Coincidence timing of femtosecond optical pulses in an X-ray free electron laser

Alvaro Sanchez-Gonzalez1, Allan S. Johnson1, Ann Fitzpatrick2, Christopher D.M. Hutchison3, Clyde Fare3, Violeta Cordon-Preciado3, Gabriel Dorlhiac3, Josie L. Ferreira3, Rhodri M. Morgan4, Jon P. Marangos1, Shigeki Owada5, Takanori Nakane6, Rie Tanaka5, Kensuke Tono5,7, So Iwata5,8 & Jasper J. van Thor*3

1Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College, London, SW7 2AZ, UK

2Diamond Light Source Ltd, Diamond House, Harwell Science & Innovation Campus, Didcot, Oxon, UK

3Molecular Biophysics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK

4Protein Crystallography Facility, Centre for Structural Biology, Flowers Building, Department of Life Sciences, Imperial College London, London, SW7 2AZ, UK

5RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-cho, Hyogo, 679-5148, Japan

6Department of Biological Sciences, Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0032, Japan

7Japan Synchrotron Radiation Research Institute, 1-1-1 Kouto, Sayo-cho, Hyogo, 679-5948 Japan

8Department of Cell Biology, Graduate School of Medicine, Kyoto University, Yoshidakonoe-cho, Sakyo-ku, Kyoto 606-8501, Japan

*Corresponding author. Email j.vanthor@imperial.ac.uk

Abstract

Femtosecond resolution pump-probe experiments are now routinely carried out at X-ray Free Electron Lasers, enabled by the development of cross-correlation ‘time-tools’ which correct the picosecond-level jitter between the optical and X-ray pulses. These tools provide very accurate, <10 fs, measurement of the relative arrival time, but do not provide a measure of the absolute coincidence time in the interaction. Cross-correlation experiments using transient reflectivity in a crystal are commonly used for this purpose, to date no quantitative analysis of the accuracy or stability of absolute coincidence time determination has been performed. We have performed a quantitative analysis of coincidence timing at the SACLA facility through a cross-correlation of 100 ± 10 fs, 400 nm optical pulses with 7 fs, 10.5 KeV X-ray pulses via transient reflectivity in a Ce:YAG crystal. We have modelled and fit the transient reflectivity, which required a convolution with a 226 ± 12 fs uncertainty which was believed to be dominated by X-ray and laser intensity fluctuations, or assuming an extinction depth 13.3 μm greater than the literature value of 66.7 μm. Despite this, we are able to determine the absolute coincidence time to an accuracy of 30 fs. We discuss the physical contributions to the uncertainty of coincidence time determination, which may include an uncharacterised off-set delay in the development of transient reflectivity, including cascading Auger decays, secondary ionisation and cooling processes. Additionally, we present measurements of the intrinsic short-term and long-term drifts between the X-rays and the optical laser timing from time-tool analysis, which is dominated by thermal expansion of the 25 m optical path between tool and the interaction region, seen to be ~60 fs over a period of 5 hours.

Introduction

The success of hard X-ray Free Electron Lasers (XFELs)1,2 over the last decade has led to their global adoption and development, with number of facilities set to more than double in the next few years3–5. The unique ability of XFELs to generate intense femtosecond hard X-ray pulses have facilitated, among other novel experiments, significant advances in ultrafast time resolved studies of charge transfer6, photocatalysis7 and biological structural dynamics8–10. In most cases these studies are pump-probe measurements involving use of an optical laser which is synchronized to the XFEL. The majority of XFELs operate in the Self-Amplified Spontaneous Emission (SASE) regime although seeding schemes have also been implemented. For the Linear Coherent Light Source (LCLS), USA and SPring-8 Angstrom Compact free electron LAser (SACLA), Japan XFELs, methods have been developed to quantify the intrinsic instability (or ‘jitter’) of the arrival time of the X-ray pulse. It has been shown the seeding can improve timing jitter at lower photon energy FEL’s11,12, however the challenge of generating a hard X-ray seed from an optical source has prevented this being extended to XFELs. Self-seeding schemes implemented for hard X-ray FELs can greatly improve the spectral purity of the pulses but do not improve the temporal jitter significantly.

Jitters are typically on the order of 100-300 fs RMS depending on the facility13,14, far greater than the typical pulse length (~10 fs). This has led to the development of diagnostics that measure the shot-to-shot timing with respect to the optical pulses used in the experiment.

