Alonso et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 933
Spatiotemporal amplitude-and-phasereconstruction by Fourier-transform of
interference spectra of high-complex-beams
Benjamín Alonso,1,* Íñigo J. Sola,1 Óscar Varela,1 Juan Hernández-Toro,1 Cruz Méndez,2 Julio San Román,1
Amelle Zaïr,1 and Luis Roso2
1Universidad de Salamanca, Área de Óptica, Departamento de Física Aplicada, E-37008 Salamanca, Spain2Centro de Láseres Pulsados Ultracortos Ultraintensos (CLPU), E-37008 Salamanca, Spain
. INTRODUCTIONaser pulse characterization is a fundamental issue forhe extensive community of laser users. Knowledge of thetructure of laser beams before they are used in certainxperiments or applications and the study of the pulse af-er it has undergone a given process, are essential. How-ver, the complexity of the electric field requires certainpecific approaches when addressing it. The electric fieldf the light is in general a vector that depends on timend three spatial variables (two transverse directions andhe longitudinal direction of propagation). We shall ne-lect the vectorial character of the electric field becauseere we are dealing with linearly polarized pulses.The temporal variation and the spatial profile of ul-
rashort laser pulses are often characterized separately.hus, the temporal characterization of pulses has a fairly
ong history. Pulse autocorrelation can afford an idea ofhe laser pulse duration and form [1]. Several robust andeliable techniques are now well established for themplitude-and-phase retrieval, such as the frequency re-olved optical gating (FROG) [2], spectral phase interfer-metry for direct electric-field reconstruction (SPIDER)3], spectral interferometry (SI) [4,5], and its combinationith the FROG, known as the temporal analysis by dis-ersing a pair of light electric fields [6]. Additionally, thepatial profile can be measured with a standard detectora charge-coupled device (CCD)], but the spatial phasei.e., the wave-front) can also be measured with differentechniques, e.g., Hartmann–Shack [7].
One approach to the study of spatiotemporal evolutiononsists of measuring the temporal pulse profile at differ-nt spatial positions with SPIDER [8,9], and the grating-liminated no-nonsense observation of ultrafast incidentaser light electric fields [GRENOUILLE (single-shotROG)] can also be used to measure the pulse-front tilt
10]. Nevertheless, in spite of being very useful theseechniques cannot measure the full spatiotemporal infor-ation and preserve the coupling. The aim of our work is
o develop a simple and robust system capable of recon-tructing the spatiotemporal amplitude and phase of la-er pulses at a fixed propagation distance. Our underlyingotivation is the broad variety of application fields of spa-
iotemporal characterization, such as the study of opticalberrations [11,12] or nonlinear propagation [13–15].Initial schemes for this purpose were based on the spa-
ially resolved SI and did not characterize the referenceeam, thus only measuring phase differences and beingnable to perform complete spatiotemporal reconstruc-ions [11,15]. The spatially resolved SI consists of measur-ng the spectral interferences of a test and a referenceeam across the spatial profile. If the spatio-spectralhase of the reference beam is known, the spatiotemporaloupled amplitude and the phase of the test pulse can beetrieved. In the scheme of Diddams et al. [16], a Mach–ehnder interferometer was used. Those authors filteredhe reference beam spatially and measured it in a singleosition, assuming a constant spectral phase. In a previ-us work, we implemented the same scheme and found
010 Optical Society of America
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934 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Alonso et al.
hat the spatial cleaning of the reference beam is not easynd does not ensure a perfectly homogeneous reference,specially when filtering complex pulses [17].
