full pulse spectrum. No additional mea-
surements are required due to the fact that the
FROG traces are measured at different spatial coor-
dinates using a CCD.
In conclusion, we propose a new technique based
on the combination of FROG ?r,?? and ??,?? spectra
measurements to completely characterize in space
and time the amplitude and phase of an ultrashort
pulse. The novelty of the method consists mainly in
linking the two spatial dimensions, i.e., near and far
field, exploiting the two integrated measurements.
By means of these we can reconstruct the phase pro-
file through a fast error-reduction algorithm. More-
over, the same experimental arrangement can be
used to characterize shorter pulses, which, having
broader spectra, would benefit from the high reso-
lution achievable through the use of the imaging
The authors gratefully acknowledge the financial
support from CNISM, the INNESCO project, the
RBIN04NYLH project, and the Lithuanian Science
and Technology Foundation Project ConTex. P. Di
Trapani acknowledges the EU Marie Curie Chair ac-
tion STELLA, contract MEXC-2005-025710.
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phase of the TP in function of time and space. The energy of
the pulse lies on a straight line r=cost·t and the phase
shows a quadratic chirp.
(Color online) (a) Retrieved amplitude and (b)
OPTICS LETTERS / Vol. 33, No. 24 / December 15, 2008