One of the most common methods used is to exploit the ultrafast transient change in reflectivity exhibited by semiconductor materials when exposed X-ray irradiation14–23. This technique, first pioneered at FLASH15,16 and further developed at the LCLS13,18–22, and SACLA14,23, has given rise to permanent diagnostic instruments at XFEL facilities providing measurements of shot-to-shot relative delay. While there have been several different schemes implemented, in each case a thin piece of dielectric material (e.g. GaAs) is exposed to the X-ray pulse which induces a reflectivity change, which is then probed by the femtosecond optical laser. Temporal information about the relative arrival time of the X-ray pulses can be encoded and recovered spatially16,19,21, spectrally18,24 or using a hybrid of the two25. In the spatial scheme both the X-ray and optical pulses are cylindrically focused on to the semiconductor plate to produce a line focus. By using different angles of incidence for each beam, relative delay temporal delay is mapped to a spatial position. By imaging the optical beam onto a CCD camera a sharp change in transmission along the line focus corresponding to the interaction between pulses can be seen. Temporal jitter is monitored by the spatial position of this edge. However, as the time-tool is not usually in the same chamber as the experiment, there may be additional drift or jitter between the two regions not addressed by the time-tool. Here, we show measurements of the magnitude of the associated timing drift for the first time.

A complimentary method that is often employed in tandem with reflectivity methods is X-ray/THz streaking26–29 in which the ‘dressing’ X-ray photoelectrons by a THz field are used to monitor relative delay. These methods have successfully corrected XFEL timing jitter to <10 fs levels, but provide only a measure of the relative timing of the X-ray and optical pulses. During an ultrafast XFEL experiment it is also important to determine the absolute arrival time of the X-ray pulses, so called “time zero”. While for materials with relatively simple dynamics or experimental signatures such as monoexponential kinetics, time zero may be inferred directly from the material response, this is not the case for more complex materials, which may exhibit phenomena time-delayed onsets of the dynamics. Furthermore, the SNR of the photoinduced difference signal would determine the accuracy of any extracted timing information, and many types of pump-probe signals are significantly smaller than the cross-correlation response of transient reflectivity. Therefore, time zero is usually determined by a simple cross-correlation technique exploiting the same transient change in reflectivity in a crystal used in the timing tool. Streaking methods are an alternative option but are not universally suitable, as streaking must be conducted under high vacuum, which can preclude certain experiments. A large number of XFEL experiments, particularly Serial Femtosecond Crystallography (SFX), often employ X-ray detectors and sample delivery systems that are not vacuum compatible in these cases are performed under atmospheric pressure helium. Furthermore, streaking methods are intrinsically slow, which may preclude experiments of complex systems which require long measurements and good statistics when considering finite beamtimes.

To measure the transient reflectivity cross-correlation method, optical and XFEL beams are overlapped in a semiconductor material and the transmission (or reflection) of the optical pulses are monitored with a photodiode while the optical delay is scanned. As this is an absolute measurement rather than a relative one, there is no encoding scheme and the resolution is nominally limited by the physical response time of the transient reflectivity process. Despite the importance of this method for determination of time zero in experiments with complex dynamics, to date there has been little analysis of the accuracy of the method or the stability of time zero over extended time periods. While there have been numerous publications discussing the precision of such methods19,20,30,31, the accuracy of the time zero determination has in fact, to the best of our knowledge, not been explicitly examined to date. While, there are still some unaddressed questions regarding this mechanism, and we offer some insight on the absolute time zero, limited by detector response and dynamic range.

While the change in absorption at different optical wavelengths varies in different materials24 the underlying mechanism is believed to be the same. The absorption of hard X-rays leads to a rapid change in the free carrier density in the bulk material due to deep core photoionisation and the subsequent cascade ionization via Auger decays and electron-electron scattering. The presence of the resulting electron-hole plasma modifies the complex refractive index of the medium, resulting in a transient change in optical reflectivity30. Estimations of the time scale of the secondary electron processes vary from 15 fs 32 to 100 fs 19, depending on theoretical model and target material. As pulse lengths of XFEL pulses decrease and synchronization with optical laser systems improves, more careful determination of time zero becomes increasingly important. This is especially significant with experiments such as TR-SFX where the probing mechanism is instantaneous elastic scattering of X-rays by electrons which does not involve any secondary processes, and therefore has a higher time resolution than the time zero determination.

Here we present for the first time a thorough analysis of the determination of coincidence time via the transient reflectivity change in a 300 micron thick Ce:YAG crystal in a configuration compatible with SFX measurements of systems with complex dynamics. The temporal response of the crystal is dominated by the temporal walk-off of the X-ray and optical pulses caused by the different refractive indices; as show later, this will be significant for all but the thinnest (<5 micron) crystals, which present signal to noise problems. We develop a simple physical model including the walk-off and perform a comparison to experiment making full use of the relative timing information from the timing tool instrument on a shot-to-shot basis. We find an additional uncertainty on the time zero determination best modelled by a convolution of the response function with a temporal broadening of 226 ± 12 fs, which may be attributed to intensity fluctuations, cumulative bleaching of the X-ray absorption, or to the uncharacterized rise time of the reflectivity. Taking this convolution into account we determine the absolute time zero with an estimated uncertainty of 30 fs. We additionally examine the stability of the coincidence time; previous work has been performed by Katayama et al.23 at the SACLA XFEL by comparing two timing tools, one using the main XFEL beam and one a 1st order diffraction from a transmission grating, and showed that measurements for each correlate very well on a 30-minute time-scale. We build on this work and present a characterization of the slow temporal drift that occurs between the main experimental chamber and the time tool (i.e. drift in absolute coincidence time) over the course of ~24 hour period. We find that over the longer time frames the drift in timing becomes much more significant, being around 60 fs over 5 hours. For experiments aiming to obtain sub-100 fs resolution periodic measurements of time zero will thus be essential, while the drift will present an even more significant barrier for studies of few-femtosecond dynamics. With improvements in timing jitter compensation and optical synchronization it is clear that few-femtosecond XFEL experiments are imminent, and therefore it is important these questions relating to absolute time zero determination are addressed in detail to ensure the successful collection and interpretation of sub-100 fs resolution time resolved data.