The amplitude-and-phase reconstruction of spatiotem-oral coupling was studied by Rivet et al. [18], using aombination of Hartmann–Shack and FROG or SPIDER.his scheme, known as the Shackled-FROG, has recentlyeen demonstrated by Rubino et al. [19]. A robust tech-ique used by Dorrer et al. consisted of two-dimensional
spatial and spectral) shearing interferometry [20]. Moreecently, a holographic method reported by Gabolde andrebino, spatially and temporally resolved intensity andhase evaluation device: full information from a singleologram (STRIPED FISH) [21] has demonstrated thebility to measure three-dimensional �x ,y , t� electric fieldsn single-shot. To study nonlinear propagation, Trull et al.eveloped a technique that measures the spatially re-olved temporal cross-correlation [13], measuring the sumrequency with a short probe. This technique provides anmage of the spatiotemporal intensity (but not the phase),hich is valuable information that has been used for mea-
urements of X-waves [14].Another recent technique is based on the spatially en-
oded arrangement temporal analysis by dispersing aair of light electric fields (SEA TADPOLE) [22] scanninghe test beam as proposed by Bowlan et al. [12]. The SEAADPOLE technique measures the spectrally resolvedpatial interferences of two non-delayed crossed beams.his idea had already been implemented by Meshulach etl. [23] in a primitive scheme involving crossing theeams directly and has been adapted by Bowlan et al. [12]y guiding a spatial selection of each beam (test and ref-rence) with equal-length single-mode optical fibers. Theemporal profile of the test beam is recovered from thepatio-spectral trace, hence with the advantage of usinghe full spectrometer resolution. The extension to spa-iotemporal characterization merely consists of scanninghe test beam profile with the fiber.
In this contribution, we report a novel scheme for thepatially resolved SI based on a fiber-optic coupler inter-erometer that has certain advantages in comparison withtandard interferometers. We refer to it as the spatiotem-oral amplitude-and-phase reconstruction by Fourier-ransform of interference spectra of high-complex-beamsSTARFISH).
Our system bears some similarities to the SEA TAD-OLE since the test and reference beams are collectedith optical fiber inputs: both systems are free of align-ent and use a single reference pulse. The main differ-
nces are that our system is based on SI instead of spatialnterferences [spatially encoded arrangement (SEA)] andhat the STARFISH only uses a fiber coupler and a stan-ard spectrometer. This makes our system very simplend robust in experimental terms, and its implementa-ion or upgrading in a laboratory in a plug-and-playcheme is easier (simply plugging the coupler to the spec-rometer). Moreover, the STARFISH measures a singlepectrum for each spatial point (instead of a spatio-pectral trace), thereby reducing the data processing,hich would be more interesting—for example—wheneasuring many spatial points at different propagation
istances. Despite this, the use of the standard SI instead
f spatial arrangements (SEA) involves a loss of spec-rometer resolution, thus limiting the pulse length ca-able of being measured with the STARFISH in compari-on with the SEA TADPOLE [22] and SEA SPIDER [24],lthough we shall show that this is not a problem.
n SI, two beams (test and reference) are delayed with re-pect to each other by a time � and propagate collinearly.he resulting spectrum of both beams is the sum of thepectra plus an interference term containing the informa-ion of the phase difference between the beams, as seen inq. (1). The interference fringes have a period given by
he inverse of the delay. For the sign, we chose the crite-ion that the reference always goes before the test and theelay is positive. The resulting spectrum is
S��� = Stest��� + Sref���
+ 2�Stest���Sref���cos��test��� − �ref��� − ���.
�1�
he fringe-inversion technique, the so-called Fourier-ransform spectral interferometry (FTSI) [25], can be ap-lied to retrieve the phase information. The inverseourier-transform affords three peaks in the temporal do-ain: one centered at t=0, coming from the continuum
pectra (test and reference spectra), and two side-peaksentered at t= ±�, corresponding to the interference term.he continuum contribution �t=0� can be depleted by sub-
racting the test and reference spectra from the interfer-nce spectrum. One side-peak is filtered and returned tohe spectral domain by direct Fourier-transform, fromhere the phase difference between the test and refer-nce beams is extracted. The time delay � must be highnough to prevent central- and side-peak overlap, butmall enough to allow the fringes to be resolved by thepectrometer. The reference spectrum must at least com-rise the whole test spectrum and the spectral amplitudeshould be comparable to have well contrasted fringes. Ifhe reference phase is known, the test spectral phase cane calculated and, together with the test spectrum, theeam can be fully characterized in the temporal domainimply by applying a Fourier-transform. The delay is cal-ulated from the side-peak position at a certain point andhe term �� is added to the retrieved phase.
The extension of the SI to spatiotemporal characteriza-ion in standard interferometer schemes is achieved by apatial reference that is delayed with regard to the testeam as shown in Fig. 1. The reference beam must be spa-ially homogeneous (a flat wave-front and a spectralhase independent of the transverse position) because ineneral it can only be characterized at a single point. Inhis scheme, the test beam is referenced at each spatialosition by a known pulse (thanks to the homogeneouspatiotemporal reference beam) and hence the connectionetween different spatial points in the test profile can bebtained. Accordingly, the experimental measurementsonsist of measuring the spectral interferences as a func-ion of the scanned transverse position of the beam pro-
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Alonso et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 935
le, that is, spatio-spectral traces depending on the posi-ion and the wavelength. In Eq. (1), this involves thepectral amplitude and phase as functions of the wave-ength and the transverse position. The spatially resolvedpectrum (spectral trace) of the test arm is also measured,hich—together with the spectral phase retrieved by theTSI—allows the test beam to be characterized.
