In-situ diagnostic of femtosecond probes for high resolution ultrafast imaging
Chen Xie, Remi Meyer, Luc Froehly, Remo Giust, Francois Courvoisier
IIn-situ diagnostic of femtosecond probes for high resolution ultra-fast imaging.
Chen Xie, , ∗ Remi Meyer, ∗ Luc Froehly, Remo Giust, Francois Courvoisier, ∗∗ Ultrafast Laser Laboratory,Key Laboratory of Opto-electronic Information Technology of Ministry of Education,School of Precision Instruments and Opto-electronics Engineering,Tianjin University, 300072 Tianjin, China FEMTO-ST institute, Univ. Bourgogne Franche-Comt´e, CNRS,15B avenue des Montboucons, 25030, Besanc¸on Cedex, France ∗ These authors equally contributed ∗∗ Corresponding author [email protected]
Ultrafast imaging is essential in physics and chemistry to investigate the femtosecond dynamicsof nonuniform samples or of phenomena with strong spatial variations. It relies on observing thephenomena induced by an ultrashort laser pump pulse using an ultrashort probe pulse at a latertime. Recent years have seen the emergence of very successful ultrafast imaging techniques ofsingle non-reproducible events with extremely high frame rate, based on wavelength or spatial fre-quency encoding. However, further progress in ultrafast imaging towards high spatial resolution ishampered by the lack of characterization of weak probe beams. Because of the difference in groupvelocities between pump and probe in the bulk of the material, the determination of the absolutepump-probe delay depends on the sample position. In addition, pulse-front tilt is a widespreadissue, unacceptable for ultrafast imaging, but which is conventionally very difficult to evaluate forthe low-intensity probe pulses. Here we show that a pump-induced micro-grating generated fromthe electronic Kerr effect provides a detailed in-situ characterization of a weak probe pulse. Itallows solving the two issues. Our approach is valid whatever the transparent medium, whateverthe probe pulse polarization and wavelength. Because it is nondestructive and fast to implement,this in-situ probe diagnostic can be repeated to calibrate experimental conditions, particularly inthe case where complex wavelength, spatial frequency or polarization encoding is used. We antic-ipate that this technique will enable previously inaccessible spatiotemporal imaging in all fields ofultrafast science and high field physics at the micro- and nanoscale.1 Introduction
The fundamental understanding of laser matter interaction in several fields of ultrafast physics and chem-istry requires imaging with both high spatial resolution (typ. sub 1 µm), and high temporal resolution(typ. sub-100 fs). This is the case for instance for laser wakefield acceleration , amplification in laser-excited dielectrics , ultrafast ionization and plasma formation , THz radiation
4, 5 , high harmonic gen-eration , new material synthesis via laser-induced microexplosion or laser nanoscale processing
8, 9 .The initial concepts of ultrafast imaging based on repetitive pump-probe measurements have beenrecently complemented by a large number of different schemes allowing the imaging of non-reproducibleevents. This is performed via a sophisticated probe sequence or compressed photography , wherethe temporal information is encoded in the probe wavelength and/or in the spatial spectrum.1 a r X i v : . [ phy s i c s . op ti c s ] F e b owever, further progress in ultrafast imaging at high resolution is still impeded by two problems.First, a key information is the absolute delay between pump and probe. This is crucial to link the excita-tion dynamics to the actual pump pulse intensity . Although synchronizing pump and probe pulses ata sample surface seems reasonably easy, the case of the synchronization of pump and probes in the bulkof a sample remains unaddressed. The problem is particularly acute when bulky microscope objectivesimpose pump and probe beams to pass through the same optic, in a nearly collinear geometry. In thiscase, a sample longitudinal shift by only 100 micrometers, such as the one needed to image througha 150 µm microscope glass slide, shifts the relative delay between colinear 800 nm pump and 400 nmprobe pulses by 40 fs because of the difference between their group velocities in the dielectric material.In other words, the absolute delay is intrinsically bound to the exact position of the focus inside the bulkof the solid or liquid medium under study. The pump-probe absolute delay must be determined using apump-probe interaction on a scale of a few tens of micrometers to obtain a temporal accuracy in pump-probe measurement below the 10 fs scale. Unfortunately, conventional pulse synchronization techniquesare inoperable in this context. Sum frequency generation and other nonlinear frequency mixing schemesrequire high intensities in the probe pulse. Frequency mixing can be operated only in specific crystals andwith only a limited number of probe wavelengths. Polarization gating usually requires several 100’s µmto several mm of pump-probe overlap distance for phase accumulation
23, 24 . Transient-Grating cross-correlation frequency resolved optical gating (TG-XFROG) technique was successfully used to measurelow energy probe pulses from a supercontinuum or even in the UV . However, the conventionalconfiguration is non-phase matched and the phase matching is only reached when the interacting waveshave a sufficiently wide angular spectrum, i.e when they are focused, which is incompatible with theultrafast imaging techniques mentioned above . Optical Kerr Effect cross-correlation was used in aspectral interferometry setup which is again incompatible with imaging.A second particularly difficult issue in ultrafast imaging with high spatial resolution is the removalof pulse front tilt due to the angular dispersion of the probe pulse. This problem arises because of thenecessary dispersion compensation of the temporal dispersion induced by microscope objectives . (Asan example, a ×
50 microscope objective induces a dispersion in excess of ∼ fs at 400 nm,which stretches a 70 fs pulse to nearly 500 fs.) Most of the compensation schemes rely on spatiallyspreading the pulse spectrum such that unavoidable small misalignment creates angular dispersion .This can be usually neglected for very low numerical aperture. In contrast, in the case of high resolutionimaging, the magnification of the setup also multiplies the coefficient of angular dispersion . As wewill see below, after a ×
50 microscope objective, the pulse front tilt can easily exceed 70 ◦ (equivalentto a delay of 90 fs in 10 µm field-of-view) for a misalignment of only 10 mrad in a prism compressor.Pulse front tilt can be measured using frequency conversion autocorrelation in various spatial schemes or using spatio-spectral interferometry
35, 36 which are both unpractical for low-intensity pulses atarbitrary central wavelengths with <