MethodsFEL configuration

The experiment was performed using the hard X-ray free-electron laser SACLA2 at the beam line 3 (BL3)33,34. The machine was operated at 30 Hz, emitting X-ray pulses of 10.5 KeV photon energy with a nominal pulse duration of 7 fs full width half maximum (FWHM) and pulse energies of 0.5 mJ. Four fast photodiodes were used to monitor the single shot x-ray intensity fluctuations reporting values proportional to the intensity but with arbitrary units.

Optical laser configuration

Optical pulses were provided by the “Synchronized Ti:Sapphire chirped-pulse-amplification system” installed at SACLA. This system provides pulses of 40 fs duration, 5 mJ energy and 800 nm wavelength at 1 kHz repetition rate which was reduced to match the repetition rate of the XFEL (60 Hz) using of a Pockels cell and further reduced to the SFX experiments’ rate (30 Hz) using a rotating mirror. As this experiment was performed in parallel with a TR-SFX experiment the 800 nm light was frequency doubled to 400 nm and attenuated to give pump pulses of ~10 µJ with 100±10 fs FWHM, these also served as the probe for time zero diagnostic. The optical pulses were characterized using a combination of SHG-FROG, X-FROG and cross-correlation (See S2.1). The laser compressor was optimized for maximum second harmonic generation. The pulse duration at the interaction region was increased by the transmissive optics between the non-linear crystal and interaction region (lens, helium chamber windows, etc.). The presence of these optics was emulated in the pulse characterization beam line using a plate of fused silica with an equivalent thickness. Similarly to the x-rays, single-shot fluctuations of the optical pulse intensity were characterized using a fast photodiode. A simplified diagram of the optical and x-ray setup is shown in Fig. 1.

Experimental setup

Fig 1. The simplified optical and X-ray beam setup showing the 800 nm femtosecond fundamental (red), the second harmonic 400 nm probe (blue) generated from it and SACLA X-ray beam (pink). The grey boxes and labels denote the rooms of SACLA experimental hall where the different components were located. Optical pulses were characterized using SHG-FROG and X-FROG (See Supplementary material for further details).

Time-tool

The first diagnostic consists of a permanently installed arrival timing monitor (time-tool)23 situated in the EH1 experimental station (Fig.1). About 2% of X-ray beam energy is separated from the main X-ray beam using the first order of a transmission grating (located in OH2) and focused with an elliptical mirror into a 10 μm gallium arsenide (GaAs) crystal with an incidence angle of 45 degrees. The X-ray beam rapidly increases the density of free-carriers in the crystal, switching it between semiconductor and metallic behavior, in a process known as metallization. Simultaneously, a small amount of the optical laser beam (<1 mJ) is separated from the main beam and focused with a cylindrical lens into the same GaAs crystal at normal incidence. This optical beam serves as a probe of the metallization process: as the density of free-carriers quickly increases, the transmittance of the crystal for the optical beam quickly decreases, decreasing the transmission of the optical pulse if arriving after the X-ray pulse. Furthermore, due to the 45 degree angle between the X-ray beam and the optical beam, different parts of the crystal corresponds linearly to different relative arrival times between the optical and X-ray pulses, mapping time into space. By imaging the back of the crystal into a CCD camera, a time dependent transmission image of the optical pulse is obtained, with a conversion constant of 2.6 fs/pixel14, and a time window of approximately 2 ps. These single shot images can be compared to a reference image obtained without the X-ray pulses, and used to extract a time position in pixels. For more details of this tool, please see reference 23.