. Experimental Setup for STARFISH: The Fiber-Opticouplero avoid the complications of standard interferometer-ased systems (homogeneous reference, precise align-ent, etc.), here we propose an interferometer based on a
ber-optic coupler for the SI (STARFISH). The fiber cou-ler must be single-mode to avoid different mode disper-ions. The fiber coupler was designed to work within aroadband spectral region ranging from 680 to 900 nm,llowing the characterization of ultrashort pulses. Theonfiguration of the fiber coupler comprises two inputorts and a common output port. The reference and testeams enter through each input port, are coupled in theransition, and exit the fiber, delayed, through the sameort. In the experimental scheme (Fig. 2), the unknowneam is split, with each replica being sent to each fiberrm. The test arm fiber input has a motorized stage forhe spatial scan (transverse), whereas the position of theber for the reference arm controls the delay between theulses (longitudinal). The reference arm selects a suitablepatial position containing all spectral components thathe test beam has at any position. This reference is char-cterized temporally with the FROG or SPIDER tech-ique. Since the reference beam is not spatially scanned
ig. 1. (Color online) Scheme of spatiotemporal reference forpatially resolved SI, consisting of using a homogeneous flat ref-rence beam delayed with respect to the test beam and scanninghe position (transverse), and measuring their respective spec-ral interferences. Thus, each spatial position is referenced by anown pulse.
ig. 2. (Color online) Setup based on the fiber-optic coupler in-erferometer for spatially resolved SI. The longitudinal positionf one fiber arm controls the relative delay between reference andest beams. The test beam is scanned transversely (spatial) withts corresponding input fiber arm.
ith the fiber input, only one point of the reference profiles selected with the fiber, and this allows a constant ref-rence to be used instead of a possibly inhomogeneous ref-rence profile. Thus, here we did not implement any spa-ial filtering because it was not necessary. Following thecheme shown in Fig. 1, our proposal consisted of refer-ncing all the positions of the test beam by a single spa-ial point of the reference beam.
The arms of the fiber coupler should be of equal lengthsthis does not exactly occur in reality) in order to avoid in-roducing different dispersions on the beams. Neverthe-ess, we consistently calibrated the global phase differ-nce of the fiber coupler (due to slight differences in theengths of the arms of the fiber coupler) and the beamplitter. This calibration was accomplished by taking a SInterferometry measurement using the same input beamn both arms. It was then taken into account in the recon-truction algorithm to retrieve the correct spectral phase.
The main advantages of this system are its simplicity,ompactness, the fact that alignment is not necessary, thease with which it can be upgraded simply by connectinghe coupler to a different spectrometer, and the fact that aonstant reference is used (we selected the broadest spec-rum region). This means that we do not need a spatiallter for the reference, and approximations concerningeference homogeneity are required. In comparison withtandard interferometers, we do not need a spatially ho-ogeneous reference beam with at least the same spatial
ection and spectrum as the test beam. Moreover, the spa-ial resolution is given by the mode-field diameter of theingle-mode fiber: in our case 4 �m.
. EXPERIMENTAL MEASUREMENTS. Introductionhe experiments were carried out using two different
erawatt-class Ti:Sapphire chirped pulse amplificationCPA) laser systems (both at a 10 Hz repetition rate). Therst system (Spectra Physics, Inc.) delivers laser pulses of20 fs (Fourier limit) with its spectrum centered at 795m. The second system (Amplitude Technologies) pro-ides 35 fs pulses centered at 805 nm. We worked withwo different lasers to test the STARFISH with pulses ofifferent durations and bandwidths. For the temporalharacterization of the reference beam we used the GRE-OUILLE (single-shot FROG, Swamp Optics) and SPI-ER (APE GmbH) devices, whereas for the spectra wesed a commercial spectrometer (AVANTES, Inc.). De-ending on the duration of the pulses, we characterizedhem with the SPIDER (35 fs pulses), where there is nombiguity in the time direction, or with the GRE-OUILLE (120 fs pulses). In the case of the GRE-OUILLE device, we identified the temporal direction byerforming a second measurement with additional knownispersion, as is usually done when using this apparatus.e observed that the GRENOUILLE spatial homogeneity
equirements were fulfilled for the 120 fs laser by measur-ng the profile with a CCD. The spatial scan was per-ormed with a motorized micrometric stage at the sameime as the spectrum was acquired.