100 µm beam diameter. Pulse front tilt is commonly regarded asvery difficult to evaluate, particularly for low-intensity broadband pulses such as those used in the recentultrafast photography techniques based on wavelength encoding.Here we demonstrate a highly sensitive in-situ diagnostic for weak probe pulses which solvesthese two issues and is applicable to a large number of ultrafast imaging scenarios. The concept isshown in Figure 1(a) and will be detailed in the next section. The pump pulse, shaped by a Spatial2
20 µm Λ α x z PumpPumpProbe
Sample c) f = 750 mmUltrafast oscillator+ Amplifier BBO crystal Prism compressor Delayline Camera Probe arm (400 nm)Pump arm (800 nm) S L M TransientKerr Gratingx50NA 0.8 x50NA 0.8Pump beam(shaped)f = 1000 mm f = 200 mmRelay lensDichroic mirror Sample t h o r de r s t o r de r CameraProbebeam d) Translation stage a) b)
Figure 1: (a) Concept of the transient grating induced by the shaped infrared pump pulse in a transparentdielectric. The probe signal diffracted by the transient grating is collected in the far-field. (b) The Kerr-induced transient grating has a period Λ and is tilted in respect to the probe axis by an angle α . The lengthof the transient grating is a few tens of microns while the sample can be much thicker. (c) Experimentalsetup design. (d) Magnified view of the setup in the dashed box of (c) to show the interacting beams andthe imaging configuration.Light Modulator, creates a micrometric transient grating via the optical Kerr effect oriented in the Braggcondition for the probe beam. The micrometric size of the transient grating generates a pump-probeinteraction that is highly localized which also allows for preserving a uniform diffraction efficiency overthe broadband spectra of ultrashort probes. The diffracted signal provides a localized characterization3f the absolute pump-probe delay. In addition, we have designed a way to straightforwardly visualizethe angular dispersion using temporal stretching of the probe, so as to to efficiently remove pulse fronttilt. Our diagnostic is valid whatever the probe wavelength and polarization and uses intensities that aresufficiently low to avoid optical breakdown or damage. We have used probe pulse energy down to pJlevel in the multishot integration regime. Therefore, this diagnostic can be repeated as many times asrequired, for instance at each sample replacement, to enable highly reproducible pump-probe imagingexperiments.The paper is organized as follows. We first derive the diffracted signal and present the optical setup.We then demonstrate that it can be used whatever the polarization configuration. We experimentallydemonstrate we can retrieve the absolute pump-probe delay when the sample is longitudinally displacedand that the delay variation actually follows the difference of the group velocities between pump andprobe. Last, we solve the second issue of pulse front tilt removal using a visualization tool based onpulse temporal stretching and observation of the diffracted signal in the far-field. We form a two-wave interference field inside a dielectric sample(fused silica, sapphire or glass) from a single pump beam, using a single Spatial Light Modulator, whichautomatically ensures the synchronization between the two pump waves. The instantaneous electronicKerr effect transforms the interference intensity pattern into a grating with a period Λ (see Fig. 1(b)). Werotate the transient grating by an angle α to match the Bragg incidence condition for a probe pulse whichis a collimated beam propagating along the optical axis. The rotation is simply performed by adding thesame tilt angle α on the two interfering pump beams using the SLM. Typically, the crossing half-angleof the pump is θ = 12 ◦ and the rotation angle is α = 6 ◦ . The ratio between these two angles is simplylinked by the ratio between the probe and pump wavelengths (see Methods section). The interferencepattern formed extends typically over a propagation distance of 40 µm (see Suppl. Material Fig. S.1)and can be easily reduced by shrinking down the area of the SLM that directs light to the first order ofdiffraction. We can limit the interference pattern to below 10 µm in length.In the Methods section, we derive the diffraction efficiency, based on the coupled wave theory forthick gratings. The diffracted signal is a cross-correlation between the squared pump intensity and theprobe intensity. The intensity in the first diffraction order, for a pump-probe delay τ , reads : I st ( τ ) ∝ (cid:18) n cos α (cid:19) (cid:90) I ( t ) I probe ( t − τ )d t (1)with n the Kerr index related to the relative polarizations states of pump and probe pulses.This signal will be key to characterize the probe, since the high-intensity pump pulse can be in-dependently characterized with another technique. We note that the interaction is based on three plane4aves, in contrast with conventional TG-XFROG where the phase matching is reached by crossing fo-cused beams . Preserving a probe beam as close as possible to a plane wave is important for furtheruse in pump-probe imaging. The rotation of the interference field of the pump can be adapted to matchthe Bragg angle for any probe incidence, for instance to meet the requirements of ultrafast imaging withstructured illumination . In addition, this Bragg angle can be adjusted to any incident probe wavelengthwhen encoding is based on the probe central wavelength in a very wide spectrum .Our experimental setup is described in Figure 1(c) and detailed in the Methods section. We usea Ti:Sapphire Chirped Pulse Amplifier (CPA) laser source which delivers ∼ fs pulses at 790 nmcentral wavelength and all measurements are performed by integrating the signal over 50 shots at 1 kHzrepetition rate.We split the beam in a pump and a probe, which is frequency doubled with a β -Barium-Borate(BBO) crystal generating 60 fs pulses Full-Width at Half Maximum (FWHM). The pump pulse is thenspectrally filtered to reduce its bandwidth to 12 nm FWHM, avoiding chromatic dispersion in the beamshaping stage. We spatially shape the pump beam using a Spatial Light Modulator (SLM) in near-normal incidence. The illuminated SLM is imaged by a 2f-2f telescope with a de-magnification factorof 208. The pump pulse duration has been evaluated to ∼
115 fs at sample site, after the first microscopeobjective. This element is, in contrast, highly dispersive for the probe beam at 395 nm central wavelength(on the order of ∼ ). We therefore compensate the linear dispersion on the 395 nm probe witha folded two-prisms compressor . The probe beam is also de-magnified by a factor of 278 so that theprobe beam has a waist of 12 µm (Rayleigh range of 1.1 mm) in the sample. The polarization state of thepump and probe pulses are independently controlled by the rotation of half-waveplates.After interaction in the sample, we collect the pulses with a second ×
50 (N.A. 0.8) microscopeobjective (MO). A relay lens images the Fourier plane of the second microscope objective onto a camera,which consequently records the far-field of the diffracted beams, which spatially separates the differentorders of diffraction. Figure S.1 in Supplementary Material shows the characterization of the generatedpump interference field and of the unperturbed probe beam.