Ce:YAG Transmission

The second diagnostic, used to measure the absolute coincidence time, was situated in the EH2 experimental station where the TR-SFX experiment was being conducted. For the purposes of this paper we define time-zero as the time when the peak intensities of the X-ray and optical pulses coincide at the front surface of the screen. The measurement relied on a similar metallization process in a 300 μm thick cerium-doped yttrium aluminum garnet (Ce:YAG) crystal screen situated at the interaction region of the experimental station, measured in the transmission geometry. This is considerably thicker than used in some previous experiments, and was chosen for a number of reasons, including improved structural stability, thermal capacity, and higher signal levels. Because of the absorption length is approximately 60 microns, most of the signal originates from the first few tens of microns in the sample, and thus similar results should be expected for thinner crystals as well. The transmission geometry was selected because it is most easily compatible with SFX experiments, meaning that when switching between experiment and time zero determination it is necessary only to ensure that the targets occupy the same plane. The main X-ray beam (Fig. 1), and the second harmonic of the optical beam were delivered almost collinearly to the screen at an angle smaller than 6º. The optical path from the beam splitter (which links to the timing tool) to the interaction region was ~25 m. Similarly to the GaAs in the time-tool, the interaction of the X-ray pulse with the Ce:YAG screen produces an alteration in the transmission of the 400 nm laser light24; this alteration in the transmission lasts for 100s of picoseconds15. Monitoring the transmitted energy of the laser light as a function of laser pulse delay using a photodiode reveals a transition (Fig. 2), the dynamics of which are indicative of the change in refractive index and hence the formation of the electron-hole plasma.

In order to maximize the contrast of the transition signal the X-ray beam was expanded to match the size of the laser beam focus. This expansion was achieved by changing a series of beryllium lenses from the focusing stack in the X-ray optical path. The X-ray beam was expanded from a FWHM diameter of 2 μm, which was the beam size used in regular SFX experiments, to ~50 μm diameter. Spatial overlap of the expanded beam with the optical laser was accomplished using the X-ray induced fluorescence in the Ce:YAG screen. Due to the relatively long separation of sequential XFEL pulses this fluorescence decays long before the next pulse and there is no cumulative signal. Except for the removal of the Beryllium lenses, which is known to have very limited (few femtoseconds) effect on timing, the geometry of the beams in this setup is the same as for most time resolved experiments, making any timing information extracted directly applicable as an absolute timing reference for other experiments.

Reference signal levels were recorded for each of the scans to be able to convert the photodiode values into optical pulse transmission. These included: a room background reference without the optical and X-ray pulses, an X-ray-only reference to measure the contribution of the X-ray fluorescence in the YAG to the photodiode values, an optical-only reference to measure the photodiode signal generated by the optical pulse, and a reference for long positive time delays with both the X-ray and optical beam.

Data analysisTiming retrieval

The corrected time delay () at the interaction region for each event was calculated as:

(1)

where the signs were chosen such that negative delays represent situations where the optical pulse arrives first. The first term () corresponds to the displacement of the experimental translation stage:

(2)

where represents the absolute position of the translation stage, is a reference position of the translation stage obtained once at the beginning of the experiment, c is the speed of light, and the factor of two accounts for the double change in optical distance per unit of displacement of the translation stage.

The second term () corresponds to the correction from the time tool:

(3)

Where was the position in pixels of the absorption edge in the time-tool images (see ref 23) retrieved using custom software, is a reference pixel of the image obtained once at the beginning of the experiment, is the conversion constant obtained from reference23 ,verified by delay stage scans, and the -1 is used to adapt the result to the criteria chosen for the sign of the time delay.

The third term () corresponds to the changes made to the translation stage of the time tool, which does not affect directly the timing at the interaction region. It removes the influence of manual changes applied to that translation stage from the time tool corrections. It is calculated as:

(4)

where represents the absolute position of the translation stage, is a reference position of the translation stage obtained once at the beginning of the experiment, c is the speed of light, and the factor of two accounts for double change in optical distance per unit of displacement of the translation stage.

As an example of time correction using time-tool information, Fig. 2 shows data from the photodiode signal (Ce:YAG transmission) for an experimental translation stage scan. In Fig. 2a, the values of the photodiode are plotted against the delay obtained from the translation of the experimental stage (). We can observe the data points separated by time steps of 66 fs corresponding to the discrete stage steps of 9.9 μm, and systematic lower photodiode values for negative delays due to the depletion of the optical transmission of the Ce:YAG crystal when the X-ray pulses arriving first. However, the transition between negative delays and positive delays is blurred, primarily due to the jitter of the FEL. Plotting the same data against the corrected time-delay values (), a much sharper transition is revealed.

Fig 2. Time dependent dynamics in Ce:YAG (a) without jitter correction and (b) with jitter correction using the time-tool. An X-ray pulse initiates a process of metallization in the Ce:YAG, and an optical pulse probes the metallization with a photodiode measuring the energy of the transmitted optical pulse. Positive delays refer to the optical pulse arriving first, however, the time axis may have an arbitrary offset.