We first tested our system for one-dimensional SI,hecking the reconstruction of laser pulses in comparison
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936 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Alonso et al.
ith the SPIDER and GRENOUILLE characterizations.mong others, we performed a test in which the delayas varied from �5 to +5 ps in 50 fs steps, and we ob-
erved that the pulse intensity and phase were recon-tructed independently of the delay over a wide range: upo several picoseconds. We also checked our reconstruc-ion algorithm with simulations involving the character-zation of complex pulses.
In general, interferometry is affected by small fluctua-ions due to system instabilities and, in particular, our fi-er coupler approach was indeed expected to exhibit ahase drift. The consequence is a variation in the relativehase term between the two interferometer arms. Sincehis variation is almost independent of the wavelength,his means a loss of the constant zero-order relative phasef the pulses, thus preventing precise knowledge of theulse wave-front and introducing a small error in theulse front. For our purposes, this was not a problem andndeed there are techniques for overcoming this drawbacknd retrieving the wave-front from this kind of measure-ent [26]. We studied the stability of the interferences for
ingle-shot measurements (acquired continuously with-ut average), tracking the zero-order phase, the full spec-ral phase, the delay, and the time-width of the recon-truction. We measured a zero-order phase drift in thenterferometer of 1.4 rad (peak to peak) during the timesually taken for a measurement to be made (about 1in). The phase drift �0.45�� slightly affected the pulse
ront, with the corresponding temporal shift thus beingimited to 0.60 fs (0.225T, with T being the laser period).or the delay, we calculated a standard deviation of 0.40
s, and for the time-width of the measured pulse we cal-ulated 0.09 fs, whereas the whole spectral phase wasery stable between shots (except for the effect of theero-order phase drift). We also studied stability for re-eated multi-shot averaging measurements, for which webtained blurry and reduced contrast interferences due tohe shift of the fringes. As a result, more stable delays butoorer reconstructions were obtained due to incorrecthase retrieval.
. Linear Chirp Experimentsn order to explore the limitations of our setup for the SI,e performed an experiment to measure the linear chirp.he test pulse was chirped through two passes in aiffraction-grating pair compressor using the 35 fs laser.e negatively chirped the pulse with group delay disper-
ion (GDD) varying from �7000 to −1000 fs2 becausehese were the compressor limits for our setup. The linearhirp stretches the test pulse and this implies that theide-peaks in time of the Fourier-transform of the inter-erences broaden and decrease in amplitude. In our GDDcan, we varied the grating distance L and hence theDD calculated as in [27] is
GDD�L� � −�3
�c2
L
d2 cos2 �m, �2�
here � is the central wavelength, c is the speed of light,/d=300 grooves/mm gives the groove density, and �m ishe output angle calculated from the grating equationin � −sin � =m� /d (for the first order m=1 and the in-
m 0
idence angle in the grating �0=15°). The GDD is linearlyependent on the grating distance and from Eq. (2) wealculated the estimated slope of GDD�L� /L=212.8 fs2/mm. We measured the chirped pulses usinghe fiber coupler interferometer at 81 grating distancesnd reconstructed them with the FTSI. Thus, we obtainedhe spectral phase and calculated the experimental GDDrom a quadratic fit as shown in Fig. 3(a), which corre-ponds to GDD=−5200 fs2. In Fig. 3(b) the GDD is repre-ented as a function of the grating distance. The linear re-ression of these data afforded a slope of −210.2 fs2/mm,n very good agreement with the estimated value. Ex-rapolation of the fit to zero distance gives an acceptableeviation, GDD�L=0�=−28.7 fs2, and the correlation coef-cient was R=0.999 84, revealing the good fit to the data.e also checked that the possible third-order dispersion
TOD) was completely negligible as compared to the GDD.inally, we studied the instantaneous wavelength (as a
unction of the time) of the pulses, calculated from thelectric field phase. In Fig. 3(c), using a false color scale,e plot the instantaneous wavelength of the pulses as a
unction of the grating distance. We have cropped the plotor the decrease in the pulse intensity greater than 3 or-ers of magnitude (shown in white). In this figure, wehow the linear dependence on time of the instantaneousavelength, explaining the pulse stretching. In Fig. 3(d),e represent the temporal reconstruction and instanta-
eous wavelength of the pulse corresponding to GDD=5200 fs2. We measured chirped pulses as long as 1.3 ps
1/e2 width, decrease in the intensity to 13.5%) for theighest GDD. We also explored the intensity profileaused by the GDD (available at Media 1) and found thator the lowest chirps pulse splitting occurred due to thepectrum profile, but not to the TOD (negligible), whiche also observed with the SPIDER measurements of the
est pulse.To further complete the results, we performed simula-
ions using the experimental spectrum (bandwidth of 70m) and chirping the pulse with negative and positiveDDs up to 40,000 fs2, always imposing our spectrometer
esolution, and found that we could measure 7 ps longulses (1/e2 width). We also carried out simulations withhe 120 fs laser spectrum with a GDD from �80,000 to0,000 fs2 and retrieved the input GDD and recon-tructed 4 ps pulses (1/e2 width).