A Kerr-based transient grating valid for all combinations of pump-probe polarizations
Here, wevalidate that the measured diffracted signal effectively follows Eq.1 and demonstrate that the measure-ment is valid for all combinations of input pump and probe polarizations. We note that for sake of clarity,we report on the validation first, while in fact the experiments required to start with the removal of pulsefront tilt followed by the optimization of the probe pulse duration, as will be explained below. Here,the probe pulse duration is 60 fs. In all figures below, the pump-probe delay is positive when the probearrives after the pump pulse at the position of the transient grating in the sample.Figure 2(a) shows in the inset the recorded cross-correlation signal measured on the camera as afunction of the relative pump-probe delay, for different pump pulse energies, indicated on top of eachcurve. Here, pump and probe pulses have the same horizontal polarization state. We observe that allcurves have identical profiles, peaked at the same position. The main Figure 2(a) shows the evolution5 elay (fs) x y
PumpProbe x y
PumpProbe x y
PumpProbe
Pump energy (µJ)0 0.1 0.2 0.3 0.4 0.5
ExperimentQuadratic fit
Delay (fs) I n t en s i t y ( c oun t s ) x y PumpProbe b)a)
Figure 2: (a) Peak cross-correlation signal as a function of pump intensity. Crosses show experimentaldata and a quadratic fit is shown as dashed line. (inset) Cross-correlation signal as a function of pump-probe delay for different pump intensities, showing the peak position and shape are invariant with pumppower. (b) Cross-correlation signal as a function of pump-probe delay for the 4 combinations of pumpand probe polarization orientations. The parameters are provided in the Methods section.of the peak signal as a function of the pump pulse energy. It fits very well with a quadratic curveof the input pump energy as expected from Eq. 1. The measurements have been performed in glass(Schott D263 microscope glass slide) and the results were also reproduced with identical conclusions insapphire. Therefore, at this input power level ( to W · cm − ), no contribution from higher ordernonlinearities or plasma formation is observable.In Figure 2(b), we show the evolution of the cross-correlation signal for the four different linearpolarization configurations: both pump and probe can be either horizontally or vertically polarized. Thegrating period is oriented vertically, as shown in Fig.1(a) . We checked that its orientation has no impacton the signal.The effective Kerr index n depends on the relative direction between pump and probe polariza-tions
37, 38 . Indeed, for an isotropic medium like glass, n // = 3 n ⊥ where n // is the Kerr index whenpump and probe polarizations are parallel, and n ⊥ corresponds to the case where these polarizations areorthogonal to each other. Therefore, the signal efficiencies between parallel polarizations and orthogonalpolarizations follow the ratio (cid:0) n // n ⊥ (cid:1) = 3 . In our measurements, the signal ratio is in a range 6-10when varying the grating period Λ . This ratio is highly sensitive to the background subtraction. Despitethe relatively large error bar, the experimental ratio is in very good agreement with the expected one.These results overall confirm that the transient grating signal is effectively generated by Kerr6ffect. We therefore obtain the pump-probe synchronization using the barycenter of the curve. Thecross-correlation curve also straightforwardly allows the measurement of the compression of the probepulse while tuning the prism compressor. This is shown in Suppl. Fig. S.3. It provides a direct evidenceof the optimal compression for the probe at the sample site. In our case, the cross-correlation curveallows us to retrieve the probe pulse duration of (cid:39)
60 fs FWHM knowing the pump pulse duration of100 fs with 2300 fs of second order dispersion (see Suppl. Mat.). We note that this in-situ diagnostic isalso particularly useful when the sample itself is highly dispersive.Finally, it is important to note that the technique is adaptable to characterize both polarizations.This is very useful to detect spectral phase differences in the optical path of the pump and probe beams.In Fig. 2(b), in all four polarization cases, the signal is effectively peaked at the same delay whatever thecombination of input pulses polarizations. However, in preliminary experiments, a non-optimal dichroicfilter used to recombine pump and probe had a different spectral reflectivity for vertical and horizontalpump polarizations, as shown in Supplementary Fig. S.2. For the horizontal pump polarization, itwas inducing a spectral phase distortion. Our technique has identified this bias: a temporal shift ashigh as 100 fs and profile distortion was apparent from the cross-correlation curve. This highlights theeffectiveness of the diagnostic even for the pump pulse. Spatial confinement of the synchronization
Since pump and probe pulses usually have different groupvelocities in the sample, the synchronization criterion, i.e. the absolute zero pump-probe delay, has tobe defined for a precise location of the focus in the sample. In contrast with other synchronization orpulse measurement techniques, here, the interaction region between pump and probe is highly localized,down to some tens of micrometers. We successfully determined the pump-probe synchronization evenfor a transient grating length below 10 micrometers, obviously compromising on higher integration timeto maintain an acceptable signal-to-noise ratio. The length of the transient grating can be adjusted usingthe SLM.When we shift the position of the transient grating within the sample, we observe that the cross-correlation curve is shifted in delay. Experimentally, a first cross-correlation curve (Fig.3(a, top curve))is acquired for a transient grating position starting at 50 µm from the entrance surface of a 400 µm thicksapphire sample, with refractive index n g = 1 . . The anisotropy of C-cut sapphire is negligible incomparison with the other effects. When the