Transmission model

The transmission signal measured on the photodiode was separated from background x-ray induced fluorescence (See Supplementary material for further details). As can be seen in Fig. 2, even with time tool correction the transition from full transmission to minimum transmission takes place over several hundred femtoseconds. The aid in the accurate determination of the time zero, we have built a simple model of the transient absorption to extract time zero from such curves. We modelled the transmission of optical pulses through the 300 μm Ce:YAG crystal travelling together with an X-ray pulse for different values of the time-delay between the two. We propagated 7 fs FWHM Gaussian X-ray pulses with unit amplitude through the crystal. We considered the refractive index to be exactly 1 for the X-ray pulses, and the absorption coefficient to be , obtained from ref 35. The time dependent intensity of the X-ray pulses as function of the penetration depth of in the crystal was calculated as:

(5)

Simultaneously, we propagated a 100 fs FWHM Gaussian optical pulse with unit amplitude at different delays with respect to the X-ray pulse. We considered the refractive index to be 1.9 in this case, causing the optical pulse travel almost at half the speed of the X-ray pulse inside the crystal. We discretized the propagation of the pulse into 120 spatial slices within the width of the crystal. Instead of choosing equally spaced steps, we chose the slices to be distributed proportionally to the amount of X-ray absorption occurring within each spatial slice, allowing a higher density of slices in the first 66.7 μm of the crystal, where most of the X-ray energy is deposited. We propagated the electric field optical pulse through each slice using the following recursive formula:

(6)

where is the accumulated X-ray energy arriving at each spatial slice per unit of time, is a free parameter of the model which represents the exponential absorption of the optical pulse per unit of accumulated X-ray energy, is the width of the spatial slice, and we divide by two due to the calculation being performed in terms of electric field instead of intensity. The existence of the free parameter justifies using unit amplitude for the X-ray pulse, as they are multiplied together as contributing to the results in exactly the same way.

The accumulated X-ray energy is calculated as:

(7)

where is the time delay between the optical pulse and the X-ray pulse at each position inside the crystal. This is calculated by adding the initial time delay and the accumulated time delay due to the difference refractive indices:

(8)

Finally the transmission of the optical pulse was calculated as:

(9)

Which removes the effect of the chosen unit amplitude for the optical pulse. Not included is the effect of the rise time in the crystal due to the cascade process which generates the carriers in the Ce:YAG screen. Rise times are discussed in greater depth later. We fitted the only free parameter of our model to match the optical absorption for long negative time delays.

ResultsModelled coincidence time

In order to provide some insights about the location of the absolute time zero, we compared our experimental data to the results from our model (Fig. 3). The first comparison between the model and the experimental data was obtained by studying the effect of the x-ray fluctuations in the optical transmission for long negative delays (<-100 fs) for which the optical pulses arrive well after the x-ray pulses (Fig. 3a). The binned experimental data shows a good agreement with the model, validating it well. The time-dependent comparison (Fig. 3b) was also performed by binning experimental data (80 fs bins, using the bin centers and median photodiode value from each bin of the experimental data points), and horizontally adjusting experimental data by applying a constant time offset to best match the model, showing a good qualitative agreement.

Fig 3. Comparison of the modelled time dependent transmission (red) with the experimental data (blue). (a) Intensity dependence. The transmission values for delays smaller than -100 fs are shown as a function of the normalized x-ray intensity. The experimental data was binned in 0.026 intervals of the normalized intensity, plotting the median for each bin. The error bars were obtained by bootstrapping. (b) Time-dependence of transient reflectivity of Ce:YAG. The experimental data was binned in 80 fs bins, plotting the median photodiode value for each bin, and adjusted horizontally. The error bars shown are the standard deviation of data points >900 fs divided by the square root the number of data point in each bin (See Supplementary material for further details). While the model follows the data qualitatively, it underestimates the transmission for delays around 0, and overestimates the transmission for delays around 500 fs.

We observe that the absolute time zero corresponds to the onset of the transition, as opposed to the center of the transition as has been used in other studies which would be the case if the transition was instantaneous and pulse duration limited.10. The reason this occurs is due to the difference in propagation speeds between the X-ray pulses and the optical pulses inside the Ce:YAG crystal, combined with the relatively large penetration depth of the X-ray pulses (66.7 μm) inside the crystal. The penetration depth was calculated using the tabulated atomic scattering factors35 along with the manufacturer provided doping concentration (0.2 wt%) and density (4.57g/cm3). Even when the optical pulse enters the crystal first (positive delays), the X-ray pulse, which propagates at almost twice the speed inside the crystal, may overtake the optical pulse before the X-ray pulse is completely absorbed. As a consequence, even for long positive time delays (~800 fs in Fig. 3a), the X-ray pulse may cause some optical absorption. On the other hand, it the X-ray pulse enters the crystal first, the time order of the pulses is maintained through the crystal, causing a flat response for negative delays (~-200 fs in Fig. 3). Thinner crystals with smaller walk-off effects will exhibit a sharper transition, but at the expense of signal levels and therefore the accuracy of time zero determination.