One of the main advantages of the technique is its sim-licity and robustness. These characteristics allow the re-onstruction system to be adapted immediately to spec-rometers or monochromators with much more resolutionimply by plugging the fiber coupler to the input port ofhe device (common in most systems). Even though thehole spectral resolution of the spectrometer is not used,
t is very easy to upgrade the STARFISH with commercialevices, with resolutions of around 0.02 nm for portablend small spectrometers in the visible and the infraredcapable of measuring Fourier-transform-limited narrow-and pulses of 5 ps full width at half-maximum (FWHM),nd even longer in the case of broadband chirped pulses],round 0.004 nm for optical spectrum analyzers andonochromators in the visible and the infrared (compat-
ble with pulses of 25 ps FWHM), and even below 100 fmn the mid-infrared range, such as in the BOSA High-
Alonso et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 937
esolution Optical Spectrum Analyzer (allowing un-hirped pulses of 1 ns FWHM to be reconstructed).
. Convergent Wavehe first spatiotemporal result reported here correspondso a convergent wave using the 35 fs pulse duration laserocused by a 50 cm focal length lens (Fig. 4). The testeam (unknown beam) is scanned transversely 31 cm af-er the lens, that is, before the focus. In experiments with5 fs pulses, the reference was calibrated with the SPI-ER device. The delay between the reference and testeams was 550 fs. In Fig. 4(a), we show the spectral in-erference trace of the reference and test beams as a func-ion of the wavelength and the transverse position. Wecanned 4 mm of the beam profile in 20 �m steps (201oints). The evolution of the fringes with the position isuadratic, in agreement with the curvature of the wave-ront and the pulse front of the test beam. The spatiotem-oral intensity reconstruction is shown in Fig. 4(b), inhich the convergence of the beam may be observed: theeripheral region of the beam arrives before the centralegion at a certain propagation distance. We fitted the re-rieved pulse-front curvature of the beam [see fit in blueashed line in Fig. 4(b)] and obtained a value of 18.6 cmor the radius of curvature, in agreement with the ex-ected value of 19 cm, if Gaussian beam propagation isssumed.
ig. 3. (Color online) (a) Experimental spectrum and phase of ab) GDD retrieved from FTSI and (c) instantaneous wavelength ontensity and instantaneous wavelength of the chirped pulse. TDD can be seen in a video available at Media 1.
Since we used a terawatt laser, the beam profile was in-omogeneous and the pulse energy fluctuated. To removehe energy instability, we have averaged the test beampectrum taken at each point in the measurements pre-ented in this work. We checked that the spectral phaseetrieved was not affected by this instability. The inten-ity reconstruction showed in Fig. 4(b) exhibits spatialodulations that can be explained in terms of the spatial
nhomogeneity of the beam. We checked that it was onlyue to the spatial profile by directly comparing the recon-truction with the Fourier-transform limit of the spatially
ig. 4. (Color online) (a) Spatio-spectral interference trace andb) spatiotemporal intensity reconstruction of a convergent waveexperimental, 35 fs pulses). The amplitude of the plots is in lin-ar scale (see color scale on right).
ively chirped pulse. Experimental scan on negative linear chirp:hirped pulses as a function of the grating distance. (d) Temporalnsity profile and instantaneous wavelength variation with the
938 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Alonso et al.
esolved spectrum (where the FTSI cannot cause them).or further proof, we checked the reproducibility of theeam profile reconstruction, thus discarding the SI or la-er instability as being the origin of the inhomogeneity ofhe reconstructed profile.