sample is then shifted upstream by a distance d = 50 µ m,the fringe pattern is shifted downstream by d ( n g − = 37 µm because of refraction. When we repeatthe cross-correlation measurement for different positions of the transient grating, we observe the cross-correlation shift in delay by 13 fs every 50 µm shift. This corresponds to the difference in group velocitiesbetween 790 and 395 nm wavelength. Analytically, the delay induced by the group velocity differencebetween red and blue pulses is (see Methods): ∆ t = ( n g − n g ) dn g c (2)where n g = 1 . and n g = 1 . are the group indices of sapphire at the central wavelengths of395 and 790 nm . We plot this curve in Fig. 3(b) as a dashed line and see that it perfectly fits withthe experimental data of the position of the barycenter of the cross-correlation curves reported from Fig.7igure 3: (a) Evolution of the TG signal as a function of sample position in sapphire (from 0 to 200 µm).(b) Barycenter of TG signal as a function of sample displacement; experimental data are is excellentagreement with the model of Eq.(2). The parameters are provided in the Methods section.3(a). Similarly, we obtained an excellent agreement in Fused silica (see Suppl. Fig. S.4), where thetemporal delay is 22.6 fs every 100 µm longitudinal shift. In microscope glass, the same shift induces adelay as high as 37 fs. We therefore demonstrate here that the strong localization of our measurementallows for retrieving the effect of the difference in group velocities on the pump-probe synchronization. Diagnostic for the pulse front tilt of the probe beam
A prism compressor is aberration-free only whenthe two prisms are perfectly parallel. But it is experimentally unavoidable that the parallelism deviates byseveral milliradians. This deviation has however a dramatic impact on the probe pulse since it generatespulse front tilt, which is highly detrimental for the imaging of ultrafast phenomena. We will see herethat the transient grating offers a straightforward visualization of the pulse front tilt such that it can beeffectively canceled with the correct adjustment of the parallelism between the compressor prisms.To evaluate how critical the problem is, we have evaluated the impact of a deviation angle fromperfect parallelism between the two prisms in the prism compressor using Z
EMAX TM software (seeMethods section). After the prism compressor, for a deviation angle of 10 mrad of the second prism andcompensation of the pointing direction with the folding mirror, the angular dispersion is 0.0045 mrad/nmwith a negligible spatial chirp. However, the telescope used afterwards to decrease the probe beam waistto 12 µm, increases the angular dispersion by the inverse of the magnification , i.e. a factor of 278.Quantitatively, at the focus of the microscope objective, the angular dispersion becomes 12 mrad/nm.Overall, a positioning error of only 10 mrad generates a significant pulse front tilt as high as 78 degreeswhich would dramatically blur the dynamics of ultrafast phenomena imaged. In the following, we will8ee how the transient grating can be used to detect and remove this strong pulse front tilt. x ( p i x e l s ) y (pixels) τ = -150 fs -100 fs 50 fs 100 fs 150 fs a) b) Figure 4: (a) Concept of the diffraction of an angularly dispersed probe pulse by the transient grating.The transient grating effectively samples the chirped pulse at the pump-probe delay and diffracts thecorresponding sub-pulse on the ROI (Region of Interest) in the first order of diffraction. (b) Typicalexperimental result. Diffracted signal as a function of delay and deviation angle in y direction.To detect angular dispersion and remove pulse front tilt, we develop a technique based on the factthat angular dispersion acts as a spectrometer. It spreads the spectral content of the probe pulse on thehorizontal y axis, which corresponds to the direction of the angular mismatch in our prism compressor.Since the camera is placed in the Fourier space, each direction k y is mapped onto a single columnof pixels. When the probe pulse is temporally chirped, the transient grating samples the probe pulsespectrum in time. This is sketched as a concept in Fig.4(a): the different wavelengths are sampled bythe transient grating at different moments (because of temporal chirp) and are diffracted into differentdirections (because of angular dispersion). Figure 4(b) shows a set of experimental images of the firstdiffracted order at different pump-probe delays when the probe pulse is slightly away from the optimaltemporal compression. We observe the lateral shift of the diffracted spot along y direction with thepump-probe delay, similarly as in the concept Fig. 4(a). In Fig. 5, we show the diffracted intensityas a function of k y (converted in wavelength by angular dispersion) and pump-probe delay for differentvalues of angular dispersion (adjusted with prism angle) and temporal dispersion φ (adjusted with prisminsertion). A single trace is obtained by summing for each delay the diffracted signal shown in 4(b) along x -direction. Because of the angular dispersion acting as a spectrometer, these maps are similar to TG-XFROG signal maps, except this time in a phase-matched configuration because of the Bragg orientation.When the spectral phase of the probe pulse is purely of second order, we have analytically demon-strated, in the Suppl. Material, that the signal in ( k y , t ) space appears as an ellipse. We have derived theslope of its major axis in the limit of high chirp and small angular deviation. The slope of the major axisof the ellipse expresses as (see Suppl. materials): 9 .5×10 fs²5.8×10 fs²0 fs² D e l a y ( f s ) Wavelength (nm) -6.5 mrad (9.4 mrad/nm) -3.3 mrad (4.7 mrad/nm) (0 mrad/nm) (4.6 mrad/nm)
390 400395 390 400395400 390395 400 390395
Temporal chirp
Simulation
AB C ABC
390 400395
Experiment
Prism angle (Angular dispersion) -5005000-500
Figure 5: In the table, each trace shows the diffraction efficiency in arbitrary units as a function of delay(vertical axis) and spatial direction k y (horizontal axis, k y = [ − .