While the model follows the time-dependent data qualitatively, there are still differences, especially around 0 fs time delay, where the model underestimates the transmission, and around 500 fs where the model overestimates the transmission. This could be due to many effects, including the lack of time-response physics in the model, potential bleaching of the Ce:YAG absorbers that could effectively increase the x-ray penetration depth, inaccuracies in the time delays measured by the timing tool or shot noise in the photodiode and/or in the optical and x-ray intensity diagnostics. We attempted to modify our base model to account for those effects, with the best results shown in Fig. 4. With the exception of the bleaching these effects will be present even in the case of thin crystals, reducing the accuracy of time zero determination.

We first treated these effects phenomenologically by applying a Gaussian convolution of 226 fs FWHM fitted with an accuracy of 12 fs (Fig. 4a). This accuracy was obtained for the error bars of the binned data calculated as described in Fig. 3b. Other methods to calculate the error bars were calculated (See Table A4.2 and Fig. A4.1), however, the final error bars were more realistic for the chosen method. By performing this Gaussian convolution, the model shows a much better agreement with the experimental data around 0 fs and 500 fs.

Fig 4. Modified models (green) compared the experimental data (blue). (a) Applying a Gaussian convolution to the model fitted to a FWHM of 226 fs obtained with 12 fs accuracy. The location of time zero is shifted by -4.03 fs when compared to the base model. (b) Modifying the x-ray penetration length from 66.7 to 82 μm. The location of time zero is shifted by -35.36 fs when compared to the base model.

We believe this Gaussian convolution could account for two different effects. Firstly, any potential imprecision in the time delay as determined by the timing tool. The time resolution of the time tool is 2.6 fs per pixel, but the actual precision of these measurements, determined by the precision of the fit to the edge, is probably on the order of a few tens of femtoseconds23. Nevertheless, this is still well below the width of the convolution. The second effect could be the presence of X-ray and optical pulse intensity fluctuations or shot noise. In Fig. 3b we can observe that there is considerable uncertainty in the measured transmission intensities even for positive >1000 fs or negative <200 fs delays, which would indicate pulse energy instability. It is noted that these measurements were for fewer observations than those collected for the 0-800 fs delay range. We attempted to include this effect explicitly in our model using a Monte Carlo approach using information about the single-shot fluctuations from diode based measurements of single-shot x-ray intensity, single-shot optical intensity and single-shot transmission intensity, however, the accuracy of the model did not improve. This is possibly due to limitations in the dynamic range of X-ray detection with photodiodes or inherent shot noise in the diode. The uncertainty can be approximately modelled as a time-delay convolution, as for example having a slightly lower (higher) X-ray energy would produce a very similar effect to the X-ray pulse arriving slightly later (earlier) in time than the optical pulse.

Finally, we also modified the base model by trying different values of the X-ray penetration length with the optical value found at around 80 μm (Fig. 4b). In this case, the model shows an excellent agreement, even better than using the Gaussian approach. However, it seems that the required change in penetration length of approximately 20% from the calculated value of 66.7 μm may be too large to justify in terms of bleaching for the experimental conditions (3x1012 x-ray photons per pulse and 1.85x1015 yttrium absorbers which dominate the absorption within the volume defined by the 2500 μm2 focal size, and the 66.7 um penetration depth). Therefore, bleaching could only measurably contribute as a cumulative effect of low probability processes for the duration of the timing experiment. By these considerations non-linear modification of the imaginary part of the material response is however unlikely.

The optical model may alternatively include both types of convolution in addition to changes in the absorption length. While this would improve the correspondence between modelled and experimental data, we did not pursue this method. Nevertheless, the difference in the location of time zero within the experimental data when using each of the modified models (Gaussian convolution compared to bleaching) is 31 fs, which can be seen as an upper bound of the accuracy of the model when determining coincidence time. This value represents, to the best of our knowledge, the first estimate of the accuracy of absolute coincidence time determination using transient reflectivity changes in a crystal.

While the particular location of the coincidence time in the transition that we found may not be directly applied to experiments that use different dielectric materials and different pulse characteristics, this work shows the need to model the transmission process for each case, instead of just taking the middle point of the curve. In this case in particular, taking the middle point would have yielded an error of more than 200 fs with respect to the time zero obtained from the simulations.

We consider the statistical error with which the transient reflectivity determines the coincidence time at the front of the Ce:YAG screen within our model. While different criteria could be considered, in case the Gaussian convolution method is chosen with the literature value of 66.7 µm for the extinction depth, we propose to evaluate the half-width at half maximum of the time-convolution function that broadens the theoretical curve shown in Fig. 3, as it contains the majority of physical contributions to the measurement of onset. Considering the error of 12 fs when estimating the width of the convolution for the modified model, we estimate the coincidence time standard error to be half that, at 6.4 fs. Within this model, this indicates that this error is comparable to the intrinsic accuracy of the time-tool relative timing, but less than the estimated dwell-time transient reflectivity process.