. Spatiotemporal Interference of Two Plane Waveshe spatiotemporal interference of two ultrashort waves
ig. 5. (Color online) Experimental and simulated spatio-pectral interference traces [(a) experimental; (c) simulation] andpatiotemporal intensity reconstruction [(b) experimental; (d)imulation] of the interference between two crossing waves for20 fs pulses.
onstitutes a more complex situation. To create the test s
eam, we formed a double-beam structure using a Mach–ehnder interferometer before the test beam input arm ofhe fiber coupler. Both beams were first aligned and tem-orally overlapped. Then, we slightly crossed one beamith respect to the other, thus obtaining the spatial inter-
erences of two crossing plane waves. In Fig. 5, we showhe experimental results and simulations for this case us-ng a different laser system of 120 fs pulses, also enablings to reconstruct pulses with a narrower spectrumFWHM of �9 nm). In this case, we used the GRE-OUILLE technique to characterize the reference thatreserves the spatial homogeneity (required by the GRE-OUILLE), because we split the laser beam before theach–Zehnder device. Figure 5(a) shows the interference
pectrum trace, which displays two different fringe pat-erns: first, the fringes in the spectral dimension corre-ponding to the spectral interferences between the testeam and a 2.0 ps delayed reference beam, and second theringes in the spatial dimension (13 maxima and minima)rising from the spatial interference of the two crossingaves that form part of the test beam. In this case, we
canned 5 mm of the beam profile in 20 �m steps (251oints). In the spatiotemporal intensity reconstructionFig. 5(b)] the two waves of the test beam had a delay ofround zero and had slightly crossing pulse fronts (rela-ive tilt of 36 fs for the 5 mm profile). The maxima andinima of the double wave reconstruction are due to the
patial interference of the beams. We performed the simu-
ig. 6. (Color online) (a) Experimental spatio-spectral interference trace and spatiotemporal intensity reconstruction [(b) experimental;d) simulation] of the interference of two crossing waves for 35 fs pulses. (c) Experimental temporal profile and instantaneous wavelengthor the 7040 �m position in comparison with the simulated data (dashed line).
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Alonso et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 939
ations using parameters (spectrum, angle, and delay) ex-racted from the experimental conditions. The interfer-nce trace is shown in Fig. 5(c), and the intensityeconstruction is shown in Fig. 5(d). The simulations andhe experiments are in good agreement, showing theame behavior.
We also implemented the previous experiment with the5 fs pulse duration laser, obtaining interferences in apectral bandwidth of 70 nm. We created the double beamith the Mach–Zehnder interferometer and controlled the
elative angle and delay between the beams. In this casehe delay between the test and reference beams was 2.0s. We then scanned 10,000 �m on a transverse axis ofhe beam in 20 �m steps (501 points). The experimentalesults and the corresponding simulations are shown inig. 6. The spatially resolved interference spectrum inig. 6(a) clearly shows the spectral interferences with theeference beam and the spatial interferences of the doubleave forming the test pulse. The reconstruction of the
patiotemporal intensity [Fig. 6(b)] reveals two relativelyrossed plane waves. The intensity has the characteristictructure of maxima and minima due to the spatial inter-erences of the two beams. In this experiment, the angleetween the beams was sufficiently high to have 100 fseparated double pulses on both sides of the beam. In Fig.(c), we show the temporal profile of the double pulse cor-esponding to the position of 7040 �m colored with the in-tantaneous wavelength and compare with the simulatedrofile (dashed line), checking that there is no importanthirp. Finally, we show the simulated intensity [Fig. 6(d)]ith the parameters involved in the experiment. The
imulations match the experimental reconstruction veryell.