03; 1 . µm − ). The left table showsexperimental results for 15 different combinations of temporal chirp φ and angular dispersion. Theangular dispersion has been numerically evaluated from the prism angle mismatch. The value of secondorder phase φ has been evaluated from the prism insertions in the prism compressor (first row 3 mm,second row 2 mm and last row 0 mm. The latter is the position for optimal pulse compression). For eachtrace, the horizontal axis scale has been converted to wavelength using the angular dispersion coefficient.When the angular dispersion is removed (central column), all wavelengths have the same direction k y .In this case, the lateral width of the spot is simply determined by the Gaussian beam size. To show theconsistency of the results, the rightmost column show three cases (A,B,C) where analytical formula forthe diffraction efficiency of the transient grating has been integrated using the parameters extracted fromthe Z EMAX simulations of the misaligned prism compressor (see Methods).10 t d k y = φ p (3)where φ is the second order dispersion and p the pulse front tilt .The top row of Fig. 5 shows the signal traces for a dispersion φ (cid:39) . × fs at the samplesite. The influence of the angular dispersion p , and the corresponding pulse front tilt, is highly apparenton the orientation of the major axis of the ellipse, which has been traced in dashed black line as a guideto the eye. The angular dispersion can be decreased to 0 with a sensitivity on the angular position ofthe prism as high as 0.5 mrad. The general symmetry observed between positive and negative values ofangular dispersion is due to the mapping of the different wavelengths in increasing or decreasing order.We also note that a slight asymmetry in the value of angular dispersion between positive and negativevalues of prism angle is due to the spectral dispersion imposed by the first prism.Therefore, our procedure to accurately visualize the angular dispersion is as follows. Since angulardispersion is proportional to the pulse front tilt p , we increase in a first step the temporal dispersion tomake the inclination of the ellipse more apparent. We can then accurately remove the pulse front tilt.Finally, the prism insertion is adjusted to minimize the second order dispersion as shown for the signalframed in green ( φ = 0 , p = 0 ).In more detail, our numerical simulations show that the misalignment from perfect parallelismof the prisms not only affects the angular dispersion, but also introduces slight second and third orderphases in the probe pulse. However, the variation of φ over the range of prism misalignments (a linein Fig.5) is typically of ± fs , reasonably smaller than the values of second order dispersion thatwe introduced using the prism insertion. The third order phase is typically fs and is apparent as aparabolic-shaped intensity pattern in the ( t, k y ) traces.We have numerically simulated the experimental cases A,B,C of Fig.5. The analytical expressionof the diffracted signal for a temporally chirped probe with higher-order phase is provided in Methods.We used for the probe’s parameters the second and third order phases and the angular dispersion deter-mined by the Z EMAX modelling of the misaligned compressor. The results of our simulations are shownas the rightmost column in Fig.5. We find an excellent agreement between simulations and experiments.We note that when the transient grating is rotated by 90 degrees, no variation in the arrival timeis observable, since no element in our setup generates angular dispersion in this direction. Therefore,the transient grating diagnostic is particularly helpful to accurately remove pulse front tilt even for faintchanges in the deviation angle of the prism compressor.11
Discussion
We have developed an extremely localized in-situ diagnostic that allows a characterization and synchro-nization of a weak probe pulse with a higher intensity pump: the synchronization between pump andprobe can be defined in a spatial domain of less than 10 µm longitudinally inside the sample; we findthe optimal point of probe pulse compression and the pulse front tilt can be removed. Therefore, thisapproach is extremely valuable for providing well-characterized ultrashort probe pulses for pump-probeimaging of ultrashort events under high magnification using single or multiple ultrashort probes over awide spectrum and with different directions whatever the polarization.The use of an SLM is highly beneficial since the convenient switch between arbitrary phase profilesmakes it possible to combine our diagnostic with pump-probe techniques where structured beam areinvolved. These structured beams indeed have a number of applications, such as high aspect ratio micro-nano structuring , laser welding among many others, and opens up new perspectives for studyinglaser-dielectric interaction in the ultrafast regime
3, 47, 48 .Our results have therefore a wide range of applicability, and we anticipate they will be particularlyuseful to characterize transient phenomena at micron-scale and laser-matter interaction within condensedmatter.
Funding
The authors acknowledge the financial supports of: European Research Council (ERC) 682032-PULSAR,Region Franche-Comte council (support to FRILIGHT platform), Labex ACTION ANR-11-LABX-0001-01; French RENATECH network and the EIPHI Graduate School ANR-17-EURE-0002.
Author contributions
R.M., R.G. and F.C. developed the setup on an initial concept by C.X and F.C., C. X. built the exper-imental interface for data acquisition, R.M. built the experimental setup, C.X. and F.C. performed thenumerical simulations of beam propagation and diffracted signal. L.F. performed the numerical analysisof the pulse front tilt and dispersion using Z
EMAX . R.M., R.G. and F.C. analyzed the experimental data.F.C.made the analytical derivations. R.M. designed the figures. The manuscript was jointly written byC.X., R.M. and F.C. and revised by all authors.