The precise placement of coincidence time for the experimental data in Fig. 3 depends on the physical model for the time convolution. However, due the very nature of using a temporal convolution to fit the uncertainty X-ray power the time axis of time of the model has also be convolved with the 226 fs Gaussian, meaning that actual time zero is within 113 fs of the zero of Fig 4a. We can make a quick sanity check to the validity of the models time zero. Assuming a linear response of the screen and no saturation, the earliest response seen will correspond to the strongest X-ray pulses catching up to and interacting with the weakest optical pulses at the maximum penetration depth inside the screen. Taking a penetration depth of 66.7 μm and the refractive index for the X-ray pulses as 1 this corresponds to a time delay of ≈-200 fs, in quite good agreement with the position of time zero from the model. A more accurate determination of time zero will require shot to shot knowledge of the X-ray intensity as well as variable X-ray attenuation conditions to characterize the response of the screen to pulse intensity including saturation effects.

Long-term stability of the coincidence

In order to test the stability of the coincidence time we repeated the same scan four times throughout the duration of our experiment (Fig. 5). Under the exact same experimental conditions we observed a significant absolute timing drift across the different scans (Fig. 5a). When comparing the location of the edge to the location in the first scan by fitting our model, (Fig 5b) we observed a change 60 fs in the absolute timing for our second scan, taken 16 hours after the first one. Having observed this, we repeated the scans at 3 hour intervals, observing in this case the timing drift go back towards the initial value in steps of 40 fs and 15 fs approximately. The same effect can be observed in Fig. 5c where a shift in the delay histograms for a particular slice of the photodiode values (highlighted in grey in Fig. 5a) can be observed.

Fig 5. Stability of the absolute timing between the time-tool and the interaction region. (a) Four different 30 minute scans were recorded at different times. (b) Shift compared to the first scan for each of the subsequent scans. (c) Histograms of the data points lying within the gray highlighted region in (a) for each of the scans.

This timing drift is independent from the global timing drift of SACLA, which is monitored by the timing tool, and most likely indicates instabilities in the long very long optical beam path (~25 m) after the beam splitter that separates the time-tool branch and the experimental branch. In fact, a 60 fs change would indicate only a difference of 18 μm in path length.

We did not attempt to fit a model into to drift as only four data points were available. However, considering that the first and last point, showing the most similar profiles, were taken at similar times of the day, the data would be compatible with daily periodic behavior, due probably mainly to changes in temperature. However, if this is true, it may imply that points taken at other times in the middle of the day may have shown a shift larger than the observed 60 fs difference, perhaps up to 120 fs if we assume something similar to a sinusoidal behavior.

The best solution to eliminate drift would be to reduce the optical path to only a few meters, however, this is not possible in practice in many cases. Alternatively, any time resolved experiment aiming to obtain a resolution below 120 fs should regularly monitor this drift by repeating time zero measurements in the interaction region periodically. During the experiment thermostats inside EH2 indicated a temperature variation of 0.4-0.6 ºC inside the hutch over the course over a 24 period. A crude estimate for the change in path length using the thermal expansion coefficients of a steel optical table (≈0.24 μm m-1 K-1)36, indicates that a change of 0.5 K across the 25 m optical path length would result in a difference of 3 μm or a drift of 10 fs. This value is smaller than the experimentally measured value, however it assumes the entire beam path was on a single optical table, which was not the case. The much larger expansion coefficient of the concrete floor (typically between 8-13 μm m-1 K-1)37 would result in 330-540 fs drift if present over the full 25 m path, could easily account for the difference by increasing the gap between tables. The fact the measured drift is so small is testament to the quality of the optical systems employed at SACLA. Increased temperature stability, which at the time of writing is underway, should reduce the drifts further.

Discussion

The analysis of the transient reflectivity response has indicated that the determination of the absolute time-zero, defined as the coincidence of the X-ray and optical pulses at the front face of a Ce:YAG screen, is affected by the intensity instability of both pulses. We show that a simple model of the response that uses the refractive indexes, the pulse durations and the extinction depth at 10.5 keV, must either be convolved with an additional 226 ± 12 fs uncertainty to describe the observed broadening, or modified to account for a larger x-ray extinction depth. For the latter possibility we consider that given an estimated ratio of 1.6x10-3 for the X-ray photons and absorbing centers, if the bleaching probability was unity it would require 123 pulses to modify the extinction depth to the apparent 80 µm. While the bleaching yield is not known this does indicate that this model is possible given the duration of the timing experiment that we conducted (see Methods). Nevertheless, the precision of the time zero depends on the number of observations and statistics, and is shown here to be of the same magnitude as the intrinsic error of the time-tool measurement of relative timing. The slow transmission change combined with a relatively long extinction depth does not necessarily reduce the certainty with which the coincidence time is determined, if sufficient measurements are available to provide satisfactory statistics.