. Spatiotemporal Interference of a Plane and apherical Waveinally, we measured the spatiotemporal interference of apherical and a plane wave structure (Fig. 7). In this ex-eriment, we used 35 fs laser pulses. We used a 50 cm fo-al lens in one arm of the Mach–Zehnder interferometero obtain the spherical beam, whereas the other arm con-rolled the delay between the spherical and plane waves.he delay between the test and reference beams was 600
s, whereas the spherical and plane waves overlapped inhe central region. The spatio-spectral interference pat-ern of the spherical and plane waves can be seen in Fig.(a) (test beam trace, without the reference). We show thexperimental interference spectral trace of the test andeference beams in Fig. 7(b), where the quadratic varia-ion of the spectral fringes position due to the curvature ofhe spherical beam contribution (convergent) can be seen.his trace is the same as the test beam spectrum, withhe only difference lying in the spectral fringes with theelayed reference pulse. The transverse scan of 4 mm waserformed with 8 �m steps (501 points). The spatiotem-oral intensity reconstruction [Fig. 7(c)] shows the inter-erence of the spherical and plane waves: a modulatedonvergent beam is retrieved with maxima and minima inhe profile. The spacing of this modulation is larger in theentral than in the peripheral region, as corresponds topherical and plane wave interference, and the same pat-ern was obtained in the simulation [Fig. 7(d)]. The rela-
ive delay between the spherical and plane beams wasero. We repeated this measurement for different relativeelays between the plane and spherical waves up to 100 fsabove and below), such that in the reconstruction we seeow both beams separate in time and the spatial interfer-nces decrease. We also tested this situation with higherelays between the test and reference beams (1.0, 1.5, and.0 ps) and obtained the same intensity reconstructions.As discussed above, here we demonstrate the stability
nd reliability of the fiber coupler interferometer. Theuctuations in the experimental measurements and re-onstructions in comparison with the simulations are notue to STARFISH limitations but to the laser beam shot-o-shot variations and the beam profile inhomogeneity,hich are a consequence of using a terawatt laser with anmplification stage. The acquisition of a full spectral tracesually takes about 1 min, such that these variations ofhe laser pulses may introduce some noise into the recon-tructions. Moreover, the use of a Mach–Zehnder interfer-meter for the double-beam structure has inherent insta-ility and may cause fluctuations in the spatiotemporalnterference experiments.
. CONCLUSIONSe propose a novel scheme for the spatiotemporal char-
cterization of ultrashort laser pulses based on a fiber-ptic coupler interferometer (STARFISH). The device hashe advantages of being alignment-free, the fact that onlyne reference is used, and simplicity; only a fiber couplernd a standard spectrometer are necessary. The time di-ection is determined with this technique whenever it isnown for the reference characterization, as in our case.We have shown it to measure 1.3 ps long (1/e2 width)
egatively chirped pulses. According to the simulations, itould measure 1 ps FWHM unchirped pulses or, in thease of broadband chirped pulses, even longer ones. More-ver, the use of better resolution spectrometers allows theeasurement of much longer pulses, conserving the ad-
antages of the fiber coupler since it is only necessary toonnect the fiber output to the spectrometer entrance. Wepplied the STARFISH not only to the characterization ofconverging beam but also to more complex structures,
ig. 7. (Color online) Spherical and plane wave interference for5 fs pulses. Experimental (a) spatio-spectral test beam and (b)nterference traces. (c) Experimental and (d) simulated spa-iotemporal intensity reconstructions.
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940 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Alonso et al.
uch as measurement of the spatiotemporal interferencef plane-plane and spherical-plane waves, with the re-ults obtained being in agreement with the simulations.e reconstructed laser beams using two laser systemsith different pulse durations (35 and 120 fs) and spectralandwidths. Despite working with terawatt lasers at 10z, which do not have a perfectly homogeneous profilend are unstable, we demonstrate the ability of ourethod to reconstruct complex pulses. We expect this sys-
em to be used to characterize laser beams after they havendergone certain nonlinear processes or have passedhrough certain optical systems.
CKNOWLEDGMENTShe authors are grateful to S. Jarabo from the Universityf Zaragoza (Spain) for his assistance regarding the fiberoupler. We acknowledge support from Spanish Ministe-io de Ciencia e Innovación (MICINN) through the Con-olider Program SAUUL (CSD2007-00013), Researchroject FIS2009-09522 and from the Junta de Castilla yeón through the Program for Groups of Excellence
GR27). We also acknowledge support from the Centro deaseres Pulsados, CLPU, Salamanca, Spain. B. Alonsond I. J. Sola acknowledge support from the MICINNhrough the Formación de Profesorado Universitario andRamón y Cajal” programs, respectively.
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