Acknowledgments
Technical assistance by L. Furfaro, C. Billet and B. Guichardaz as well as fruitful discussions with D.Brunner and J.M. Dudley from FEMTO-ST (Besanc¸on, France) are gratefully acknowledged.12 onflict of interest
The authors declare no competing interests.
The phase mask applied to our SLM is split in two equal parts: the top half generatesa plane-wave-like beam propagating at an angle α − θ and the lower half of the SLM mask symmetricallygenerates a beam propagating in direction α + θ toward the optical axis. The two beamlets cross at anangle of θ . We perform spatial filtering in the Fourier plane of the first lens ( f = 750 mm) after theSLM to select only the first diffraction order due to the SLM mask. The fringe period of the interferingtwo pump waves is Λ = λ pump n sin θ . (For ease of reading, all angles are expressed in the dielectric mediumof refractive index n , but wavelengths are expressed in vacuum) Bragg angle
The rotation angle α applied to match the Bragg incidence condition for the probe pulse,is determined by λ probe = 2 n Λ sin α , such that: sin ( α ) = sin ( θ ) λ probe λ pump . Diffraction Efficiency
The diffraction efficiency of the probe pulse on the pump-induced grating can be derived usingcoupled-wave theory describing thick gratings, since the grating is much longer than its period Λ (seeSuppl. Materials Fig.S.1). We use in the following the work by Kogelnik which provides analyticallythe effect of detuning . We note that identical results could be obtained using Four-Wave Mixing(FWM) theory. The samples investigated here (fused silica and sapphire) possess a large bandgap. Thisensures that resonant 3-photon absorption is negligible and therefore that the nonlinear Kerr response isinstantaneous . We note that for TiO and smaller bandgap dielectrics, the ∼ fs retardance observedin reference is close to negligible in comparison with the pulse durations used here.The coupled wave theory allows for deriving the signal diffracted in the first order of diffraction.The assumptions are that the incident beam and diffracted one are monochromatic plane waves, incidenton an infinitely wide grating of thickness d . Those conditions are reasonably fulfilled in our experiments.In this framework, the diffraction efficiency, i.e. the ratio between the diffracted intensity in the first orderat the exit of the grating | A ( d ) | and the incident beam intensity | A i | , can be expressed as : η ( ξ, φ ) = sin (cid:112) ξ + φ (cid:18) ξ φ (cid:19) (4)13here φ = πλ (1 − α ) dδn (5) ξ = 2 πn sin αλ (1 − α ) δλ (6) λ is the probe central wavelength. ξ expresses the detuning, i.e. expresses how the diffraction efficiencyreduces when the probe wavelength differs from the central wavelength at which the Bragg incidenceis met. The wavelength detuning is written δλ . A similar relationship can also be expressed for theillumination angle detuning .Therefore, in our experimental conditions, where d ∼
30 µm, δn (cid:39) − and λ = 0 . µ m, α = 6 ◦ the diffraction efficiency is on the order of − , which varies over a 30 fs probe pulse spectrum by lessthan . (The diffraction efficiency peak width exceeds 50 nm FWHM, i.e about one order of magnitudelarger than the bandwidth of a 30 fs pulse centered at 395 nm).Since the wavelength detuning is negligible over the probe pulse spectrum, the diffraction effi-ciency becomes: η = | A ( d ) | | A i | = sin (cid:18) π dδnλ | cos α | (cid:19) (7)Then we can express the time-integrated diffracted intensity, with τ being the delay of the probewith regard to the pump pulse: Σ( τ ) = (cid:90) I ( t )d t (8) = (cid:90) sin (cid:18) π dn I pump ( t ) λ cos α (cid:19) I probe ( t − τ )d t (9) ∝ (cid:18) n λ cos α (cid:19) (cid:90) I ( t ) I probe ( t − τ )d t (10)provided that dδn (cid:28) λ cos α which is fulfilled in our experimental conditions. Hence, the diffractedsignal is proportional to the correlation function between I and I probe , as in TG-XFROG
25, 51 . Setup
The Ti:Sapphire Chirped Pulse Amplifier (CPA) laser source (Coherent Legend) delivers ∼ fspulses at 790 nm central wavelength and repetition rate 1 kHz. We split the beam in a pump and a probe,14he latter is frequency doubled with a 50 µm thick β Barium Borate (BBO) crystal. The pump pulse isthen spectrally filtered to reduce its bandwidth to 12 nm FWHM, avoiding chromatic dispersion in thebeam shaping stage. The spectral transmission curve of the filter is nearly Gaussian so as to ensure theabsence of pre-/post pulses. We spatially shape the pump beam using a Spatial Light Modulator (SLM)in near-normal incidence. The input beam is expanded to quasi-uniformly illuminate the full active areaof the SLM. We de-magnify the resulting shaped beam by a factor 208 using a 2f-2f arrangement usinga first lens of focal length f = 750 mm and a second of focal length f = 3 . mm (MicroscopeObjective Olympus MPLFLN ×
50 with Numerical Aperture (NA) 0.8). Spatial filtering of the first orderof diffraction is performed in the Fourier plane of the first lens (not shown in the figure).We pre-compensate the linear dispersion of the 395 nm probe with a folded prism compressor .The probe beam is then de-magnified by a × f = 1000 mm and the same ×
50 microscope objective) so that the probe beam in the sample has a waist of 12 µm (Rayleigh rangeof 1.1 mm) at the focus of the microscope objective.Pump and probe energies and polarization states are controlled by half-waveplates and polarizedbeam splitters. Polarizations are controlled using independent half-waveplates placed on each beampath. The relative pump-probe delay is controlled with a motorized delay line in the probe beam, witha resolution of 3.3 fs. Figure S.1 in the Supplementary Materials shows the pump and probe beamexperimental characterizations.After interaction in the sample, we collect the pulses with a second ×
50 (N.A. 0.8) microscopeobjective (MO), as shown in Figure 1(d). When recording the diffracted signal from the transient grating,we filter out the residual pump signal. A relay lens images the back focal plane of the second MO on aCCD camera, which consequently records the far-field of the diffracted beams, which spatially separatesthe different orders of diffraction. We select out the +1 order by summing the signal over a regionof ∼ ×