A source of uncertainty that is not considered in our model is the rise time of optical response following X-ray excitation, which we took to be instantaneous. A delay in the optical response will, however, follow after the absorption of an X-ray photon in the Ce:YAG that results in the generation of a single high energy free electron, which then produces numerous lower energy free electrons through an avalanche process. The effect of the free electrons upon the propagation of the optical pulse is usually determined through the Drude model, which considers the free-electrons to form a plasma, the strength of which is related to density of the free electrons25,30. Within this framework, it is the dynamics of the cascading generation of free electrons that determines the response time. A full quantum treatment of this process is clearly beyond current computational capabilities, and state-of-the-art treatments have used classical Monte-Carlo methods together with realistic electron scattering cross sections in order to model the electron distribution. While such models agree on sub-100 fs rise times, there is variation in the specifics. For X-ray photons in the range of 9-10 keV in diamond, for example, estimates of free-electron rise time range from below 10 fs 32 to greater than 30 fs 38.

A more significant error may result from the neglect of the thermal distribution of free carriers and of band structure in the Drude model. When the electron cascade inside a material is complete, the free electrons no longer have sufficient energy to excite new free carriers. This still leaves the free carriers with energies up to the band-gap energy, generally on the order of a few-eV 32, leading to a hot carrier distribution. A hot carrier distribution impacts the optical properties of a material in a variety of ways, and the effect of carrier temperature upon transient absorption has been extensively studied39–42. A significant change in regime occurs when considering optical frequencies either above or below the band-gap, which exhibit very different behaviors with respect to the presence and temperature of free-carriers43. Indeed it has been shown that the change in transmission or reflection of a material following excitation of hot carriers can be of different signs for optical frequencies which are above and below the band gap, prior to cooling of the carrier distribution44,45. This is because of a complex interplay of the free-electron plasma, depletion of the valence band, unoccupied states at the bottom of the conduction band, and band-gap renormalization. With cooling these effects can once again change sign. Even in the absence of a sign change, competition between these terms can result in a slower change in the absorption than predicted by electron population alone.

In Ce:YAG the situation is even more complex. Ce:YAG has a two band structure, with optical excitations at 450 nm to the 5d(2A1g) band and at 340 nm to the 5d(2B1g) band. The optical probe in our experiment at 400 nm is located between the two bands; thus the change in optical transmission and reflection will also be sensitive to which band the free electrons populate at the end of the cascade. The non-radiative transition from the 5d(2B1g) to 5d(2A1g) is expected to be much slower than the dynamics of interest here46. More precise modelling of the carrier dynamics, including cooling and complete band structures, would be necessary to accurately account for the possible effects upon the optical properties, which may result in delays of the peak absorption of several hundred femtoseconds47,48. The use of effectively single band materials, with band gaps well above the optical frequency, may alleviate some challenges, but more accurate modelling of the thermal distribution is still necessary to extract rise times with tens of femtosecond precision. At present, it is not clear that this is numerically feasible. Alternatively, accurate characterization of material rise-times using few-cycle laser pulses synchronized to few-femtosecond XFEL pulses using, for instance, gas phase timing systems, would enable robust determination of the impact of thermalization effects.

Conclusion

We have shown an experimental approach to determine the absolute coincidence time for a pump-probe experiment at an XFEL at hard X-ray energies. We have analyzed the response resulting from transient reflectivity in Ce:YAG and found a 226 ± 12 fs uncertainty believed to be due to X-ray intensity fluctuations. Alternatively bleaching of Yttrium centers must be assumed to modify the extinction depth, which would however require relatively high probability of bleaching given the duration of the experiment. While we show that an accuracy for the determination of coincidence time is attainable which is comparable to the optical pulse length, further information about the transient reflectivity dependence on X-ray power and more rigorous determination of the shot to shot X-ray intensity will be required to increase the accuracy. We conclude that from the comparison of the Gaussian convolution model and the bleaching model, the uncertainty of coincidence time is 31 fs for this particular experiment using transient reflectivity of Ce:YAG at 10.5 KeV energy. We have further shown long term drifts in the coincidence time at SACLA on the order of 60-100 fs, suggesting studies aiming for temporal resolution below this value will have to perform periodic measurements of time zero to account for this drift. One current unknown source of error is the delay-time between X-ray absorption and transient reflectivity. Envisioning few-fs pump-probe experiments in the future, it is clear that experimental determination of the ultrafast material response must determine these details for classes of experiment where cross-correlation is necessary rather than direct observation of coincidence time from pump-probe differences if amplitudes are sufficiently large.

Acknowledgements

JJvT acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) [EP/M000192/1]. We acknowledge the support from Engineering and Physical Sciences Research Council (UK) (EPSRC) Grant EP/I032517/1 and the European Research Council (ERC) ASTEX Project 290467. A.S.-G. is funded by the Science and Technology Facilities Council (STFC). ASJ acknowledges support from Marie Curie ITN EC317232.The XFEL experiments were performed at the BL3 of SACLA with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2016A8032)

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