100 pixels on the CCD. The measurement is performed in multishot regime, with a 14 bitscamera in free-run mode, with illumination time chosen in the range 10 to 50 ms so as to use the wholedynamical range of the camera over which the response is linear.Before starting our experiments, we characterized the 790 nm pump pulse by self-referenced spec-tral interferometry (Wizzler™) before the first microscope objective to measure and fully characterize theshortest pulse achievable, then we used the grating compressor of our CPA amplifier to compensate thedispersion of the MO and compress the pulse at the sample site, which was evaluated with two-photongeneration in a BBO crystal. Taking into account the dispersion of the microscope objective of 2300 fs at 790 nm central wavelength (separately characterized), the pump pulse duration is 100 fs at the samplesite. Shift in delay of the cross-correlation curve
When the sample is shifted upstream by d , the crossingpoint of the two pump beams is shifted downstream by a distance ∆ = | d | ( n g − , at first order in thesmall angle. Then, Eq.2 can be retrieved by considering the difference in optical paths between pumpand probe from the initial case, again at first order in the small angles of the waves with the optical axis.15 rism deviation measurement The relative angle of rotation of the second prism in the prism com-pressor has been monitored by measuring the deviation angle of the reflection of a laser pointer onto onefacet of the prism. The precision of this measurement was 0.3 mrad.
Parameters for the experiments shown in the figures
Fig. 2 Fig. 5 Fig. 3Material Schott D263 Sapphire Sapphire E pump (µJ) 0.05 - 0.5 0.1 0.1 θ ( ◦ ) 14.4 12 12Table 1: Parameters used in the experiments shown in the figures above: material, pump energy E pump and half-crossing angle of the pump beams θ . In all cases, to match the Bragg incidence angle, α is halfof θ because the probe central wavelength is half of the pump one.All angles are given in material and correspond to a single grating pitch Λ = 1 . µm. Differenceon angle values are related to respective material indices. Numerical simulations of the misaligned prism compressor
The prism compressor and the 2f-2f op-tical arrangement were numerically simulated using Z
EMAX software. This software enables a completeray tracing over complex imaging systems and computes the optical path length. Hence it is then possibleto model the dispersion effects up to the third order, including pulse front tilt and higher order disper-sion. We note that the precise design of the Olympus MPLFLN ×
50 microscope objective that we usedexperimentally is not available within this software. We replaced it by a ×
60 microscope objective inour simulations. Since the dispersions are not precisely identical, we have adapted the dimensions of thefolded prism compressor in the numerical simulations to precisely compensate the second order phase ofthe microscope objective and the lens. The parameters of the folded prism compressor were the follow-ing: the prisms are SF10 prisms with an apex angle of 60 ◦ degrees and a distance of 158 mm betweenthe prisms. However, we note that the results obtained using this simulation are in excellent agreementwith the experiments. The values of the higher order phases found using Z EMAX and used to simulatethe traces are respectively: (A) φ = 1 . × fs and φ = − . × fs , (B) φ = 1 . × fs and φ = − . × fs , (C) φ = 3 . × fs and φ = − . × fs . Simulations of the diffracted signal in Fig. 5
The simulations of the diffracted signal in Fig. 5 arebased on numerical integration of the following expression : Σ( k y , τ ) = (cid:90) | ˆ A +1 ( k y , t ) | d t ∼ (cid:90) (cid:90) | I pump .A probe ( y, t ) | e ik y y d t d y where ˆ A +1 is the spatial Fourier transform of the diffracted amplitude, and A probe is the amplitude ofthe probe pulse. ˜ A probe ( y, ω ) = E e − T ω / e − x /w e − ipωy e i ( φ ω + φ ω + ... ) , following the model byAkturk et al . 16he input pump pulse is modelled by a Gaussian pulse with 100 fs Full Width at Half-Maximum(FWHM). The probe is a 60 fs pulse with a small bump in the amplitude spectrum peaking at 390 nm soas to reproduce the experimental spectrum. References
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Delay (fs) -300 -200 -100 0 100 200 300Grating x y
Grating x y
130 fs135 fs
Figure S.2: Evidence of a 133 ± y -direction. This is repeated in the bottom figure for a periodicity along x direction,as in the results shown in the main text (see Fig.2(b)).The solid lines show the results for collinear pump and probe polarizations. Dashed lines are usedfor orthogonal polarizations. We observe that in all cases, the signal for a horizontal pump polarization(blue curves) leads to a delay of approximately 130 fs with respect to the vertical one. This experimentshows that the dichroic mirror induces a distortion of the spectral phase for the vertical pump polarizationimportant enough to make a retardance exceeding the pulse duration. In contrast, Fig.2(b) in the maintext shows the results in the same conditions for a different dichroic mirror (Layertech 101495) wherethe shift is absent. We note that detecting this discrepancy between vertical and horizontal polarizationswould be extremely difficult without our technique.3 Compression of the probe pulse
The cross-correlation signal provided by the transient grating allows for controlling the probe compres-sion and estimating the probe pulse duration.In figure S.3, we plot the diffracted signal as a function of delay for different positions of theprism compressor (insertion of glass in the probe beam path). We observe that for the position notedas “0 mm”, the cross-correlation signal is the most compressed. This corresponds to the shortest pulseduration achievable for the probe pulse. Using the knowledge of the pump pulse duration ( (cid:39)
110 fs), weretrieve the probe pulse duration by numerical fitting of the experimental curve, which leads to a probepulse duration of (cid:39)
60 fs.
Delay (fs) D i ff r a c t i on s i gna l ( a r b . un i t ) Figure S.3: Signal for different positions of the prism insertion.
Figure S.4 shows, as in the main text Fig 3, the evolution of the diffracted signal as a function of thepump-probe delay for three different positions of the sample. This time, it is performed for fused silicamaterial instead of sapphire. We obtain similarly an excellent agreement between experimental data andthe prediction of Eq.2, i.e. -22.6 fs for 100 µm. t - k y traces of Fig. 5 in presence of angular dispersion In this section, we analytically derive the inclination angle of the traces of Fig. 4. We start with theexpression of a pulse with a Gaussian distribution in space and time, with waist w and with pure angulardispersion with parameter p = d k y d ω in the direction y and second order phase φ . The pulse duration4 D i ff r a c t ed s i gna l ( no r m . )
100 µm -200 -150 -100 -50 0 50 100 150 200
Delay (fs)
200 µm
Depth (µm) -50-40-30-20-10010 D e l a y ( f s ) ExperimentTheory
Figure S.4: (left) Evolution of the diffracted signal as a function of sample position in fused silica (from0 to 200 µm). (right) Barycenter of the diffracted signal as a function of sample displacement;(FWHM) is (cid:112) log (2) T : ˜ E ( y, ω ) = E e − T ω / e − x /w e − ipωy e iφ / ω (11)In this equation, t corresponds to the pump probe delay.After double Fourier transformation over y and t coordinates: ˆ E ( k y , t ) = E (cid:48) e − αt − ( ky − iαpt )24(1 /w p α ) = E (cid:48) e u + iv (12)with E (cid:48) being a constant, u and v real-valued, and α = 1 T + 2 iφ = α (cid:48) + iα (cid:48)(cid:48) We isolate u : u = − α (cid:48) t − (cid:104) k y − pt ) ( α (cid:48) − α (cid:48)(cid:48) ) + 4 k y ptα (cid:48)(cid:48) (cid:105) × (cid:104) w + p α (cid:48) (cid:105) + ( p ) α (cid:48)(cid:48) ∆ (cid:104) − k y ptα (cid:48) − α (cid:48) α (cid:48)(cid:48) pt (cid:105) (13)with ∆ = 4 (cid:104)(cid:16) w + p α (cid:48) (cid:17) + (cid:16) p α (cid:48)(cid:48) (cid:17) (cid:105) . 5he location of the iso-intensity patterns detected on the camera is determined by u = K , where K is a constant. This equation can be rewritten in a quadratic form, which is the equation of an ellipse.We introduce the normalized transverse wavevector ˜ k y and time ˜ t with k y = κ ˜ k y and t = τ ˜ t . A ˜ k y + B ˜ k y ˜ t + C ˜ t + D ˜ k y + E ˜ t + F = 0 (14)with the following values: A = − κ ξ ∆ B = κτ (4 pα (cid:48)(cid:48) ξ − p α (cid:48) α (cid:48)(cid:48) )∆ C = τ (∆ α (cid:48) + 4 p α (cid:48)(cid:48) ξ − p α (cid:48) ξ − α (cid:48) α (cid:48)(cid:48) p )∆ D =0 E =0 F =0 (15) F is chosen here to zero but it can be any constant and ξ = (cid:16) w + p α (cid:48) (cid:17) .Equation 14 describes an ellipse in ( k y , t ) space, which can be rewritten in a matrix form as: T X A X + T B X + F = 0 (16)with X = (cid:18) k y t (cid:19) A = (cid:18) A B/ B/ C (cid:19) B = (cid:18) DE (cid:19) The major axis of the ellipse in ( k y , t ) space is rotated by an angle θ from the k y axis. Whatfollows is the determination of this angle. The rotation matrix R = (cid:0) cos θ − sin θ sin θ cos θ (cid:1) is chosen so that T R A R is diagonal. The two eigenvalues of A are: λ , = A + C ± √ A + B + C − AC (17)6he eigenvectors allow the construction of the rotation matrix R . Then the rotation angle can be deter-mined from: tan θ = − B/ C − λ = BA − C + √ A + B + C − AC (18)In the following, we perform a development of A , B and C assuming that the temporal dispersion φ is large and the angular dispersion p is small. We use the following adimensional parameter: ε = p T w φ After a lengthy but straightforward calculation, we get at the first order in ε : A (cid:39) κ w (cid:16) − ε (cid:17) B (cid:39) κτ (cid:16) − ε (cid:17) pw φ C (cid:39) τ T (cid:104) − T φ + ε (cid:16) − τ φ (cid:1)(cid:105) (19)We then find the denominator of Eq. 18: A − C + (cid:112) A + B + C − AC (cid:39) τ T κ w φ + ε (cid:16) τ T − τ T φ + τ κw φ (cid:17) (20)This can be further simplified using the fact that φ (cid:29) T such that A − C + (cid:112) A + B + C − AC (cid:39) τ T ε (21)We finally get: tan θ (cid:39) κτ φ pp